# Small transitive homogeneous $3$-$(v,\\{4,6\\},1)$ designs

Michael Epstein  Colorado State University, Fort Collins, Colorado, 80523 USA
<EMAIL_ADDRESS>Donald L. Kreher  Michigan Technological
University, Houghton, Michigan, 49931 USA
<EMAIL_ADDRESS>Spyros S. Magliveras  Florida Atlantic University, Boca Raton,
Florida, 33431 USA
<EMAIL_ADDRESS>
###### Abstract

A _$3$ -$(v,\\{4,6\\},1)$ design_ is a configuration of $v$ _points_ and a
collection of $4$\- and $6$-element subsets called blocks, that jointly
contain every 3-element subset exactly once. Using an exhaustive computer
search on $v\leq 28$ points we investigate the $3$-$(v,\\{4,6\\},1)$ designs
that have a transitive automorphism group and where the blocks of size 6 form
a 2-class symmetric design. A 2-class symmetric design with parameters
$(v,k{;}\lambda_{1},\lambda_{2}{;}\delta_{1},\delta_{2})$ is a set-system on
$v$ points and $v$ blocks of size $k$, where every pair of points are in
$\lambda_{1}$ or $\lambda_{2}$ blocks and every pair of blocks intersect in
$\delta_{1}$ or $\delta_{2}$ points. The 2-class symmetric designs include
biplanes, semi-biplanes, and 2-class symmetric partially balanced incomplete
block designs.

Dedicated to Doug Stinson on the occasion of his 66th birthday.

##  1 Introduction

A _$3$ -$(v,\\{4,6\\},1)$ design_ is a configuration of $v$ _points_ and a
collection $4$\- and $6$-element subsets, called the _tetrads_ and _hexads_
respectively, that jointly contain every 3-element subset exactly once. Such
systems have a long history of investigation, see for example Hanani [5],
Mills [16], Buggenhaut [2] and Kramer-Kreher-Mathon [11]. In general a _$t$
-wise balanced design_ with parameters $t$-$(v,\mathscr{K},\lambda)$ is a
configuration of _points_ and subsets of points called _blocks_ such that
every $t$ element set is contained in $\lambda$ blocks. The parameter
$\mathscr{K}$ is the set of allowed block sizes and the parameter $\lambda$ is
called the _index_. If $\mathscr{K}=\\{k\\}$ a $t$-wise balanced design is
simply called a $t$-design. A $1$-design is also called a _tactical
configuration_. A $t$-design with $b$ blocks and $v$ points is symmetric if
$b=v$. A _$3$ -$(v,\\{4,6\\},1)$ design_ was termed by Assmus and Sardi [1] to
be _homogeneous_ if the tetrads and hexads are each tactical configurations
and there are precisely $v$ hexads. (Homogeneous is an unfortunate name for
such designs, because it may be confused with homogeneous groups.) Hence for
such designs the hexads form a symmetric $1$-design with parameters
$1$-$(v,6,6)$. They prove Theorem 1.

######  Theorem 1 (Assmus and Sardi [1]).

If a homogenous $3$-$(v,\\{4,6\\},1)$ design exists, then $v\equiv 2$ or
$4\pmod{6}$ and $v\geq 16$. Moreover, there is a unique such design when
$v=16$ and it consists of the best biplane and its sixteen ovals.

There are three non-isomorphic $2$-$(16,6,2)$ designs. The one with the
largest automorphism group was discovered by E. Kummer in 1864, [14] and has
become known as the _Kummer configuration_ , [7]. It was studied by C. Jordan
in 1869, [9] and was called _the best biplane_ on 16 points in [1].

Incidentally, because a homogenous $3$-$(v,\\{4,6\\},1)$ design contains
exactly $v$ hexads, it contains $\frac{1}{4}\big{(}\binom{v}{3}-20v\big{)}$
tetrads.

The _automorphism group_ $G$ of a $3$-$(v,\\{4,6\\},1)$ design consists of
those permutations of the points that send blocks to blocks. If this group
acts transitively on the points we say that the design is _transitive_. All
designs discussed in this article will be transitive. A transitive
$3$-$(v,\\{4,6\\},1)$ design with $v$ hexads is automatically homogeneous. We
follow the standard group naming conventions that can for example be found at
the url:

https://people.maths.bris.ac.uk/~matyd/GroupNames/.

In particular $\textsc{A}_{n}$, $\textsc{C}_{n}$ and $\textsc{D}_{n}$ are the
isomorphism classes of the alternating, cyclic and dihedral groups of degree
$n$ and if $q$ is a prime power, then $\textsc{F}_{q}$ denotes a group
isomorphic to the set of all affine transformations of the field
$\mathbb{F}_{q}$:

$\textsc{F}_{q}=\\{x\mapsto\alpha x+\beta:\alpha,\beta\in\mathbb{F}_{q}\text{
and }\alpha\not=0\\}\cong\mathbb{F}_{q}\rtimes\mathbb{F}_{q}^{\times}.$

It is a Frobenius group.

##  2 2-class symmetric designs

A $1$-$(v,k,k)$ design where each pair of points is in exactly
$\lambda=\frac{k(k-1)}{v-1}$ blocks is a symmetric design with parameters
$(v,k,\lambda)$ or equivalently a $(v,k,\lambda)$ design or
$2$-$(v,k,\lambda)$ design. A $(v,k,2)$ design is also known as a _biplane_
($\textsc{bp}(v,k)$).

A $1$-$(v,k,k)$ design where every pair of points is either in $\lambda_{1}$
or $\lambda_{2}$ blocks and every pair of blocks intersect in $\delta_{1}$ or
$\delta_{2}$ points is a
$(v,k{;}\lambda_{1},\lambda_{2}{;}\delta_{1},\delta_{2})$-design. The pairs
are partitioned into two _classes_. The pairs occurring in $\lambda_{1}$
blocks are first class pairs, whereas the pairs occurring in $\lambda_{2}$
blocks are second class pairs. We call such designs _2-class symmetric
designs_.

A $2$-class symmetric design with $\lambda_{1}=\lambda_{2}=\lambda$ is a
symmetric design and has $\delta_{1}=\delta_{2}=\lambda$.

A $2$-class symmetric design with $\lambda_{1}=\delta_{1}=0$ and
$\lambda_{2}=\delta_{2}=2$ is called a _semi-biplane_ ($\textsc{sbp}(v,k)$).

A _2-class symmetric design with parameters_
$(v,k{;}\lambda_{1},\lambda_{2}{;}\delta_{1},\delta_{2})$ is also a _2-class
partially balanced incomplete block design_
($\textsc{pbibd}(v,k,\lambda_{1},\lambda_{2})$) if the two pair classes form a
2-class association scheme. In this case the first and second class pairs are
called $i$-th associates for $i=1$ and $2$ respectively. In such a scheme
there are constants $n_{1}$ and $n_{2}$ and fixed 2 by 2 matrices $P_{1}$ and
$P_{2}$. Each symbol has exactly $n_{i}$ $i$-th associates. If $x$ and $y$ are
$i$-th associates, then the number of other symbols that are $j$-th associates
of $x$ and $k$-th associates of $y$ is $P_{i}[j,k]$. Only the
$\textsc{pbibd}(v,k,\lambda_{1},\lambda_{2})$s where every pair of blocks
intersect in $\delta_{1}$ or $\delta_{2}$ points are
$(v,k{;}\lambda_{1},\lambda_{2}{;}\delta_{1},\delta_{2})$-designs. Clatworthy
in [4] identifies 8 types of 2-class partially balanced incomplete block
designs among which only two types are relevant to this article:

Singular group divisible type:

These pbibds are obtained by starting with a BIBD and then replacing each
point by a constant number of new points.

Regular group divisible type:

A pbibd is of this type if $r-\lambda_{2}>0$ and $rk-v\lambda_{1}>0$, where
$r$ is the number of blocks containing a given point. Note $r=k$ when the
design is symmetric.

In this article we enumerate transitive $3$-$(v,\\{4,6\\},1)$ designs, where
the hexads form a $2$-class symmetric design. These designs will be labeled
$D^{v}_{i}$ and presented as a list of baseblocks that can be developed by a
given group of automorphisms whose generators we provide. (Any listed
$3$-$(v,\\{4,6\\},1)$ design, where the hexads do not form a $2$-class
symmetric design are denoted by $X^{v}_{i}$.)

###  2.1 $v=16$

The unique homogeneous $3$-$(16,\\{4,6\\},1)$ design $D_{1}^{16}$ can be
constructed from the adjacent 4 by 4 square. For each point $0,1,\ldots,f$ in
the square we associate the 6-element blocks consisting of those points in its
row and column but not including it. These 6-element blocks form a 2-(16,6,2)
design that is known as the _Kummer configuration_ or as the _best biplane on
16 points_.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$a$$b$$c$$d$$e$$f$

The 4-element blocks consist of the 4 corners of each of the 36 sub-rectangles
of the square and the 4 points in each of the 24 transversals. These 60
4-element blocks are the 60 _ovals_ of the best biplane. This exact
construction of the 2-(16,6,2) biplane first appeared in the 1869 paper of C.
Jordan where he also determines its automorphism group, [9]. Assmus and Sardi
reveal in their 1981 paper [1] that the blocks of size 4 constructed by
rectangles and transversals are the biplane’s ovals.

The best biplane:

$12348c$ $02359d$ $0136ae$ $0127bf$ $05678c$ $14679d$ $2457ae$ $3456bf$
$049abc$ $158abd$ $2689be$ $3789af$ $048def$ $159cef$ $26acdf$ $37bcde$

The ovals

From rectangles

$0145$ $0246$ $0347$ $1256$ $1357$ $2367$ $0189$ $028a$ $038b$ $129a$ $139b$
$23ab$ $01cd$ $02ce$ $03cf$ $12de$ $13df$ $23ef$ $4589$ $468a$ $478b$ $569a$
$579b$ $67ab$ $45cd$ $46ce$ $47cf$ $56de$ $57df$ $67ef$ $89cd$ $8ace$ $8bcf$
$9ade$ $9bdf$ $abef$

From transversals

$05af$ $05be$ $069f$ $06bd$ $079e$ $07ad$ $14af$ $14be$ $168f$ $16bc$ $178e$
$17ac$ $249f$ $24bd$ $258f$ $25bc$ $278d$ $279c$ $349e$ $34ad$ $358e$ $35ac$
$368d$ $369c$

$D^{16}_{1}$:

Generators
(0,1,2,3)(4,5,6,7)(8,9,a,b)(c,d,e,f)
(0,4,8,c)(1,5,9,d)(2,6,a,14)(3,7,b,f)
Baseblocks
$\\{1,2,3,4,8,c\\}$,
$\\{0,1,4,5\\}$, $\\{0,2,4,6\\}$, $\\{0,1,8,9\\}$, $\\{0,2,8,a\\}$,
$\\{0,7,a,d\\}$, $\\{0,5,b,e\\}$, $\\{0,5,a,f\\}$

The full automorphism group of $D^{16}_{1}$ is
$\textsc{C}_{2}^{4}.\textsc{A}_{6}.\textsc{C}_{2}$ and is generated by:

$(0,2,a)(1,5,3,9,c,7)(6,f,8)(b,e)\quad\text{and}\quad(0,f,5,d,c,2)(1,9,e,6,a,7)(3,b,4).$

Naturally the $3$-$(16,\\{4,6\\},1)$ design can also be described bigraphicaly
by taking all subgraphs of $K_{4,4}$, isomorphic to

|  |
---|---|---
biplane | rectangles | transversals

A complete list of bigraphical $t$-$(v,K,1)$ designs can be found in [6].

Alternatively we can describe the $3$-$(16,\\{4,6\\},1)$ by embedding the
unique graphical $2$-$(15,\\{3,5\\},1)$. We take the point set to be
$E(K_{6})\cup\\{\infty\\}$. The blocks are all subgraphs of $K_{6}$ isomorphic
to the following.

$\infty$ |  | $\infty$ |
---|---|---|---

A complete list of graphical $t$-$(v,K,1)$ designs can be found in [3].

###  2.2 $v=20$

Salwach [17] studied homogeneous $3$-$(20,\\{4,6\\},1)$ designs in 1985 and
found two whose hexads form a semi-biplane. His motivation arose by
considering a possible extension of a supposed projective plane of order 10.
Should such an extension (a 3-(112, 12, 1) design) have existed, then it is
not difficult to show that there would necessarily be two blocks whose
symmetric difference would consist of 20 points, see [12]. The remaining
blocks intersect these 20 points in 2, 4 or 6 points giving rise to the
tetrads and hexads of a homogeneous 3-(20, {4, 6}, 1), see [1, 2] . However in
1989 it was shown that a projective plane of order 10 cannot exist [15].

Only one of Salwach’s two homogeneous $3$-$(20,\\{4,6\\},1)$ designs has a
transitive automorphism group. Salwach reports Andries Brouwer also
constructed a homogeneous $3$-$(20,\\{4,6\\},1)$ design, where the blocks of
size 6 do not form a semi-biplane. This was a private communication and as far
as we can determine the Brouwer homogeneous $3$-$(20,\\{4,6\\},1)$ design was
never published and we do not know the details of this construction. Andries
writes us:

> I recall the situation with this design. Ed Assmus gave a talk early in some
> conference in England, and mentioned that such a design was unknown. I sat
> down in the grass a sunny afternoon and constructed one, and gave him a
> copy.

Indeed there is a postscript to the Assmus and Sardi article [1] that confirms
this. Thus it is likely that the private communication that Salwach reports
was passed on to him from Ed Assmus or refers to this postscript.

It turns out that there are in fact exactly three transitive homogeneous
$3$-$(20,\\{4,6\\},1)$ designs which we label below as $D^{20}_{1}$,
$X^{20}_{2}$ and $X^{20}_{3}$. Only the hexads of $D^{20}_{1}$, form a 2-class
symmetric design. These hexads are in fact the semi-biplane
$\textsc{sbp}(20,6)$. The full automorphism group of these transitive
homogeneous $3$-$(20,\\{4,6\\},1)$ designs is denoted by $G$ in the following
table.

$\begin{array}[]{lcllcl}\hline\cr&&G&|G|&&\textrm{Hexads type}\\\ \hline\cr
D^{20}_{1}&&\bigg{.}\textsc{A}_{5}\bigg{.}&60&&\textsc{sbp}(20,6)\\\ \hline\cr
X^{20}_{2}&&\bigg{.}\textsc{F}_{5}\bigg{.}&20&&\textrm{Not a 2-class symmetric
design}\\\ \hline\cr
X^{20}_{3}&&\bigg{.}\textsc{C}_{2}^{2}\times\textsc{F}_{5}\bigg{.}&80&&\textrm{Not
a 2-class symmetric design}\\\ \hline\cr\end{array}$

$D^{20}_{1}$:

Generators
(0,1)(2,12)(3,13)(4,5)(6,18)(7,19)(8,14)(9,15)(10,16)(11,17)
(0,2,8)(1,3,9)(4,11,12)(5,10,13)(6)(7)(14,19,17)(15,18,16)
Baseblocks
$\\{0,1,2,14\\}$, $\\{0,1,3,11\\}$, $\\{0,2,6,8\\}$, $\\{0,2,10,13\\}$,
$\\{0,3,7,17\\}$, $\\{0,2,5,7,12,18\\}$

$X^{20}_{2}$:

Generators
(0,1)(2,19)(3,18)(4,17)(5,16)(6,14)(7,15)(8,13)(9,12)(10,11)
(0,2,8,6)(1,3,9,7)(4,15,5,14)(10,13,18,16)(11,12,19,17)
Baseblocks
$\\{0,2,11,14\\}$, $\\{0,1,10,11\\}$, $\\{0,1,2,8\\}$, $\\{0,1,5,7\\}$,
$\\{0,3,8,11\\}$, $\\{0,2,10,12\\}$, $\\{0,3,5,16\\}$, $\\{0,5,12,15\\}$,
$\\{0,3,10,18\\}$, $\\{0,4,9,16\\}$, $\\{0,5,11,19\\}$, $\\{0,1,3,4,6,12\\}$

$X^{20}_{3}$:

Generators
(0,1)(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)
(0,2)(1,3)(4,18)(5,19)(6,16)(7,17)(8,15)(9,14)(10,13)(11,12)
(0)(1)(2,7,18,15)(3,6,19,14)(4,13,17,8)(5,12,16,9)(10)(11)
Baseblocks
$\\{0,1,10,11\\}$, $\\{0,2,5,10\\}$, $\\{0,2,7,8\\}$, $\\{0,1,4,17\\}$,
$\\{0,3,11,13\\}$, $\\{0,2,11,12\\}$, $\\{0,1,2,6,14,18\\}$,

###  2.3 $v=22$

There is a unique $\textsc{sbp}(22,6)$. It is the double of the Paley biplane.
The Paley biplane is the $2$-$(11,5,2)$ design and is constructed by
developing $\\{1,3,4,5,9\\}$, (the quadratic residues) modulo $11$. If $A$ is
an incidence matrix for the $2$-$(11,5,2)$ then the incidence matrix for the
$\textsc{sbp}(22,6)$ is $\left[\begin{array}[]{cc}A&I\\\
I&A^{\textsc{t}}\end{array}\right]$. This incidence matrix construction is
known as _doubling_ and is due to Hughes and Dickey, see [8]. It can also be
obtained by developing $\\{0_{0},1_{0},2_{0},4_{0},7_{0},0_{1}\\}$ by the
group $G$ acting on $\\{x_{i}:x\in\mathbb{Z}_{11},i\in\mathbb{Z}_{2}\\}$
generated by

$\begin{array}[]{ccc}x_{i}\mapsto(x+1)_{i}&\text{ and
}&x_{i}\leftrightarrow(-x)_{i+1},\end{array}$

which is isomorphic to the dihedral group $\textsc{D}_{11}$,.

There are up to isomorphism 3 transitive $3$-$(22,\\{4,6\\},1)$ designs whose
hexads form the $\textsc{sbp}(22,6)$. These are design $D^{22}_{1}$,
$D^{22}_{2}$ and $D^{22}_{3}$ presented below as a set of 16 baseblocks with
respect to the automorphism group $G$. Designs $X^{22}_{4}$, $X^{22}_{5}$,
$X^{22}_{6}$, and $X^{22}_{7}$, are the transitive $3$-$(22,\\{4,6\\},1)$
designs that exist on 22 points whose hexads do not form a 2-class symmetric
design. They have 19, 20, 20 and 15 baseblocks respectively.

$D^{22}_{1}$:

Baseblocks
$\\{0_{0},2_{0},6_{0},1_{1}\\}$, $\\{0_{0},1_{0},2_{1},4_{1}\\}$,
$\\{0_{0},1_{0},9_{0},3_{1}\\}$, $\\{0_{0},3_{0},0_{1},3_{1}\\}$,
$\\{0_{0},4_{0},2_{1},6_{1}\\}$, $\\{0_{0},1_{0},5_{1},6_{1}\\}$,
$\\{0_{0},1_{0},1_{1},7_{1}\\}$, $\\{0_{0},2_{0},2_{1},7_{1}\\}$,
$\\{0_{0},1_{0},5_{0},8_{1}\\}$, $\\{0_{0},3_{0},4_{1},8_{1}\\}$,
$\\{0_{0},2_{0},6_{1},8_{1}\\}$, $\\{0_{0},1_{0},8_{0},9_{1}\\}$,
$\\{0_{0},4_{0},3_{1},9_{1}\\}$, $\\{0_{0},3_{0},6_{1},9_{1}\\}$,
$\\{0_{0},2_{0},8_{0},10_{1}\\}$, $\\{0_{0},1_{0},2_{0},4_{0},7_{0},0_{1}\\}$

$D^{22}_{2}$:

Baseblocks
$\\{0_{0},1_{0},9_{0},2_{1}\\}$, $\\{0_{0},2_{0},6_{0},1_{1}\\}$,
$\\{0_{0},1_{0},8_{0},3_{1}\\}$, $\\{0_{0},3_{0},0_{1},3_{1}\\}$,
$\\{0_{0},3_{0},1_{1},5_{1}\\}$, $\\{0_{0},2_{0},3_{1},5_{1}\\}$,
$\\{0_{0},1_{0},4_{1},5_{1}\\}$, $\\{0_{0},1_{0},1_{1},7_{1}\\}$,
$\\{0_{0},2_{0},2_{1},7_{1}\\}$, $\\{0_{0},1_{0},5_{0},8_{1}\\}$,
$\\{0_{0},4_{0},1_{1},8_{1}\\}$, $\\{0_{0},2_{0},6_{1},8_{1}\\}$,
$\\{0_{0},4_{0},3_{1},9_{1}\\}$, $\\{0_{0},1_{0},6_{1},9_{1}\\}$,
$\\{0_{0},2_{0},8_{0},10_{1}\\}$, $\\{0_{0},1_{0},2_{0},4_{0},7_{0},0_{1}\\}$

$D^{22}_{3}$:

Baseblocks
$\\{0_{0},1_{0},9_{0},1_{1}\\}$, $\\{0_{0},1_{0},2_{1},7_{1}\\}$,
$\\{0_{0},1_{0},5_{0},3_{1}\\}$, $\\{0_{0},3_{0},1_{1},4_{1}\\}$,
$\\{0_{0},2_{0},2_{1},4_{1}\\}$, $\\{0_{0},1_{0},8_{0},5_{1}\\}$,
$\\{0_{0},3_{0},0_{1},6_{1}\\}$, $\\{0_{0},1_{0},4_{1},6_{1}\\}$,
$\\{0_{0},2_{0},6_{0},7_{1}\\}$, $\\{0_{0},4_{0},3_{1},7_{1}\\}$,
$\\{0_{0},2_{0},1_{1},8_{1}\\}$, $\\{0_{0},3_{0},5_{1},9_{1}\\}$,
$\\{0_{0},1_{0},8_{1},9_{1}\\}$, $\\{0_{0},2_{0},8_{0},10_{1}\\}$,
$\\{0_{0},5_{0},5_{1},10_{1}\\}$, $\\{0_{0},1_{0},2_{0},4_{0},7_{0},0_{1}\\}$

$X^{22}_{4}$:

Baseblocks
$\\{0_{0},1_{0},3_{0},5_{0}\\}$, $\\{0_{0},1_{0},4_{0},7_{0}\\}$,
$\\{0_{0},2_{0},7_{0},0_{1}\\}$, $\\{0_{0},2_{0},5_{0},1_{1}\\}$,
$\\{0_{0},1_{0},2_{1},7_{1}\\}$, $\\{0_{0},3_{0},1_{1},4_{1}\\}$,
$\\{0_{0},4_{0},1_{1},5_{1}\\}$, $\\{0_{0},1_{0},6_{0},6_{1}\\}$,
$\\{0_{0},2_{0},4_{1},6_{1}\\}$, $\\{0_{0},2_{0},6_{0},8_{1}\\}$,
$\\{0_{0},5_{0},3_{1},8_{1}\\}$, $\\{0_{0},4_{0},2_{1},9_{1}\\}$,
$\\{0_{0},3_{0},6_{1},9_{1}\\}$, $\\{0_{0},2_{0},7_{1},9_{1}\\}$,
$\\{0_{0},1_{0},2_{0},10_{1}\\}$, $\\{0_{0},4_{0},3_{1},10_{1}\\}$,
$\\{0_{0},5_{0},4_{1},10_{1}\\}$, $\\{0_{0},1_{0},9_{0},0_{1},1_{1},3_{1}\\}$,
$\\{0_{0},1_{0},8_{0},4_{1},5_{1},8_{1}\\}$

$X^{22}_{5}$:

Baseblocks
$\\{0_{0},1_{0},3_{0},6_{0}\\}$, $\\{0_{0},1_{0},4_{0},5_{0}\\}$,
$\\{0_{0},2_{0},4_{0},8_{0}\\}$, $\\{0_{0},3_{0},1_{1},4_{1}\\}$,
$\\{0_{0},5_{0},0_{1},5_{1}\\}$, $\\{0_{0},1_{0},4_{1},5_{1}\\}$,
$\\{0_{0},4_{0},2_{1},6_{1}\\}$, $\\{0_{0},2_{0},0_{1},7_{1}\\}$,
$\\{0_{0},1_{0},2_{1},8_{1}\\}$, $\\{0_{0},4_{0},3_{1},7_{1}\\}$,
$\\{0_{0},1_{0},6_{1},7_{1}\\}$, $\\{0_{0},3_{0},0_{1},8_{1}\\}$,
$\\{0_{0},5_{0},3_{1},8_{1}\\}$, $\\{0_{0},4_{0},4_{1},8_{1}\\}$,
$\\{0_{0},3_{0},6_{1},9_{1}\\}$, $\\{0_{0},2_{0},4_{1},9_{1}\\}$,
$\\{0_{0},1_{0},2_{0},10_{1}\\}$, $\\{0_{0},3_{0},7_{1},10_{1}\\}$,
$\\{0_{0},1_{0},9_{0},0_{1},1_{1},3_{1}\\}$,
$\\{0_{0},2_{0},7_{0},1_{1},6_{1},8_{1}\\}$

$X^{22}_{6}$:

Baseblocks
$\\{0_{0},1_{0},3_{0},5_{0}\\}$, $\\{0_{0},1_{0},4_{0},7_{0}\\}$,
$\\{0_{0},1_{0},6_{0},8_{0}\\}$, $\\{0_{0},3_{0},1_{1},4_{1}\\}$,
$\\{0_{0},5_{0},0_{1},5_{1}\\}$, $\\{0_{0},1_{0},4_{1},5_{1}\\}$,
$\\{0_{0},4_{0},2_{1},6_{1}\\}$, $\\{0_{0},2_{0},0_{1},7_{1}\\}$,
$\\{0_{0},1_{0},2_{1},8_{1}\\}$, $\\{0_{0},4_{0},3_{1},7_{1}\\}$,
$\\{0_{0},1_{0},6_{1},7_{1}\\}$, $\\{0_{0},3_{0},0_{1},8_{1}\\}$,
$\\{0_{0},5_{0},3_{1},8_{1}\\}$, $\\{0_{0},4_{0},4_{1},8_{1}\\}$,
$\\{0_{0},3_{0},6_{1},9_{1}\\}$, $\\{0_{0},2_{0},4_{1},9_{1}\\}$,
$\\{0_{0},1_{0},2_{0},10_{1}\\}$, $\\{0_{0},3_{0},7_{1},10_{1}\\}$,
$\\{0_{0},1_{0},9_{0},0_{1},1_{1},3_{1}\\}$,,
$\\{0_{0},2_{0},7_{0},1_{1},6_{1},8_{1}\\}$

$X^{22}_{7}$:

Baseblocks
$\\{0_{0},1_{0},4_{0},5_{0}\\}$, $\\{0_{0},2_{0},5_{0},7_{0}\\}$,
$\\{0_{0},1_{0},3_{0},1_{1}\\}$, $\\{0_{0},3_{0},7_{0},3_{1}\\}$,
$\\{0_{0},1_{0},9_{0},3_{1}\\}$, $\\{0_{0},1_{0},2_{0},6_{1}\\}$,
$\\{0_{0},3_{0},0_{1},6_{1}\\}$, $\\{0_{0},4_{0},1_{1},6_{1}\\}$,
$\\{0_{0},3_{0},4_{1},8_{1}\\}$, $\\{0_{0},1_{0},7_{1},8_{1}\\}$,
$\\{0_{0},3_{0},2_{1},9_{1}\\}$, $\\{0_{0},3_{0},6_{0},10_{1}\\}$,
$\\{0_{0},2_{0},8_{1},10_{1}\\}$, $\\{0_{0},1_{0},9_{1},10_{1}\\}$,
$\\{0_{0},1_{0},6_{0},0_{1},2_{1},4_{1}\\}$,

The full automorphism group for each design except design $D^{22}_{3}$ is
$G\cong\textsc{D}_{11}$, and has order $22$. The full automorphism group for
design $D^{22}_{3}$ has order 110 and is isomorphic to the group
$\textsc{F}_{11}$. It is generated by

$\begin{array}[]{cccc}x_{i}\mapsto(x+1)_{i},&x_{i}\leftrightarrow(-x)_{i+1}&\text{
and }&x_{i}\mapsto(3x+1)_{i}\end{array}$

###  2.4 $v=26$

On $v=26$ points there are 332 nonisomorphic transitive $1$-$(26,6,6)$ designs
that do not cover a triple twice. Although there are seven that are 2-class
symmetric designs only two of them, $H_{1}$ and $H_{2}$, extend to a
transitive $3$-$(26,\\{4,6\\},1)$ design.

$H_{1}$

is a 2-class symmetric design with parameters $(26,6{;}1,2{;}1,2)$.

$H_{2}$

is a 2-class symmetric design with parameters $(26,6{;}1,6{;}0,2)$ and also a
singular group divisible type $\textsc{pbibd}(26,6,1,6)$ with parameters

$\begin{array}[]{llll}n_{1}=24,&n_{2}=1,&P_{1}=\left[\begin{array}[]{rr}22&1\\\
1&0\end{array}\right],&P_{2}=\left[\begin{array}[]{rr}24&0\\\
0&0\end{array}\right]\end{array}.$

This pbibd is recoded as S44 on pages 104, 116 and 117 of [4].

There is up to isomorphism a unique $3$-$(26,\\{4,6\\},1)$ design $D^{26}_{1}$
that extends $H_{1}$. The pbibd $H_{2}$ extends to 6 nonisomorphic
$3$-$(26,\\{4,6\\},1)$ designs $D^{26}_{2},D^{26}_{3},\ldots,D^{26}_{7}$.
Three have $\textsc{C}_{26}$ as their full automorphism group and three have
$\textsc{C}_{26}\rtimes\textsc{C}_{3}$. Consequently there are 7 nonisomorphic
transitive $3$-$(26,\\{4,6\\},1)$ designs whose hexads form a 2-class
symmetric design.

Develop the following baseblocks modulo 26 to obtain the 4 nonisomorphic
special $3$-$(26,\\{4,6\\},1)$ designs that have full automorphism group
$\textsc{C}_{26}$.

$D^{26}_{1}$:

Baseblocks
$\\{0,2,7,11\\}$, $\\{0,1,4,12\\}$, $\\{0,2,6,12\\}$, $\\{0,1,3,14\\}$,
$\\{0,2,5,14\\}$, $\\{0,4,9,15\\}$, $\\{0,1,10,15\\}$, $\\{0,1,7,16\\}$,
$\\{0,4,11,16\\}$, $\\{0,3,13,16\\}$, $\\{0,3,9,17\\}$, $\\{0,3,6,18\\}$,
$\\{0,5,13,18\\}$, $\\{0,1,8,19\\}$, $\\{0,2,10,20\\}$, $\\{0,4,13,20\\}$,
$\\{0,1,18,20\\}$, $\\{0,1,2,22\\}$, $\\{0,3,10,22\\}$, $\\{0,1,5,23\\}$,
$\\{0,1,17,24\\}$, $\\{0,1,6,9,11,13\\}$

$D^{26}_{2}$:

Baseblocks
$\\{0,2,4,10\\}$, $\\{0,1,6,7\\}$, $\\{0,1,8,9\\}$, $\\{0,3,7,10\\}$,
$\\{0,1,2,11\\}$, $\\{0,1,12,15\\}$, $\\{0,4,11,16\\}$, $\\{0,2,6,17\\}$,
$\\{0,3,9,18\\}$, $\\{0,2,12,18\\}$, $\\{0,2,9,19\\}$, $\\{0,4,12,19\\}$,
$\\{0,3,6,20\\}$, $\\{0,3,11,21\\}$, $\\{0,4,9,21\\}$, $\\{0,2,16,21\\}$,
$\\{0,2,14,22\\}$, $\\{0,1,16,22\\}$, $\\{0,1,5,23\\}$, $\\{0,1,3,24\\}$,
$\\{0,1,4,13,14,17\\}$, $\\{0,2,7,13,15,20\\}$

$D^{26}_{3}$:

Baseblocks
$\\{0,2,4,10\\}$, $\\{0,1,2,9\\}$, $\\{0,1,7,11\\}$, $\\{0,1,5,12\\}$,
$\\{0,3,9,14\\}$, $\\{0,1,3,15\\}$, $\\{0,1,6,16\\}$, $\\{0,2,11,17\\}$,
$\\{0,3,7,18\\}$, $\\{0,4,12,18\\}$, $\\{0,1,10,19\\}$, $\\{0,2,12,19\\}$,
$\\{0,3,6,20\\}$, $\\{0,3,11,21\\}$, $\\{0,4,16,21\\}$, $\\{0,2,18,21\\}$,
$\\{0,1,18,22\\}$, $\\{0,2,16,23\\}$, $\\{0,1,21,23\\}$, $\\{0,1,20,24\\}$,
$\\{0,1,4,13,14,17\\}$, $\\{0,2,7,13,15,20\\}$

$D^{26}_{4}$:

Baseblocks
$\\{0,1,3,6\\}$, $\\{0,2,4,8\\}$, $\\{0,1,5,10\\}$, $\\{0,1,8,11\\}$,
$\\{0,1,9,15\\}$, $\\{0,4,11,16\\}$, $\\{0,3,10,17\\}$, $\\{0,2,12,18\\}$,
$\\{0,1,16,18\\}$, $\\{0,4,10,19\\}$, $\\{0,2,14,19\\}$, $\\{0,1,2,21\\}$,
$\\{0,3,11,21\\}$, $\\{0,4,12,21\\}$, $\\{0,3,14,22\\}$, $\\{0,2,16,22\\}$,
$\\{0,1,19,22\\}$, $\\{0,1,12,23\\}$, $\\{0,2,17,23\\}$, $\\{0,1,7,24\\}$,
$\\{0,1,4,13,14,17\\}$, $\\{0,2,7,13,15,20\\}$

Develop the following baseblocks modulo 26 to obtain the 3 nonisomorphic
special $3$-$(26,\\{4,6\\},1)$ designs that have full automorphism group
$\textsc{C}_{26}\rtimes\textsc{C}_{3}$.

$D^{26}_{5}$:

Baseblocks
$\\{0,1,5,9\\}$, $\\{0,1,6,11\\}$, $\\{0,2,8,12\\}$, $\\{0,1,3,15\\}$,
$\\{0,2,11,16\\}$, $\\{0,3,6,17\\}$, $\\{0,2,6,18\\}$, $\\{0,3,7,18\\}$,
$\\{0,1,12,18\\}$, $\\{0,2,4,19\\}$, $\\{0,3,9,19\\}$, $\\{0,1,10,19\\}$,
$\\{0,1,16,20\\}$, $\\{0,1,7,21\\}$, $\\{0,3,11,21\\}$, $\\{0,2,9,21\\}$,
$\\{0,2,5,22\\}$, $\\{0,3,8,22\\}$, $\\{0,1,2,23\\}$, $\\{0,1,8,24\\}$,
$\\{0,1,4,13,14,17\\}$, $\\{0,2,7,13,15,20\\}$

$D^{26}_{6}$:

Baseblocks
$\\{0,1,5,6\\}$, $\\{0,1,7,8\\}$, $\\{0,1,3,9\\}$, $\\{0,2,6,10\\}$,
$\\{0,3,7,10\\}$, $\\{0,1,11,12\\}$, $\\{0,3,11,14\\}$, $\\{0,4,10,15\\}$,
$\\{0,4,11,16\\}$, $\\{0,2,11,17\\}$, $\\{0,1,10,18\\}$, $\\{0,2,12,18\\}$,
$\\{0,2,9,19\\}$, $\\{0,4,12,19\\}$, $\\{0,3,6,20\\}$, $\\{0,4,9,21\\}$,
$\\{0,2,16,21\\}$, $\\{0,2,14,22\\}$, $\\{0,2,5,23\\}$, $\\{0,1,2,24\\}$,
$\\{0,1,4,13,14,17\\}$, $\\{0,2,7,13,15,20\\}$

$D^{26}_{7}$:

Baseblocks
$\\{0,1,7,8\\}$, $\\{0,1,3,11\\}$, $\\{0,2,8,12\\}$, $\\{0,2,9,14\\}$,
$\\{0,4,10,15\\}$, $\\{0,1,12,15\\}$, $\\{0,1,6,16\\}$, $\\{0,2,11,17\\}$,
$\\{0,2,6,18\\}$, $\\{0,1,10,18\\}$, $\\{0,4,12,19\\}$, $\\{0,3,10,19\\}$,
$\\{0,3,6,20\\}$, $\\{0,1,9,21\\}$, $\\{0,2,16,21\\}$, $\\{0,1,5,22\\}$,
$\\{0,3,18,22\\}$, $\\{0,2,19,22\\}$, $\\{0,2,5,23\\}$, $\\{0,1,2,24\\}$,
$\\{0,1,4,13,14,17\\}$, $\\{0,2,7,13,15,20\\}$

###  2.5 $v=28$

On $v=28$ points there are 854 nonisomorphic transitive $1$-$(28,6,6)$ designs
that do not cover a triple twice. There are four that are 2-class symmetric
designs, but only three $H_{1}$, $H_{2}$, $H_{3}$ that extend to a transitive
$3$-$(28,\\{4,6\\},1)$ design.

$H_{1},H_{2}$

are each 2-class symmetric design with parameters $(28,6{;}1,2{;}1,2)$ and are
each also regular group divisible type $\textsc{pbibd}(28,6,1,2)$ with
parameters

$\begin{array}[]{llll}n_{1}=24,&n_{2}=3,&P_{1}=\left[\begin{array}[]{rr}20&3\\\
3&0\end{array}\right],&P_{2}=\left[\begin{array}[]{rr}24&0\\\
2&0\end{array}\right]\end{array}.$

Clatworthy records these pbibd parameters as R171 on pages 178 and 220 of [4],
but only one solution is provided.

$H_{3}$

is the semi-biplane $\textsc{sbp}(28,6)$.

The Clatworthy $\textsc{pbibd}(28,6,1,2)$ recorded as R171 extends to the
twelve $3$-$(28,\\{4,6\\},1)$ designs
$D^{28}_{3},D^{28}_{4},\ldots,D^{28}_{14}$, the other
$\textsc{pbibd}(28,6,1,2)$ extends to the two $3$-$(28,\\{4,6\\},1)$ designs
$D^{28}_{1}$ and $D^{28}_{2}$.

The group generators

$(0,2,4,\ldots,26)(1,3,5,\ldots,27)\quad\text{and}\quad(0,1)(2,3)\cdots(26,27)$
(1)

generate $\textsc{C}_{2}\times\textsc{C}_{14}$, which can be used to develop
the baseblocks below into the required $3$-$(28,\\{4,6\\},1)$ designs
$D^{28}_{1},D^{28}_{2},\ldots,D^{28}_{10}$.

$D^{28}_{1}$:

Baseblocks
$\\{0,1,4,5\\}$, $\\{0,2,8,11\\}$, $\\{0,1,10,12\\}$, $\\{0,4,9,13\\}$,
$\\{0,3,4,14\\}$, $\\{0,2,13,14\\}$, $\\{0,3,9,15\\}$, $\\{0,1,14,15\\}$,
$\\{0,2,9,16\\}$, $\\{0,3,11,18\\}$, $\\{0,4,10,19\\}$, $\\{0,4,8,20\\}$,
$\\{0,5,14,20\\}$, $\\{0,7,15,20\\}$, $\\{0,5,8,21\\}$, $\\{0,2,12,22\\}$,
$\\{0,4,17,22\\}$, $\\{0,2,4,23\\}$, $\\{0,3,12,23\\}$, $\\{0,5,17,23\\}$,
$\\{0,1,18,23\\}$, $\\{0,2,7,24\\}$, $\\{0,3,8,24\\}$, $\\{0,2,15,25\\}$,
$\\{0,1,8,26\\}$, $\\{0,2,5,27\\}$, $\\{0,1,2,6,17,20\\}$

$D^{28}_{2}$:

Baseblocks
$\\{0,1,4,5\\}$, $\\{0,2,7,10\\}$, $\\{0,3,6,12\\}$, $\\{0,2,8,12\\}$,
$\\{0,2,5,16\\}$, $\\{0,3,10,15\\}$, $\\{0,2,11,15\\}$, $\\{0,1,14,15\\}$,
$\\{0,3,8,17\\}$, $\\{0,4,11,17\\}$, $\\{0,2,14,18\\}$, $\\{0,1,8,19\\}$,
$\\{0,3,9,19\\}$, $\\{0,5,10,20\\}$, $\\{0,7,12,20\\}$, $\\{0,2,9,21\\}$,
$\\{0,6,15,21\\}$, $\\{0,1,12,22\\}$, $\\{0,3,18,22\\}$, $\\{0,3,4,23\\}$,
$\\{0,2,4,24\\}$, $\\{0,3,11,25\\}$, $\\{0,4,12,25\\}$, $\\{0,2,19,25\\}$,
$\\{0,1,10,27\\}$, $\\{0,2,23,27\\}$, $\\{0,1,2,6,17,20\\}$

$D^{28}_{3}$:

Baseblocks
$\\{0,1,4,5\\}$, $\\{0,2,6,11\\}$, $\\{0,1,10,11\\}$, $\\{0,3,8,12\\}$,
$\\{0,2,9,12\\}$, $\\{0,1,14,15\\}$, $\\{0,4,10,16\\}$, $\\{0,5,11,16\\}$,
$\\{0,3,15,16\\}$, $\\{0,5,10,17\\}$, $\\{0,2,15,17\\}$, $\\{0,2,8,18\\}$,
$\\{0,3,11,18\\}$, $\\{0,4,15,18\\}$, $\\{0,3,10,19\\}$, $\\{0,5,12,20\\}$,
$\\{0,6,15,20\\}$, $\\{0,2,4,21\\}$, $\\{0,1,8,22\\}$, $\\{0,2,13,22\\}$,
$\\{0,4,17,22\\}$, $\\{0,2,5,25\\}$, $\\{0,3,20,24\\}$, $\\{0,2,23,24\\}$,
$\\{0,3,4,25\\}$, $\\{0,1,12,26\\}$, $\\{0,2,10,27\\}$, $\\{0,1,2,7,16,20\\}$

$D^{28}_{4}$:

Baseblocks
$\\{0,1,4,5\\}$, $\\{0,2,8,11\\}$, $\\{0,1,10,11\\}$, $\\{0,3,7,12\\}$,
$\\{0,2,9,12\\}$, $\\{0,2,5,13\\}$, $\\{0,3,9,13\\}$, $\\{0,1,14,15\\}$,
$\\{0,3,15,16\\}$, $\\{0,4,10,17\\}$, $\\{0,5,11,17\\}$, $\\{0,2,15,17\\}$,
$\\{0,3,8,18\\}$, $\\{0,2,6,18\\}$, $\\{0,4,15,18\\}$, $\\{0,3,10,19\\}$,
$\\{0,4,9,20\\}$, $\\{0,5,12,20\\}$, $\\{0,6,15,20\\}$, $\\{0,2,10,21\\}$,
$\\{0,1,8,22\\}$, $\\{0,5,10,22\\}$, $\\{0,3,4,24\\}$, $\\{0,2,19,24\\}$,
$\\{0,2,4,25\\}$, $\\{0,1,12,26\\}$, $\\{0,2,22,27\\}$, $\\{0,1,2,7,16,20\\}$

$D^{28}_{5}$:

Baseblocks
$\\{0,1,4,5\\}$, $\\{0,3,4,10\\}$, $\\{0,2,5,10\\}$, $\\{0,1,10,11\\}$,
$\\{0,2,8,12\\}$, $\\{0,1,14,15\\}$, $\\{0,3,15,16\\}$, $\\{0,4,9,17\\}$,
$\\{0,2,15,17\\}$, $\\{0,2,6,18\\}$, $\\{0,3,11,18\\}$, $\\{0,4,15,18\\}$,
$\\{0,5,8,20\\}$, $\\{0,6,15,20\\}$, $\\{0,2,9,21\\}$, $\\{0,5,11,21\\}$,
$\\{0,1,8,22\\}$, $\\{0,3,7,22\\}$, $\\{0,2,13,22\\}$, $\\{0,3,13,23\\}$,
$\\{0,5,16,23\\}$, $\\{0,2,19,24\\}$, $\\{0,3,20,24\\}$, $\\{0,2,4,25\\}$,
$\\{0,3,19,25\\}$, $\\{0,1,12,26\\}$, $\\{0,2,11,27\\}$, $\\{0,1,2,7,16,20\\}$

$D^{28}_{6}$:

Baseblocks
$\\{0,1,4,5\\}$, $\\{0,2,6,11\\}$, $\\{0,1,10,11\\}$, $\\{0,3,8,12\\}$,
$\\{0,2,9,12\\}$, $\\{0,1,14,15\\}$, $\\{0,4,10,16\\}$, $\\{0,5,11,16\\}$,
$\\{0,5,10,17\\}$, $\\{0,2,14,17\\}$, $\\{0,2,8,18\\}$, $\\{0,3,11,18\\}$,
$\\{0,4,15,18\\}$, $\\{0,3,10,19\\}$, $\\{0,5,12,20\\}$, $\\{0,6,15,20\\}$,
$\\{0,2,4,21\\}$, $\\{0,1,8,22\\}$, $\\{0,2,13,22\\}$, $\\{0,4,17,22\\}$,
$\\{0,2,5,25\\}$, $\\{0,3,20,24\\}$, $\\{0,2,23,24\\}$, $\\{0,3,4,25\\}$,
$\\{0,2,10,27\\}$, $\\{0,1,12,27\\}$, $\\{0,1,2,7,16,20\\}$

$D^{28}_{7}$:

Baseblocks
$\\{0,1,4,5\\}$, $\\{0,2,8,11\\}$, $\\{0,1,10,11\\}$, $\\{0,3,7,12\\}$,
$\\{0,2,9,12\\}$, $\\{0,2,5,13\\}$, $\\{0,3,9,13\\}$, $\\{0,1,14,15\\}$,
$\\{0,4,10,17\\}$, $\\{0,5,11,17\\}$, $\\{0,2,14,17\\}$, $\\{0,3,8,18\\}$,
$\\{0,2,6,18\\}$, $\\{0,4,15,18\\}$, $\\{0,3,10,19\\}$, $\\{0,4,9,20\\}$,
$\\{0,5,12,20\\}$, $\\{0,6,15,20\\}$, $\\{0,2,10,21\\}$, $\\{0,1,8,22\\}$,
$\\{0,5,10,22\\}$, $\\{0,3,4,24\\}$, $\\{0,2,19,24\\}$, $\\{0,2,4,25\\}$,
$\\{0,1,12,27\\}$, $\\{0,2,22,27\\}$, $\\{0,1,2,7,16,20\\}$

$D^{28}_{8}$:

Baseblocks
$\\{0,1,4,5\\}$, $\\{0,3,4,10\\}$, $\\{0,2,5,10\\}$, $\\{0,1,10,11\\}$,
$\\{0,2,8,12\\}$, $\\{0,1,14,15\\}$, $\\{0,4,9,17\\}$, $\\{0,2,14,17\\}$,
$\\{0,2,6,18\\}$, $\\{0,3,11,18\\}$, $\\{0,4,15,18\\}$, $\\{0,5,8,20\\}$,
$\\{0,6,15,20\\}$, $\\{0,2,9,21\\}$, $\\{0,5,11,21\\}$, $\\{0,1,8,22\\}$,
$\\{0,3,7,22\\}$, $\\{0,2,13,22\\}$, $\\{0,3,13,23\\}$, $\\{0,5,16,23\\}$,
$\\{0,2,19,24\\}$, $\\{0,3,20,24\\}$, $\\{0,2,4,25\\}$, $\\{0,3,19,25\\}$,
$\\{0,2,11,27\\}$, $\\{0,1,12,27\\}$, $\\{0,1,2,7,16,20\\}$

$D^{28}_{9}$:

Baseblocks
$\\{0,1,4,5\\}$, $\\{0,2,6,11\\}$, $\\{0,1,10,11\\}$, $\\{0,3,8,12\\}$,
$\\{0,2,9,12\\}$, $\\{0,1,14,15\\}$, $\\{0,4,10,16\\}$, $\\{0,5,11,16\\}$,
$\\{0,3,15,16\\}$, $\\{0,5,10,17\\}$, $\\{0,2,15,17\\}$, $\\{0,2,8,18\\}$,
$\\{0,3,11,18\\}$, $\\{0,4,15,18\\}$, $\\{0,3,10,19\\}$, $\\{0,5,12,20\\}$,
$\\{0,6,14,20\\}$, $\\{0,2,4,21\\}$, $\\{0,7,14,21\\}$, $\\{0,2,13,22\\}$,
$\\{0,4,17,22\\}$, $\\{0,1,8,23\\}$, $\\{0,2,5,25\\}$, $\\{0,3,20,24\\}$,
$\\{0,2,23,24\\}$, $\\{0,3,4,25\\}$, $\\{0,1,12,26\\}$, $\\{0,2,10,27\\}$,
$\\{0,1,2,7,16,20\\}$

$D^{28}_{10}$:

Baseblocks
$\\{0,2,6,11\\}$, $\\{0,3,8,12\\}$, $\\{0,2,9,12\\}$, $\\{0,1,14,15\\}$,
$\\{0,4,10,16\\}$, $\\{0,5,11,16\\}$, $\\{0,5,10,17\\}$, $\\{0,2,14,17\\}$,
$\\{0,1,4,18\\}$, $\\{0,2,8,18\\}$, $\\{0,3,11,18\\}$, $\\{0,3,10,19\\}$,
$\\{0,5,12,20\\}$, $\\{0,6,15,20\\}$, $\\{0,2,4,21\\}$, $\\{0,1,8,22\\}$,
$\\{0,2,13,22\\}$, $\\{0,4,17,22\\}$, $\\{0,2,5,25\\}$, $\\{0,3,20,24\\}$,
$\\{0,2,23,24\\}$, $\\{0,3,4,25\\}$, $\\{0,1,10,25\\}$, $\\{0,2,10,27\\}$,
$\\{0,1,12,27\\}$, $\\{0,1,2,7,16,20\\}$

The remaining 4 transitive $3$-$(28,\\{4,6\\},1)$ are obtained by developing
the baseblocks below with the group $\textsc{F}_{8}$ generated by

${\footnotesize\left.\begin{array}[]{@{}l@{}}(0,4,8,13,17,21,25)(1,5,9,12,16,20,24)(2,6,10,15,19,23,26)(3,7,11,14,18,22,27)\\\
\text{and
}(2,3)(4,18)(5,19)(6,7)(8,23)(9,22)(10,25)(11,24)(12,26)(13,27)(16,17)(20,21)\end{array}\right\\}}$
(2)

$D^{28}_{11}$:

Baseblocks
$\\{0,1,4,5\\}$, $\\{0,2,4,10\\}$, $\\{0,2,6,11\\}$, $\\{0,5,8,17\\}$,
$\\{0,1,14,15\\}$, $\\{0,4,11,17\\}$, $\\{0,2,14,17\\}$, $\\{0,2,13,18\\}$,
$\\{0,4,15,18\\}$, $\\{0,5,15,19\\}$, $\\{0,6,15,20\\}$, $\\{0,2,9,21\\}$,
$\\{0,1,8,22\\}$, $\\{0,2,5,24\\}$, $\\{0,2,19,25\\}$, $\\{0,1,12,27\\}$,
$\\{0,2,8,27\\}$, $\\{0,1,2,7,16,20\\}$

$D^{28}_{12}$:

Baseblocks
$\\{0,1,4,5\\}$, $\\{0,2,4,10\\}$, $\\{0,2,6,11\\}$, $\\{0,5,8,17\\}$,
$\\{0,1,14,15\\}$, $\\{0,4,11,17\\}$, $\\{0,2,15,17\\}$, $\\{0,2,13,18\\}$,
$\\{0,4,15,18\\}$, $\\{0,5,15,19\\}$, $\\{0,6,15,20\\}$, $\\{0,2,9,21\\}$,
$\\{0,1,8,22\\}$, $\\{0,2,5,24\\}$, $\\{0,2,19,25\\}$, $\\{0,1,12,26\\}$,
$\\{0,2,8,27\\}$, $\\{0,1,13,27\\}$, $\\{0,1,2,7,16,20\\}$

$D^{28}_{13}$:

Baseblocks
$\\{0,2,4,10\\}$, $\\{0,2,6,11\\}$, $\\{0,5,8,17\\}$, $\\{0,1,14,15\\}$,
$\\{0,4,11,17\\}$, $\\{0,2,14,17\\}$, $\\{0,1,4,18\\}$, $\\{0,2,13,18\\}$,
$\\{0,6,15,20\\}$, $\\{0,2,9,21\\}$, $\\{0,1,8,22\\}$, $\\{0,2,5,24\\}$,
$\\{0,1,10,25\\}$, $\\{0,2,19,25\\}$, $\\{0,1,12,27\\}$, $\\{0,2,8,27\\}$,
$\\{0,1,2,7,16,20\\}$

$D^{28}_{14}$:

Baseblocks
$\\{0,1,4,5\\}$, $\\{0,2,4,10\\}$, $\\{0,2,6,11\\}$, $\\{0,5,8,17\\}$,
$\\{0,1,14,15\\}$, $\\{0,4,11,17\\}$, $\\{0,2,15,17\\}$, $\\{0,2,13,18\\}$,
$\\{0,4,15,18\\}$, $\\{0,5,15,19\\}$, $\\{0,6,14,20\\}$, $\\{0,2,9,21\\}$,
$\\{0,1,9,22\\}$, $\\{0,1,8,23\\}$, $\\{0,2,5,24\\}$, $\\{0,2,19,25\\}$,
$\\{0,1,12,26\\}$, $\\{0,2,8,27\\}$, $\\{0,1,13,27\\}$, $\\{0,1,2,7,16,20\\}$

The full automorphism group of $D^{28}_{1},D^{28}_{2},\ldots,D^{28}_{10}$ has
order 28 and is $\textsc{C}_{2}\times\textsc{C}_{14}$ with generators (1).

The full automorphism group of $D^{28}_{11}$ and $D^{28}_{12}$ has order order
56 and is $\textsc{F}_{8}$ with generators (2), but the full automorphism
group of $D^{28}_{13}$ and $D^{28}_{14}$ has order order 168. It is
$\textsc{F}_{8}\rtimes\textsc{C}_{3}=\textsc{Aut}(\textsc{F}_{8})$ with
generators (2) and (3).

${\footnotesize\left.\begin{array}[]{@{}l@{}}(1,14,15)(2,4,22)(3,18,9)(5,8,17)(6,26,25)(7,12,10)(11,21,27)(13,24,20)(16,19,23)\end{array}\right.}$
(3)

Enumerating the extensions of the semi-biplane $\textsc{sbp}(28,6)$ $H_{3}$
has proven to be very difficult, because there are apparently millions of
nonisomorphic extensions. We provide baseblocks and generators for one of the
transitive $3$-$(28,\\{4,6\\},1)$ designs, whose hexads are $H_{3}$, the semi-
biplane $\textsc{sbp}(28,6)$.

$D^{28}_{15}$:

Generators
$(4,8,17)(5,9,16)(6,10,18)(7,11,19)(12,25,20)(13,24,21)(14,27,22)(15,26,23)$
$(0,1)(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)\cdots(24,25)(26,27)$
$(0,2,1,3)(4,15,9,27,17,23,5,14,8,26,16,22)(6,13,11,25,18,21,7,12,10,24,19,20)$
$(0,4,8,12,17,20,25)(1,5,9,13,16,21,24)(2,6,10,15,18,23,26)(3,7,11,14,19,22,27)$
Baseblocks
$\\{0,2,4,22\\}$, $\\{0,1,14,15\\}$, $\\{0,1,4,5\\}$, $\\{0,2,5,15\\}$,
$\\{0,4,14,19\\}$, $\\{0,4,9,21\\}$, $\\{0,2,7,24\\}$, $\\{0,5,18,22\\}$,
$\\{0,4,12,26\\}$, $\\{0,4,13,27\\}$, $\\{0,1,2,6,10,18\\}$

##  3 Computation

Let $G$ be a transitive subgroup of $\textsc{Sym}(X)$. If $x\in X$ and $g\in
G$, then the image of $x$ under $g$ is $x^{g}$ and if $S\subseteq X$, then
$S^{g}=\\{x^{g}:x\in S\\}$ is the image of $S$. The orbit of $S$ under the
action of $G$ is $S^{G}=\\{S^{g}:\text{$g\in G$}\\}$. Let

$\begin{array}[]{c@{,\ldots,}cl}\Delta_{1}&\Delta_{m}&\text{ be the orbits of
$3$-element sets}\\\ \Gamma_{1}&\Gamma_{n}&\text{ be the orbits of $4$-element
sets}\\\ E_{1}&E_{s}&\text{ be the orbits of $6$-element sets}\end{array}$

Choose $E_{i_{1}},E_{i_{2}},\ldots,E_{i_{x}}$ such that
$\sum_{j=1}^{x}|E_{i_{j}}|=v$, then $\mathscr{H}=\cup_{j=1}^{x}E_{i_{j}}$ is a
$1$-$(v,6,6)$ design and a possible candidate set of hexads. Choose
$T_{i}\in\Delta_{i}$ to be any fixed representative, $i=1,2,..,m$ and let

$\displaystyle\mathscr{R}$
$\displaystyle=\\{\Delta_{i}:|\\{S\in\mathscr{H}:S\supset T_{i}\\}|=0\\},$
$\displaystyle\mathscr{C}$
$\displaystyle=\\{\Gamma_{j}:|\\{K\in\Gamma_{j}:K\supset T_{i}\\}|\leq 1\text{
for every }\Delta_{i}\in\mathscr{R}\\}$

Define the $(0,1)$-valued $\mathscr{R}$ by $\mathscr{C}$ matrix $A$ by

$A[\Delta,\Gamma]=|\\{K\in\Gamma:K\supset T\\}|,$

where $T\in\Delta$ is a fixed representative. Then a (0,1) solution $U$ to
$AU=J$, where $J=[1,1,\ldots,1]^{T}$, selects the tetrads that together with
the candidate hexads forms a homogeneous $3$-$(v,\\{4,6\\},1)$ on $v=|X|$
points that has $G$ as an automorphism group. The orbit representatives and
construction of the incidence matrix used tools described in [13] with
suitable modification. The matrix equation $AU=J$ was solved with a backtrack
algorithm using the dancing links data structure described in [10]. The
implementation we employed can be downloaded from

http://pottonen.kapsi.fi/libexact.html

A subgroup $G$ of $\textsc{Sym}(X)$ is _minimally transitive_ if it contains
no proper transitive subgroup. To find all transitive designs on a given
number of points $v$ only the minimal groups need be searched. The following
magma code finds generators for all minimal transitive groups on $v$ points:

    procedure MinTrans(v)
     fn:="MinTransGroupsDegree"*IntegerToString(v)*".txt";
     SetAutoColumns(false);
     SetColumns(0);
     for i in [1..NumberOfTransitiveGroups(v)] do
      G,c:=TransitiveGroup(v,i);
      max:=MaximalSubgroups(G);
      if not &or [IsTransitive(max[j]`subgroup): j in [1..#max]] then
       Write(fn, c);
       Write(fn, Sprint(#Generators(G))*" " *Sprint(v));
       for j in [1..NumberOfGenerators(G)] do
        Write(fn,G.j);
       end for;
       Write(fn, GroupName(G));
       Write(fn, "------------");
      end if;
     end for;
     SetAutoColumns(true);
    end procedure;

For each $v$ we collect all designs afforded by each of the minimal transitive
groups of degree $v$. These designs are then reduced to a list of
nonisomorphic designs using the graph isomorphism program `nauty` obtained
here:

http://users.cecs.anu.edu.au/~bdm/

Once all designs are found, `nauty` was used to find their full automorphism
groups. Then essentially the same method described above was re-run to obtain
the baseblocks for the designs with respect to their full groups.

## Acknowledgments

We thank the Fields Institute for Research in Mathematical Sciences for the
support they provided and Rosemary Bailey for her very useful comments.

## References

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