Group selection 
Choose prescribed automorphism group
If you choose the "group" button in DISCRETA you have different possibilities for
further choices. You get the following list:
Here you have to press "Choose prescribed automorphism group" to receive a
list of groups we have implemented. A new window will open:
Besides the possibility of typing in a set of generating permutations by hand
(an example is given) you have the choice between 5 kinds of groups:
The first selection is "well known". Here you can choose the symmetric, the
alternating, dieder or the cyclic group on n points (you have to choose the
parameter n first), as well as the holomorph of the cyclic group on n points,
the trivial group, the (only) subgroup of Hol(C_{n}) of index m or
finally the wreath product of S_{n} and S_{m}.


For the construction of designs very useful are the linear groups  you have to
press "linear": we consider
the following matrix groups in their permutation representation. First of all
there is the general linear group GL(n,q)  where q is the order of the
corresponding field  of regular n /times nmatrices; then SL(n,q) is the
special linear group of matrices with determinante 1.
Derived from them we have
the GGL(n,q) where GG stands for Gamma, and the SSl(n,q) where SS stands for
Sigma. The corresponding projective groups we get as factor groups modulo the
center. Then we have the affine groups of affine transformations. Finally, you
can choose the group of affine translations T(n,q), the projective special
unitary group PSU(3,q^{2}) and the Sz(q).
For all groups you have to
choose the parameters n and q.


You also can use several sporadic groups:
The Mathieu groups M_{11}, M_{12}, M_{23} and
M_{24} are simple groups. They are the only nontrivial, at least
4transitive groups:
 M_{11}: sharply 4transitive of degree 11, order: 11 · 10
· 9 · 8
 M_{12}: sharply 5transitive of degree 12, order: 12 · 11
· 10 · 9 · 8
 M_{23}: 4transitive of degree 23, order: 23 · 22 · 21
· 20 · 3 · 2^{4}
 M_{24}: 5transitive of degree 24, order: 24 · 23 · 22
· 21 · 20 · 3 · 2^{4}
The HigmanSims group HS can be described as the automorphism group of a
combinatorial geometry consisting of a set M of 176 points and a set Q of 176
quadrics. It is 2transitive in its action on M and has an equivalent
20transitive action on Q. The order of HS is 44.352.000.


There exists a catalogue of all solvable groups of degree at most 127. With the
parameter n you choose the degree and the choice of m gives you the mth group
of degree n in the catalogue.


Here you can choose the symmetry groups of regular and semiregular solids.
You start with the choice of one of the platonic solids: tetrahedron, cube,
octahedron, icosahedron and dodecahedron.
Then it is possible to modify the
chosen solid for example by
 truncating the edges (only for vizualizing
the solid the distinction between dodecahedron, cube and the others is
important  the group just the same, namely the induced group of the first
solid),
 considering the dual solid i.e. taking the center of the faces as the
new vertices (the platonic solids are pairwise dual: cube
and octahedron, icosa and dodecahedron, tetrahedron with itself); it is very
interesting to build the dual of semiregular and archimedian solids,
 taking
the midpoints of edges as the vertices of the new solid and
 adding a central point of a solid; that means adding a fixpoint of
the automorphism group.
In addition you can choose the group of the hypercube (4dimensional cube).
All those solids are drawn as 3Dgraphs. The pictures can be shown on the screen
by pressing the button "group" and "show solid" (examples).



Very useful are the combinations of the different groups. You have 2
possibilities:
 You can choose two groups and combine them by
 appending the second to the first
 building the direct sum
 building the direct product
 building the wreath product
 exponentiation of the first group by the second one
 considering the action on mappings n^{m}
 You can choose one group and
 add a fixpoint
 build a point stabilizer
 induce the action of the group on 2sets
 induce the action of the group on 2tuples
 induce the action of the group on the injective 2tuples
 induce the action of the group on 3sets
 build the ith power of the group (you have to type in i)
One example (starting point is the tetrahedron) is given by the following selection:
The resulting designs and details of the construction you can find at this page.
Remark: You have to click on the pictures to get them largely.
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Last updated: August 24, 1999, Evi Haberberger