Explanation of pages on groups

This page explains the data given in pages on individual groups.

Each page is divided into several sections:

Alternative Names

A group (or, to be pedantic, an isomorphism class of groups) may have many names. For example, "psl32" and "projective special linear(3,2)" are different ways of saying the same thing; they are also, for boring reasons, the same as "gl32"; and, for more interesting reasons, they all denote the same isomorphism class as "psl27".

The page for a group lists some of its names.


Some statistics about the group are given in a table. They are:

  1. Its order.
  2. Its GAP identifier: a string of the form "24,6" which serves as a unique identifier and is used by the GAP software package.
  3. A presentation for the group.
  4. The orders of its elements, grouped by conjugacy classes and by automorphism classes. For example, the group D8 is shown as having elements "1 of 1, 1+2*2 of 2, 2 of 4". This means that it has an identity, a unique element of order 2, four elements of order 2 which are all mapped to one another by automorphisms but form two conjugacy classes of size two, and two elements of order 4 which form a single conjugacy class.
  5. Its centre: a subgroup containing all the elements that commute with all elements of the group. The centre is necessarily Abelian.
  6. Its derived subgroup: the subgroup generated by all the commutators (elements of the form aba-1b-1) of the group. (For small groups such as those listed in these pages, the set of commutators is the whole commutator subgroup.)
  7. Its automorphism group: the group of all automorphisms of the group.
  8. Its inner automorphism group: the group of all automorphisms of the group that can be done by conjugacy with elements of the group.
  9. Its outer automorphism group. The inner automorphism group is necessarily a normal subgroup of the automorphism group; the outer automorphism group is their quotient.
  10. Its Schur multiplier: this is the largest group that could be incorporated into the group as a centre.

Permutation diagrams

A permutation diagram is a way of specifying a group as a permutation group on a set of points. Each vertex of the permutation diagram is one of the points permuted, and each colour that appears in the diagram is one of a set of permutations that between them generate the group.

For example, here are three permutation diagrams, all showing D6. Assume that the vertices are numbered clockwise, starting with "1" at the top left.

The first has two permutations:       black (1 2 3) and red (2 3)
The second has two permutations:  black (1 3) and red (2 3)
The third has two permutations:       black (1 2)(3 4)(5 6) and red (2 3)(4 5)(6 1)

Here is another example, for C3⋊D8. Assume that the vertices are numbered clockwise around the triangle, and then clockwise around the square. The group is generated by three permutations: black (1 2 3), red (2 3)(4 5 6 7), and green (4 6).

The red generator acts on the black to give a D6, while the green acts on the red to give a D8

If arrowheads are omitted, all circuits should be assumed to go clockwise. Thus in the example to the right, for sl23, both black 4-circuits can be assumed to go clockwise; and the arrowheads on the red circuits are unnecessary and could have been omitted.

If all the points are connected together somehow (as in the three examples for D6), the group acts transitively on the points. If the diagram is in disjoint parts (as in the example for C3⋊D8), the group acts intransitively.

A Cayley diagram can be regarded as a special case of a permutation diagram. We can regard a Cayley diagram as a sharply 1-transitive permutation diagram, on a number of points equal to the order of the group.

Cayley diagrams

Some Cayley diagrams, also known as Cayley graphs, may be given.

Many of them are drawn in the plane. But some, such as the one to the right for C3⋊C8, are drawn on other surfaces. The latter have pink lines and pale pink regions, which function as "sewing instructions" for the surface. Thus the diagram to the right is in a torus.

Regular maps

If the group is the rotational symmetry group, or the full symmetry group, of some regular map, this mnay be mentioned