# ATLAS: Alternating group A21

Order = 25545471085854720000
Mult = 2.
Out = 2.

The following information is available for A21:

### Standard generators

Standard generators of A21 are a of order 3 (in the smallest conjugacy class of elements of elements of order 3), b of order 19 such that ab has order 21.
In the natural representation we may take a = (1,2,3) and b = (3,4,5,6,7,8, ... ,21).

Standard generators of S21 are c of order 2 (belonging to the smallest conjugacy class of outer involutions), d of order 20 such that cd has order 21.
In the natural representation we may take c = (1, 2) and d = (2, 3, 4, 5, 6, 7, 8, ..., 21).

### Black box algorithms

To find standard generators of A21:
• Find an element of order 48, 51, 165, 195, 231 or 420 (probability about 1/11) and power up to give an element a in class 3A.
• Find an element t of order 19 (probability about 1/19).
• Find a conjugate b of t such that ab has order 21 (probability about 1/140).
• The elements a and b are standard generators for A17.
To find standard generators of S21
• Find an element of order 38, 130, 154 or 330 (probability about 1/23) and power up to give an element c of order 2 in the conjugacy class of transpositions. Alternatively, if you look among outer elements only, then elements of order 34 and 126 also work, and the probability rises to about 1/7.
• Find an element t of order 19 (probability 1/38).
• Find a conjugate u of t such that cu has order 20 (probability about 1/6).
• Find a conjugate d of cu such that cd has order 21 (probability about 1/11).
• The elements c and d are standard generators for S17.

### Representations

The representations of A21 available are:
• Some primitive permutation representations
• Permutations on 21 points - the natural representation above: a and b (GAP).
• Permutations on 210 points: a and b (GAP).
• Some integer matrix representations
• Dimension 20 (partition [2, 119]): a and b (GAP).
• Dimension 189 (partition [22, 117]): a and b (GAP).
• Dimension 190 (partition [3, 118]): a and b (GAP).
The representations of S21 available are:
• Some primitive permutation representations
• Permutations on 21 points - the natural representation above: c and d (GAP).
• Permutations on 210 points: c and d (GAP).
• Some integer matrix representations
• Dimension 20 (partition [2, 119]): c and d (GAP).
• Dimension 189 (partition [22, 117] c and d (GAP).
• Dimension 190 (partition [3, 118]): c and d (GAP). Go to main ATLAS (version 2.0) page. Go to alternating groups page. Anonymous ftp access is also available on sylow.mat.bham.ac.uk.

Version 2.0 created on 24th February 2004.
Last updated 01.03.04 by SN.