Mult = 2.

Out = 2.

The following information is available for A_{22}:

Standard generators of A_{22} are **a** of order 3
(belonging to the smallest conjugacy class of elements of order 3),
**b** of order 21
such that **ab** has order 20 and **[a,b]**
has order 2. This forces **b** to be in the conjugacy class of
21-cycles.

In the natural
representation we may take
**a** = (1, 2, 3) and
**b** = (2, 3, 4, 5, 6, 7, 8, ..., 22).

Standard generators of S_{22} are **c** of order 2
(belonging to the smallest conjugacy class of outer involutions),
**d** of order 21 such that **cd** has order 22. This forces
**d** to be in the conjugacy class of 21-cycles.

In the natural
representation we may take
**c** = (1, 2) and
**d** = (2, 3, 4, 5, 6, 7, 8, ..., 22).

To find standard generators of A_{22}:

- Find an element of order 51, 57, 156, 195 or 231 (probability about 1/12)
and power it up to give an element
*a*in class 3A. - Find an element
*t*of order 19 (probability about 1/57). - Find a conjugate
*u*of*t*such that*au*has order 21 (probability about 1/54). Then*au*is a 21-cycle. - Find a conjugate
*b*of*au*such that*ab*has order 20 and*[a,b]*has order 2. - The elements
*a*and*b*are standard generators of A_{22}.

To find standard generators of S_{22}:

- Find an element of order 38, 102, 182 or 198 (probability about 1/21)
and power it up to give an element
*c*of order 2 in the class of transpositions. (If you only look among outer elements, then finding an element of order 34, 130 or 154 would also work, andincreases the probability of finding an appropriate element to about 1/7 in each attempt.) - Find an element
*t*of order 19 (probability 1/38). - Find a conjugate
*u*of*t*such that*cu*has order 20 (probability about 1/2). - Find a conjugate
*v*of*cu*such that*cv*has order 21 (probability about 1/6). - Find a conjugate
*d*of*cv*such that*cd*has order 22 (probability 1/11). - The elements
*c*and*d*are standard generators of S_{22}.

- Some primitive permutation representations
- Some integer matrix representations

- Some primitive permutation representations
- Some integer matrix representations

Go to main ATLAS (version 2.0) page.

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Anonymous ftp access is also available on sylow.mat.bham.ac.uk.

Version 2.0 created on 17th February 2004.

Last updated 01.03.04 by SN.