# ATLAS: Alternating group A15

Order = 653837184000
Mult = 2.
Out = 2.

### Standard generators

Standard generators of A15 are a in class 3A, b of order 13 such that ab has order 15 and abb has order 15. In the natural representation we may take a = (1, 2, 3) and b = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15).

Standard generators of S15 are c in class 2D, d in class 14E such that cd has order 15. In the natural representation we may take c = (1, 2) and d = (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15).

### Black box algorithms

To find standard generators for A15:
• Find an element of order 33 or 105 (probability 92/1155, or approximately 1/13) and power it up to give an element a in class 3A.
• Find an element t of order 13 (probability 1/13).
• Find a conjugate b of t such that ab has order 15 and abb has order 15 (probability 1/35).
• The elements a and b are standard generators of A15.
To find standard generators for S15:
• Find an element of order 26 or 70 (probability 24/455, or approximately 1/18) and power it up to give an element c in class 2D. There are other possibilities if you are able to distinguish outer and inner elements (such as powering up an outer 22 or 42 element).
• Find an element t of order 14 (probability approximately 1/14). This is probably in class 14E.
• Find a conjugate c of t such that cd has order 15 (probability 2/15, or about 1/7). If you cannot find such a d after several attempts, then t was probably in the wrong class, so go back to the second step.
• The elements c and d are standard generators of S15.

### Representations

The representations of A15 available are:
• Some primitive permutation representations
• Permutations on 15 points - the natural representation above: a and b (GAP).
• Permutations on 105 points a and b (GAP).
• Permutations on 455 points a and b (GAP).
• Permutations on 1365 points a and b (GAP).
• Some integer matrix representations
• Dimension 14 (partition [2,113]): a and b (GAP).
• Dimension 90 (partition [22,111]): a and b (GAP).
• Dimension 91 (partition [3, 112]): a and b (GAP).
• Dimension 350 (partition [23, 19]): a and b (GAP).
• Dimension 364 (partition [4, 111]): a and b (GAP).
The representations of S15 available are:
• Some primitive permutation representations
• Permutations on 15 points - the natural representation above: c and d (GAP).
• Permutations on 105 points c and d (GAP).
• Permutations on 455 points c and d (GAP).
• Permutations on 1365 points c and d (GAP).
• Some integer matrix representations
• Dimension 14 (partition [2,113]): c and d (GAP).
• Dimension 90 (partition [22,111]): c and d (GAP).
• Dimension 91 (partition [3, 112]): c and d (GAP).
• Dimension 350 (partition [23, 19]): c and d (GAP).
• Dimension 364 (partition [4, 111]): 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 17th February 2004.
Last updated 18.02.04 by SN.