# ATLAS: Alternating group A17

Order = 177843714048000
Mult = 2.
Out = 2.

The following information is available for A17:

### Standard generators

Standard generators of A17 are a in class 3A, b of order 15 such that ab has order 17 (so we must have b in class 15G).
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, 16, 17).

Standard generators of S17 are c in class 2E, d of order 16 such that cd has order 17 (so we must have b in class 16A).
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, 16, 17).

### Black box algorithms

To find standard generators of A17:
• Find an element of order 39 or 105 (probability about 1/16) and power up to give an element s in class 3A.
• Find an element t of order 17 (probability about 1/9).
• Find a conjugate u of t such that su has order 15 (probability about 1/40). Then u is in class 15G.
• Let a be the inverse of s and b be su. Then a and b are standard generators for A17.
To find standard generators of S17:
• Find an element of order 90 or 210 (probability 1/63) and power up to give an element c in class 2E. Alternatively, if you look among outer elements only, then elements of order 26 and 70 also work, and the probability rises to about 1/13.
• Find an element t of order 16 (probability 1/16).
• Find an conjugate d of t such that cd has order 17 (probability about 1/9).
• The elements c and d are standard generators for S17.

### Representations

The representations of A17 available are:
• Some primitive permutation representations
• Permutations on 17 points - the natural representation above: a and b (GAP).
• Permutations on 136 points: a and b (GAP).
• Permutations on 680 points: a and b (GAP).
• Permutations on 2380 points: a and b (GAP).
• Some integer matrix representations
• Dimension 16 (partition [2, 115]): a and b (GAP).
• Dimension 119 (partition [22, 113]): a and b (GAP).
• Dimension 120 (partition [3, 114]): a and b (GAP).
• Dimension 544 (partition [22, 111]): a and b (GAP).
• Dimension 560 (partition [4, 113]): a and b (GAP).
The representations of S17 available are:
• Some primitive permutation representations
• Permutations on 17 points - the natural representation above: c and d (GAP).
• Permutations on 136 points: c and d (GAP).
• Permutations on 680 points: c and d (GAP).
• Permutations on 2380 points: c and d (GAP).
• Some integer matrix representations
• Dimension 16 (partition [2, 115]): c and d (GAP).
• Dimension 119 (partition [22, 113]): c and d (GAP).
• Dimension 120 (partition [3, 114]): c and d (GAP).
• Dimension 544 (partition [22, 111]): c and d (GAP).
• Dimension 560 (partition [4, 113]): 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 27.02.04 by SN.