# ATLAS: Alternating group A19

Order = 60822550204416000
Mult = 2.
Out = 2.

The following information is available for A19:

### Standard generators

Standard generators of A19 are a of order 3 (in the smallest conjugacy class of elements of elements of order 3), b of order 17 such that ab has order 19.
In the natural representation we may take a = (1,2,3) and b = (3,4,5,6,7,8, ... ,19).

Standard generators of S19 are c of order 2 (belonging to the smallest conjugacy class of outer involutions), d of order 18 such that cd has order 19. This forces d to belong to the conjugacy class of 18-cycles.
In the natural representation we may take c = (1, 2) and d = (2, 3, 4, 5, 6, 7, 8, ..., 19).

### Black box algorithms

To find standard generators of A19:

• Find an element of order 165 or 210 (probability about 1/69) and power up to give an element a in class 3A.
• Find an element t of order 17 (probability about 1/17).
• Find a conjugate b of t such that ab has order 19 (probability about 1/114).
• The elements a and b are standard generators for A19.

To find standard generators of S19:

• Find an element of order 33, 110 or 126 (probability 1/22) and power up to give an element c in the class of transpositions.
• Find an element t of order 19 (probability 1/19).
• Find an conjugate u of t such that cu has order 18 (probability 1/9).
• The elements c and d = cu are standard generators for S19.

### Representations

The representations of A19 available are:
• Some primitive permutation representations
• Permutations on 19 points - the natural representation above: a and b (GAP).
• Permutations on 171 points: a and b (GAP).
• Permutations on 969 points: a and b (GAP).
• Some integer matrix representations
• Dimension 18 (partition [2, 117]): a and b (GAP).
• Dimension 152 (partition [22, 115]): a and b (GAP).
• Dimension 153 (partition [3, 116]): a and b (GAP).
The representations of S19 available are:
• Some primitive permutation representations
• Permutations on 19 points - the natural representation above: c and d (GAP).
• Permutations on 171 points: c and d (GAP).
• Permutations on 969 points: c and d (GAP).
• Some integer matrix representations
• Dimension 18 (partition [2, 117]): c and d (GAP).
• Dimension 152 (partition [22, 115]): c and d (GAP).
• Dimension 153 (partition [3, 116]): 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.