ATLAS: Alternating group A₂₃

The following information is available for A₂₃:

• Standard generators.
• Black box algorithms.
• Representations.

Standard generators

Standard generators of A₂₃ are a of order 3 (in the smallest conjugacy class of elements of elements of order 3), b of order 21 such that ab has order 23. 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 = (3,4,5,6,7,8, ... ,23).

Standard generators of S₂₃ are c of order 2 (belonging to the smallest conjugacy class of outer involutions), d of order 22 such that cd has order 23. This forces d to be in the conjugacy class of 22-cycles.
In the natural representation we may take c = (1, 2) and d = (2, 3, 4, 5, 6, 7, 8, ..., 23).

Black box algorithms

To find standard generators of A₂₃:

• Find an element of order 57, 195, 231, 273 or 330 (probability about 1/19) and power up to give an element s in class 3A.
• Find an element t of order 23 (probability about 1/12).
• Find a conjugate u of t such that su has order 21 (probability 1/140).
• Let a = s^-1 and b = su
• The elements a and b are standard generators for A₂₃.

To find standard generators of S₂₃:

• Find an element of order 182, 198, 390, 462 or 630 (probability about 1/59) and power up to give an element c in the class of transpositions. (Alternatively, if you restrict your search to outer elements then an element of order 38 or 154 would also work, and would increase the probability at each attempt to about 1/13.
• Find an element t of order 23 (probability 1/23).
• Find an conjugate u of t such that cu has order 22 (probability 1/11).
• The elements c and d = cu are standard generators for S₂₃.

Representations

• Some primitive permutation representations
  • Permutations on 23 points - the natural representation above: a and b (GAP).
  • Permutations on 253 points: a and b (GAP).
• Some integer matrix representations
  • Dimension 22 (partition [2, 1²¹]): a and b (GAP).
  • Dimension 230 (partition [2², 1¹⁹]): a and b (GAP).
  • Dimension 231 (partition [3, 1²⁰]): a and b (GAP).

• Some primitive permutation representations
  • Permutations on 23 points - the natural representation above: c and d (GAP).
  • Permutations on 253 points: c and d (GAP).
• Some integer matrix representations
  • Dimension 22 (partition [2, 1²¹]): c and d (GAP).
  • Dimension 230 (partition [2², 1¹⁹]): c and d (GAP).
  • Dimension 231 (partition [3, 1²⁰]): c and d (GAP).

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