ATLAS: Alternating group A₂₂

The following information is available for A₂₂:

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

Standard generators

Standard generators of A₂₂ 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 S₂₂ 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).

Black box algorithms

To find standard generators of A₂₂:

• 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 A₂₂.

To find standard generators of S₂₂:

• 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 S₂₂.

Representations

• Some primitive permutation representations
  • Permutations on 22 points - the natural representation above: a and b (GAP).
  • Permutations on 231 points: a and b (GAP).
• Some integer matrix representations
  • Dimension 21 (partition [2, 1²⁰]): a and b (GAP).
  • Dimension 209 (partition [2², 1¹⁸]): a and b (GAP).
  • Dimension 210 (partition [3, 1¹⁹]): a and b (GAP).

• Some primitive permutation representations
  • Permutations on 22 points - the natural representation above: c and d (GAP).
  • Permutations on 231 points: c and d (GAP).
• Some integer matrix representations
  • Dimension 21 (partition [2, 1²⁰]): c and d (GAP).
  • Dimension 209 (partition [2², 1¹⁸]): c and d (GAP).
  • Dimension 210 (partition [3, 1¹⁹]): c and d (GAP).

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