ATLAS: Alternating group A₂₀

The following information is available for A₂₀:

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

Standard generators

Standard generators of A₂₀ are a of order 3 (belonging to the smallest conjugacy class of elements of order 3), b of order 19 such that ab has order 18 and [a,b] has order 2.
In the natural representation we may take a = (1, 2, 3) and b = (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,19,20).

Standard generators of S₂₀ are c of order 2 (belonging to the smallest conjugacy class of outer involutions), d of order 19 such that cd has order 20.
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, 18, 19, 20).

Black box algorithms

To find standard generators of A₂₀:

• Find an element of order 51, 78, 132, 165 or 168 (probability about 1/14) and power up to give an element a in class 3A.
• Find an element t of order 19 (probability about 1/10).
• Find a conjugate b of t such that ab has order 17 and a^-1b^-1ab has order 2 (probability about 1/127).
• The elements a and b are standard generators for A₂₀.

To find standard generators of S₂₀:

• Find an element of order 34, 130 or 154 (probability about 1/23) and power up to give an element c in the class of transpositions. (Alternatively, if you restrict your search to outer elements, then elements of order 120 and 126 also work, and the probability increases to about 1/9.)
• Find an element t of order 19 (probability 1/19).
• Find an conjugate d of t such that cd has order 20 (probability 1/10).
• The elements c and d are standard generators for S₂₀.

Representations

• Some primitive permutation representations
  • Permutations on 20 points - the natural representation above: a and b (GAP).
  • Permutations on 190 points: a and b (GAP).
• Some integer matrix representations
  • Dimension 19 (partition [2, 1¹⁷]): a and b (GAP).
  • Dimension 170 (partition [2², 1¹⁶]): a and b (GAP).
  • Dimension 171 (partition [3, 1¹⁷]): a and b (GAP).

• Some primitive permutation representations
  • Permutations on 20 points - the natural representation above: c and d (GAP).
  • Permutations on 190 points: c and d (GAP).
• Some integer matrix representations
  • Dimension 19 (partition [2, 1¹⁷]): c and d (GAP).
  • Dimension 170 (partition [2², 1¹⁶]): c and d (GAP).
  • Dimension 171 (partition [3, 1¹⁷]): c and d (GAP).

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