ATLAS: Alternating group A₁₈

The following information is available for A₁₈:

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

Standard generators

Standard generators of A₁₈ are a of order 3 (belonging to the smallest conjugacy class of elements of order 3), b of order 17 such that ab has order 16 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).

Standard generators of S₁₈ are c of order 2 (belonging to the smallest conjugacy class of outer involutions), d of order 17 such that cd has order 18.
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).

Black box algorithms

To find standard generators of A₁₈:

• Find an element of order 39, 66 or 120 (probability about 1/24) and power it up to give an element a in class 3A.
• Find an element t of order 17 (probability 2/17, or approximately 1/9).
• Find a conjugate b of t such that ab has order 16 and [a,b] has order 2 (probability 1/96).
• The elements a and b are standard generators of A₁₆.

To find standard generators of S₁₈:

• Find an element of order 78, 110 or 126 (probability 1/34) and power it up to give an element c in class 2A. (If you only look among outer elements, then finding an element of order 26, 78, or 90 would also work, and increases the probability of finding an appropriate element to about 1/9.)
• Find an element t of order 17 (probability 1/17).
• Find a conjugate d of t such that cd has order 18 (probability 1/9).
• The elements c and d are standard generators of S₁₈.

Representations

• Some primitive permutation representations
  • Permutations on 18 points - the natural representation above: a and b (GAP).
  • Permutations on 153 points: a and b (GAP).
  • Permutations on 816 points: a and b (GAP).
  • Permutations on 3060 points: a and b (GAP).
• Some integer matrix representations
  • Dimension 17 (partition [2, 1¹⁶]): a and b (GAP).
  • Dimension 135 (partition [2², 1¹⁴]): a and b (GAP).
  • Dimension 136 (partition [3, 1¹⁵]): a and b (GAP).

• Some primitive permutation representations
  • Permutations on 18 points - the natural representation above: c and d (GAP).
  • Permutations on 153 points: c and d (GAP).
  • Permutations on 816 points: c and d (GAP).
  • Permutations on 3060 points: c and d (GAP).
• Some integer matrix representations
  • Dimension 17 (partition [2, 1¹⁶]): c and d (GAP).
  • Dimension 135 (partition [2², 1¹⁴]): c and d (GAP).
  • Dimension 136 (partition [3, 1¹⁵]): c and d (GAP).

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