ATLAS: Alternating group A₁₉

The following information is available for A₁₉:

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

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

• 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).

• 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).

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