ATLAS: Alternating group A₁₆

The following information is available for A₁₆:

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

Standard generators

Standard generators of A₁₆ are a in class 3A, b in class 15F/G such that ab has order 14 and abb has order 63. This last condition can be replaced with [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).
Standard generators of the double cover 2.A₁₆ are not yet defined.

Standard generators of S₁₆ are c in class 2E, d in class 15FG such that cd has order 16.
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).
Standard generators of the double covers 2.S₁₆ are not yet defined.

Black box algorithms

To find standard generators of A₁₆:

• Find an element of order 33, 39, 84 or 105 (probability about 1/9) and power it up to give an element a in class 3A.
• Find an element t of order 15 (probability about 1/15). It is probably in class 15F or 15G.
• Find a conjugate b of t such that ab has order 14 and abb has order 63 (probability about 1/6). If you cannot find such a conjugate, then t was probably in the wrong class, so go back to step 2.
• The elements a and b are standard generators of A₁₆.

To find standard generators of S₁₆:

• Find an element of order 26, 66 or 90 (probability about 1/15) 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 22 or 70 would also work, and increases the probability of finding an appropriate element to about 1/6.)
• Find an element t of order 15 (probability about 1/15). It is probably in class 15FG.
• Find a conjugate d of t such that cd has order 16 (probability 1/8). If you cannot find such a conjugate after several attempts, then t was probably in the wrong class, so go back to step 2.
• The elements c and d are standard generators of S₁₆.

Representations

• Some primitive permutation representations
  • Permutations on 16 points - the natural representation above: a and b (GAP).
  • Permutations on 120 points: a and b (GAP).
  • Permutations on 560 points: a and b (GAP).
  • Permutations on 1820 points: a and b (GAP).
• Some integer matrix representations
  • Dimension 15 (partition [2, 1¹⁴]): a and b (GAP).
  • Dimension 104 (partition [2², 1¹²]): a and b (GAP).
  • Dimension 105 (partition [3, 1¹³]): a and b (GAP).
  • Dimension 440 (partition [2³, 1¹⁰]): a and b (GAP).
  • Dimension 455 (partition [4, 1¹²]): a and b (GAP).

• Some primitive permutation representations
  • Permutations on 16 points - the natural representation above: c and d (GAP).
  • Permutations on 120 points: c and d (GAP).
  • Permutations on 560 points: c and d (GAP).
  • Permutations on 1820 points: c and d (GAP).
• Some integer matrix representations
  • Dimension 15 (partition [2, 1¹⁴]): c and d (GAP).
  • Dimension 104 (partition [2², 1¹²]): c and d (GAP).
  • Dimension 105 (partition [3, 1¹³]): c and d (GAP).
  • Dimension 440 (partition [2³, 1¹⁰]): c and d (GAP).
  • Dimension 455 (partition [4, 1¹²]): c and d (GAP).

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