ATLAS: Unitary group U₄(4)

The following information is available for U₄(4):

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

Standard generators

Black box algorithms

Finding generators

To find standard generators of U₄(4):

• Find an element of order 20 or 30 and power it up to give an element x in class 2A.
• Find an element t of order 4 (not by powering up). Hopefully t is in class 4B.
• Look for a conjugate y of t such that xy has order 20. If all the orders of xy seem to be 15 or less, t is probably in the wrong class, so go back to step 2.
• The elements x and y are standard generators.

Checking generators

To check that elements x and y of U₄(4) are standard generators:

• Check o(x) = 2
• Check o(y) = 4
• Check o(xy) = 20
• Check o(x(xy)¹⁰) = 3

Representations

• Some primitive permutation representations.
  • Permutations on 325 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Permutations on 1040 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Permutations on 1105 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Permutations on 3264 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).

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