# ATLAS: Symplectic group S6(5)

Order = 228501000000000 = 29.34.59.7.13.31.
Mult = 2.
Out = 2.

The following information is available for S6(5):

### Standard generators

Standard generators of S6(5) are a and b where a is in class 2A, b is in class 8A, ab has order 9 and abababbabb has order 15.
Standard generators of the double cover 2.S6(5) = Sp6(5) are preimages A and B such that AB has order 9 and ABB has order 31. (NOTE: extra condition added 7/2/05).

Standard generators of S6(5):2 are not defined.
Standard generators of either of the double covers 2.S6(5):2 are not defined.

### Black box algorithms

To find standard generators of S6(5):
• Find an element x in class 2A, say by powering up an element of order 4, 8, 12, 20, 24, 26, 30, 40, 52, 60, 78, 120 or 130.
• Find an element t in class 8A, say by powering up an element of order 40 or 120.
• Find a conjugate y of t such that xy has order 9 and xyxyxyyxyy has order 15.
• The elements x and y are standard generators.

### Representations

The representations of S6(5) available are:
• Some primitive permutation representations.
• Permutations on 3906 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).

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Version 2.0 file created on 4th August 2004.
Last updated 7.2.05 by SJN.