ATLAS: Symplectic group S4(7)

Order = 138297600 = 117602 = 28.32.52.74 = (24.3.5.72)2.
Mult = 2.
Out = 2.
FACT: This is the smallest simple group whose order is a proper power.

Standard generators

Standard generators of S4(7) are a and b where a is in class 2A, b has order 5 and ab has order 7.
Standard generators of the double cover 2.S4(7) = Sp4(7) are preimages A and B where A has order 5 and AB has order 7.

Standard generators of S4(7):2 are c and d where c is in class ??, d has order ? and cd has order ?? ... etc.


Presentations

S4(7): 2-generator, 6-relator, length 91.

< a, b | a2 = b5 = (ab)7 = [a, b2]4 = (ababab2abab2)2 = [a, babab-2abab] = 1 >

Remark: Adding in the redundant relation [a, babab-1]2 = 1 of length 24 (giving a 2-generator, 7-relator, length 115 presentation) eases coset enumeration.


Representations

The representations of S4(7) available are: The representations of 2.S4(7) = Sp4(7) available are: The representations of S4(7):2 available are:

Maximal subgroups

The maximal subgroups of S4(7) include the following. The specifications refer to the orthogonal construction unless otherwise stated.
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Last updated 27th August 1999,
R.A.Wilson, R.A.Parker and J.N.Bray