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 B has order 5 and AB has order 7.

Standard generators of S4(7):2 are c and d where c is in class 2C, d has order 5 and cd has order 12.
Standard generators of either group 2.S4(7):2 are preimages C and D where D has order 5.


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: The representations of 2.S4(7):2 [with o(C) = 4] available are: The representations of 2.S4(7):2 [with o(C) = 2] available are:

Maximal subgroups

The maximal subgroups of S4(7) include the following. The specifications refer to the orthogonal construction unless otherwise stated.
Main ATLAS page Go to main ATLAS (version 2.0) page.
Classical groups page Go to classical groups page.
Old S4(7) page Go to old S4(7) page - ATLAS version 1.
ftp access Anonymous ftp access is also available. See here for details.

Version 2.0 created on 13th June 2000.
Last updated 27.06.04 by SJN.
Information checked to Level 0 on 27.06.00 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.