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:

a and b as permutations on 400 points - action on points (in the natural representation as Sp4(7)).
a and b as permutations on 400 points - action on isotropic lines of the symplectic space.
a and b as permutations on 1176 points.
a and b as permutations on 1225 points.
a and b as 5 × 5 matrices over GF(7) - the natural representation as O5(7).

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

A and B as 4 × 4 matrices over GF(7) - the natural representation as Sp4(7).

The representations of S₄(7):2 available are:

none.

Maximal subgroups

The maximal subgroups of S₄(7) include the following. The specifications refer to the orthogonal construction unless otherwise stated.

7¹⁺²:(3 × 2.L₂(7)), the point stabiliser (symplectic); the isotropic line stabiliser (orthogonal).
Order: 345744.
Index: 400.
7³:(3 × L₂(7):2), the isotropic line stabiliser (symplectic); the point stabiliser (orthogonal).
Order: 345744.
Index: 400.
L₂(49):2₂, the minus-point stabiliser.
Order: 117600.
Index: 1176.
2.(L₂(7) × L₂(7)):2, the plus-point stabiliser.
Order: 112896.
Index: 1225.
(D₈ × L₂(7)):2, the minus-line stabiliser.
Order: 2688.
Index: 51450.
A₇.
Order: 2520.
Index: 54880.
S₃ × L₂(7):2, the plus-line stabiliser.
Order: 2016.
Index: 68600.
2⁴:S₅, the base stabiliser.
Order: 1920.
Index: 72030.
2⁴:S₅, the base stabiliser.
Order: 1920.
Index: 72030.

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Last updated 27th August 1999,
R.A.Wilson, R.A.Parker and J.N.Bray