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.