# 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:
• Permutations on 400[a] points - action on points (in the natural representation as Sp4(7)): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 400[b] points - action on isotropic lines of the symplectic space: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 1176 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 1225 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 5 over GF(7) - the natural representation as O5(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some faithful irreducibles in characteristic 0
• Dimension 25 over Z(b7): a and b (GAP).
• Dimension 126 over Z: a and b (GAP).
• Dimension 175(a) over Z: a and b (GAP).
• Dimension 175(b) over Z: a and b (GAP).
• Dimension 224 over Z: a and b (GAP).
The representations of 2.S4(7) = Sp4(7) available are:
• Dimension 4 over GF(7) - the natural representation as Sp4(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
The representations of S4(7):2 available are:
• Permutations on 400[a] points - action on cosets of N(7^{1+2}): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 400[b] points - action on cosets of N(7^3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 5 over GF(7): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
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:
• 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 = C(2C).
Order: 117600.
Index: 1176.

• 2.(L2(7) × L2(7)):2, the plus-point stabiliser = N(2A).
Order: 112896.
Index: 1225.

• (D8 × L2(7)):2, the minus-line stabiliser = N(2B).
Order: 2688.
Index: 51450.

• A7.
Order: 2520.
Index: 54880.

• S3 × L2(7):2, the plus-line stabiliser = C(2D).
Order: 2016.
Index: 68600.

• 24:S5, the base stabiliser.
Order: 1920.
Index: 72030.

• 24:S5, the base stabiliser.
Order: 1920.
Index: 72030. Go to main ATLAS (version 2.0) page. Go to classical groups page. Go to old S4(7) page - ATLAS version 1. 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.