ATLAS: Symplectic group S_{4}(7)
Order = 138297600 = 11760^{2} = 2^{8}.3^{2}.5^{2}.7^{4} = (2^{4}.3.5.7^{2})^{2}.
Mult = 2.
Out = 2.
FACT: This is the smallest simple group whose order is a proper power.
Standard generators of S_{4}(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.S_{4}(7) = Sp_{4}(7)
are preimages A and B where B has order 5 and AB
has order 7.
Standard generators of S_{4}(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.S_{4}(7):2 are preimages
C and D where D has order 5.
S_{4}(7): 2generator, 6relator, length 91.
< a, b  a^{2} = b^{5} = (ab)^{7} = [a, b^{2}]^{4} = (ababab^{2}abab^{2})^{2} = [a, babab^{2}abab] = 1 >
Remark: Adding in the redundant relation [a, babab^{1}]^{2} = 1 of length 24 (giving a 2generator, 7relator, length 115 presentation) eases coset enumeration.
The representations of S_{4}(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.S_{4}(7) = Sp_{4}(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 S_{4}(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.S_{4}(7):2 [with o(C) = 4] available are:

Dimension 4 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 2.S_{4}(7):2 [with o(C) = 2] available are:
The maximal subgroups of S_{4}(7) include the following. The specifications refer to the orthogonal construction unless otherwise stated.

7^{1+2}:(3 × 2.L_{2}(7)), the point stabiliser (symplectic); the isotropic line stabiliser (orthogonal).
Order: 345744.
Index: 400.

7^{3}:(3 × L_{2}(7):2), the isotropic line stabiliser (symplectic); the point stabiliser (orthogonal).
Order: 345744.
Index: 400.

L_{2}(49):2_{2}, the minuspoint stabiliser = C(2C).
Order: 117600.
Index: 1176.
 2.(L_{2}(7) × L_{2}(7)):2, the pluspoint stabiliser = N(2A).
Order: 112896.
Index: 1225.
 (D_{8} × L_{2}(7)):2, the minusline stabiliser = N(2B).
Order: 2688.
Index: 51450.
 A_{7}.
Order: 2520.
Index: 54880.
 S_{3} × L_{2}(7):2, the plusline stabiliser = C(2D).
Order: 2016.
Index: 68600.
 2^{4}:S_{5}, the base stabiliser.
Order: 1920.
Index: 72030.
 2^{4}:S_{5}, 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.