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 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.
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.
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:
-
Dimension 4 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) = 2] available are:
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.