ATLAS: Symplectic group S4(4)
Order = 979200 = 28.32.52.17.
Mult = 1.
Out = 4.
See also ATLAS of Finite Groups, pp 44-45.
Standard generators of S4(4) are a and b where
a is in class 2A or 2B, b is in class 5E, ab has order 17
and ababb has order 15.
Standard generators of S4(4):2 are c and d where
c is in class 2D, d is in class 4C or 4D, cd has order 17
and cdd has order 4.
Standard generators of S4(4):4 are e and f where
e is in class 2AB, f is in class 4F or 4F', ef has
order 16, eff had order 6, efeff has order 8 and
efeffefff has order 6.
NB: The above conditions distinguish classes 4F and 4F'. The condition that eff had order 6 is redundant.
Also, (ab)f is conjugate to (ab)3 in the simple group S4(4).
An outer automorphism of S4(4) of order 2 maps (a, b) to
(a, (abb)^-3b(abb)^3).
An outer automorphism of S4(4):2 of order 2 maps (c, d) to
((cd)^-2c(cd)^2, (cdcdd)^-1(cdcdcddcd)^3cdcdd).
As usual, when we give the order of an outer automorphism of G, this order is its order in [the image] Out(G), NOT its order in Aut(G).
The representations of S4(4) available are:
-
Permutations on 85[a] points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 120[b] points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 4[c] over GF(4) - the natural representation:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 18 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 18 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 33[b] over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 18 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Some faithful irreducibles in characteristic 0
- Dimension 18 over Z:
a and b (GAP).
- Dimension 34[a] over Z:
a and b (GAP).
- Dimension 50 over Z:
a and b (GAP).
- Dimension 102 over Z (reducible over Z[z5]):
a and b (GAP).
- Dimension 85[a] over Z:
a and b (GAP).
- Dimension 153 over Z:
a and b (GAP).
- Dimension 204 over Z[b5]:
a and b (GAP).
The representations of S4(4):2 available are:
-
Dimension 8[a] over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of S4(4):4 available are:
-
Permutations on 170 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).
-
Dimension 16 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).
NB: Unlike in v1, the representations of S4(4):4 in v2.0 are
on standard generators.
The maximal subgroups of S4(4) are as follows. Words calculated by Ibrahim Suleiman. Some of these groups have been reordered since v1 in order
to satisfy the generality requirements.
At the moment, we are only going to give enough class representatives so that
we can sort out some of our generality problems.
Representatives of some of the 27 conjugacy classes of S4(4) are given below.
- 1A: identity [or a2].
- 2A: a.
- 2B: .
- 2C: .
- 3A: .
- 3B: (ababb)5.
- 5C: (ababb)3.
- 15D: abab2.
Representatives of some of the 30 conjugacy classes of S4(4):2 are given below.
- 1A: identity [or c2].
- 2A: d2.
- 2B: .
- 2C: .
- 3A: (cdcdd)4 or (cdcdcd3)5.
- 3B: .
- 5AB: (cdcdcd3)3.
- 6A: (cdcdd)2.
- 15AB: cdcdcd3.
- 2D: c.
- 4C: d.
- 4D: .
- 12A: cdcd2.
Representatives of some of the 30 conjugacy classes of S4(4):4 are given below.
- 1A: identity [or e2].
- 2AB: e.
- 2C: .
- 17ABCD: (ef)5ef3.
Go to main ATLAS (version 2.0) page.
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Go to old S4(4) page - ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 27th June 2000.
Last updated 02.07.04 by JNB.
Information checked to
Level 0 on 27.06.00 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.