# ATLAS: Symplectic group S4(4)

Order = 979200 = 28.32.52.17.
Mult = 1.
Out = 4.

### Standard generators

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).

### Automorphisms

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).

### Representations

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:
The representations of S4(4):4 available are:
NB: Unlike in v1, the representations of S4(4):4 in v2.0 are on standard generators.

### Maximal subgroups

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.

### Conjugacy classes

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. Go to classical groups page. 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.