# ATLAS: Non-split extension 23.L3(2)

Order = 1344 = 26.3.7.
Mult = 2.
Out = 2.

### Note

This group is one of just three non-split extensions 2n.Ln(2). (The other two are 24.L4(2) and 25.L5(2).) This group occurs as the `base' stabiliser in G2(q) for q odd, and is maximal if q is prime. The group also occurs in many other groups including HS (as a subgroup of 43:L3(2)) and S6(2) (as a subgroup of 26:L3(2)).

### Standard generators

Standard generators of 23.L3(2) are a and b where a is in class 2B, b has order 3, ab has order 7 and abababbababbabb has order 3. The last condition distinguishes classes 7A and 7B.
Standard generators of the double cover 2.23.L3(2) = 23.SL2(7) are preimages A and B where B has order 3 and AB has order 7.

Standard generators of 23.L3(2).2 = 23.(L3(2) × 2) = 23+1.L3(2) are c and d where c is in class 2B, b is in class 6B/C, cd has order 14, cdcddd has order 3 and cdcd5cd4cd2 has order 2. These conditions are sufficient to distinguish classes 6B from 6C and 14A from 14B.
Standard generators of either of the double covers 2.23+1.L3(2) are preimages C and D where CDD has order 7.

### Automorphisms

An outer automorphism of 23.L3(2) of order 2 may be obtained by mapping (a, b) to (abbabbabababbabb, b).
We may take c = a and d = ub, where u is the above automorphism. This implies that a = c and b = d-2.

### Presentations

The presentations of 23.L3(2) and Aut(23.L3(2)) on their standard generators are given below.

< a, b | a2 = b3 = (ab)7 = (ababab-1abab-1ab-1)3 = 1 >.

< c, d | c2 = d6 = (cdcd3)3 = cdcdcdcd-2cd-2cdcd-2 = (cdcd-1cd-2cd2)2 = 1 >.

### Representations

The representations of 23.L3(2) available are:
• a and b as permutations on 14 points.
• a and b as permutations on 14 points - the image of the above under an outer automorphism.
• a and b as 6 × 6 matrices over GF(2) - showing the inclusion in S6(2).
• All irreducible representations in characteristic 0.
• a and b as 7 × 7 monomial matrices over Z.
• a and b as 7 × 7 monomial matrices over Z.
• a and b as 14 × 14 matrices over Z.
• a and b as 21 × 21 monomial matrices over Z.
• a and b as 21 × 21 monomial matrices over Z.
• a and b as 4 × 4 matrices over Z4 (the integers modulo 4).

### Conjugacy classes

The following tables give some information about the conjugacy classes of 23.L3(2) and 23+1.L3(2) respectively. Please note that classes 8A, 8B, 14A and 14B square into classes 4A, 4B, 7A and 7B respectively. All other power maps are easily deduced.

 Class 1A 2A 2B 4A 4B 3A 6A 8A 8B 7A B** |Centraliser| 1344 192 16 32 32 6 6 8 8 7 7 Image in L3(2) 1A 1A 2A 2A 2A 3A 3A 4A 4A 7A 7B

 Class 1A 2A 2B 4AB 3A 6A 8AB 7A B** 2C 4C 6B C** 4D 14A B** |Centraliser| 2688 384 32 32 12 12 8 14 14 336 16 12 12 8 14 14 Image in L3(2) 1A 1A 2A 2A 3A 3A 4A 7A 7B 1A 2A 3A 3A 4A 7A 7B Image in C2 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1

The following are representatives of the conjugacy classes of 23.L3(2).

• 1A: identity.
• 2A: (ababb)^4 or [a, b]^4.
• 2B: a.
• 4A: ababbababb or [a, b]^2.
• 4B: (abababbababb)^2 or abababbabababbababbabb.
• 3A: b.
• 6A: abababbabb or [a, bab].
• 8A: ababb or [a, b].
• 8B: abababbababb.
• 7A: ab.
• 7B: abb.
The following are representatives of the conjugacy classes of 23+1.L3(2) = Aut(23.L3(2)).
• 1A: identity.
• 2A: cd3cd3.
• 2B: c.
• 4AB: cdcd2cd3 or [c, d]2.
• 3A: d2.
• 6A: cdcdcd-1cd-1 or [c, dcd].
• 8AB: cdcd-1 or [c, d].
• 7A: cdcd or cd-2.
• 7B: cd2 or cd-1cd-1.
• 2C: d3.
• 4C: cd3.
• 6B: d.
• 6C: d-1.
• 4D: cdcd2.
• 14A: cd.
• 14B: cd-1.