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

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


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.


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.


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 >.


The representations of 23.L3(2) available are:

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.

Class1A2A 2B4A4B3A6A 8A8B7AB**
|Centraliser|1344192 16323266 8877
Image in L3(2)1A1A 2A2A2A3A3A 4A4A7A7B

Class1A2A 2B4AB3A6A 8AB7AB** 2C4C6BC** 4D14AB**
|Centraliser|2688384 32321212 81414 336161212 81414
Image in L3(2)1A1A 2A2A3A3A 4A7A7B 1A2A3A3A 4A7A7B
Image in C211 1111 111 -1-1-1-1 -1-1-1

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

The following are representatives of the conjugacy classes of 23+1.L3(2) = Aut(23.L3(2)).
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Last updated 26th September 1998,
R.A.Wilson, R.A.Parker and J.N.Bray