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.
Return to main ATLAS page.
Last updated 26th September 1998,
R.A.Wilson, R.A.Parker and J.N.Bray