ATLAS: Non-split extension 2^{3}.L_{3}(2)
Order = 1344 = 2^{6}.3.7.
Mult = 2.
Out = 2.
Note
This group is one of just three non-split extensions 2^{n}.L_{n}(2). (The other two are 2^{4}.L_{4}(2) and 2^{5}.L_{5}(2).) This group occurs as the `base' stabiliser in G_{2}(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 4^{3}:L_{3}(2)) and S_{6}(2) (as a subgroup of 2^{6}:L_{3}(2)).
Standard generators
Standard generators of 2^{3}.L_{3}(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 2.2^{3}.L_{3}(2) = 2^{3}.SL_{2}(7) are preimages
A and B where
B has order 3 and AB has order 7.
Standard generators of 2^{3}.L_{3}(2).2 = 2^{3}.(L_{3}(2) × 2) = 2^{3+1}.L_{3}(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 cdcd^{5}cd^{4}cd^{2} 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 2.2^{3+1}.L_{3}(2) are preimages
C and D where CDD has order 7.
Automorphisms
An outer automorphism of 2^{3}.L_{3}(2) of order 2 may be obtained by mapping (a, b) to (a^{bbabbabababbabb}, 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 2^{3}.L_{3}(2) and Aut(2^{3}.L_{3}(2)) on their standard generators are given below.
< a, b | a^{2} = b^{3} = (ab)^{7} = (ababab^{-1}abab^{-1}ab^{-1})^{3} = 1 >.
< c, d | c^{2} = d^{6} = (cdcd^{3})^{3} = cdcdcdcd^{-2}cd^{-2}cdcd^{-2} = (cdcd^{-1}cd^{-2}cd^{2})^{2} = 1 >.
Representations
The representations of 2^{3}.L_{3}(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 Z_{4} (the integers modulo 4).
Conjugacy classes
The following tables give some information about the conjugacy classes of 2^{3}.L_{3}(2) and 2^{3+1}.L_{3}(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 L_{3}(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 L_{3}(2) | 1A | 1A |
2A | 2A | 3A | 3A |
4A | 7A | 7B |
1A | 2A | 3A | 3A |
4A | 7A | 7B |
Image in C_{2} | 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 2^{3}.L_{3}(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 2^{3+1}.L_{3}(2) = Aut(2^{3}.L_{3}(2)).
- 1A: identity.
- 2A: cd^{3}cd^{3}.
- 2B: c.
- 4AB: cdcd^{2}cd^{3} or [c, d]^{2}.
- 3A: d^{2}.
- 6A: cdcdcd^{-1}cd^{-1} or [c, dcd].
- 8AB: cdcd^{-1} or [c, d].
- 7A: cdcd or cd^{-2}.
- 7B: cd^{2} or cd^{-1}cd^{-1}.
- 2C: d^{3}.
- 4C: cd^{3}.
- 6B: d.
- 6C: d^{-1}.
- 4D: cdcd^{2}.
- 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