ATLAS: Linear group L2(7),
Linear group L3(2)
Order = 168 = 23.3.7.
Mult = 2.
Out = 2.
See also the ATLAS of Finite Groups, page 3.
The page for the group 23.L3(2) (nonsplit extension)
is available here
[but beware of incompatibilities with this page].
The following information is available for L2(7) = L3(2):
Standard generators of L2(7) = L3(2) are
a and b where
a has order 2, b has order 3
and ab has order 7.
Standard generators of the double cover 2.L2(7)
= SL2(7) = 2.L3(2) are preimages
A and B where
B has order 3 and AB has order 7.
Standard generators of L2(7):2 = PGL2(7) = L3(2):2 are
c and d where
c is in class 2B, d has order 3,
cd has order 8 and cdcdd has order 4. These conditions imply that cd is in class 8A.
Standard generators of either of the double covers 2.PGL2(7) are
preimages C and D where
D has order 3.
An outer automorphism, u say, of L2(7) = L3(2) of order 2 may be obtained by mapping (a, b) to (a, b-1).
The lift of u to an automorphism, U say, of SL2(7) = 2.L3(2) maps (A, B) to (A-1, B-1).
To obtain our standard generators for L2(7):2 = L3(2):2 we may take
c = u and d = bababb.
This forces a = [c, d]2 = (cddcd)2 = (ddcdc)2 and b = (dc)3(ddc)3 (and u = c).
Alternatively, we can take
c = ubabbab and d = b, in which case we force
a = ((cd)4)dcdc and b = d (and u = cdcdccdcd).
To find standard generators for L2(7) = L3(2):
- Find an element of order 2 or 4. This powers up to x in class 2A.
[The probability of success at each attempt is 3 in 8 (about 1 in 3).]
- Find an element y of order 3.
[The probability of success at each attempt is 1 in 3.]
- Find conjugates a of x and b of y such that ab has order 7.
[The probability of success at each attempt is 2 in 7 (about 1 in 4).]
To find standard generators for L2(7).2 = L3(2).2:
- Find an element of order 6. This cubes to x in class 2B.
[The probability of success at each attempt is 1 in 6 (or 1 in 3 if you look through outer elements only).]
- Find an element of order 3 or 6. This powers up to y of order 3.
[The probability of success at each attempt is 1 in 3.]
- Find conjugates a of x and b of y such that ab has order 8 and ababb has order 4.
[The probability of success at each attempt is 3 in 14 (about 1 in 5).]
Presentations for L2(7) = L3(2) and L2(7):2 = L3(2):2 in terms of their standard generators are given below.
< a, b | a2 = b3 = (ab)7 = [a, b]4 = 1 >.
< c, d | c2 = d3 = (cd)8 = [c, d]4 = 1 >.
These presentations, and those of the covering groups, are available in Magma format as follows:
L2(7) = L3(2) on a and b;
SL2(7) = 2.L3(2) on A and B;
PGL2(7) = L3(2):2 on c and d;
Representations are available for the following decorations of L2(7) = L3(2).
The representations of L2(7) = L3(2) available are:
- All faithful transitive permutation representations.
-
Permutations on 7a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- the action on points; primitive.
-
Permutations on 7b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- the action on lines; primitive.
-
Permutations on 8 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- primitive.
-
Permutations on 14a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- on the cosets of A4 fixing a point.
-
Permutations on 14b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- on the cosets of A4 fixing a line.
-
Permutations on 21 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 24 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 28 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 42a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- on the cosets of O2(point stab).
-
Permutations on 42b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- on the cosets of O2(line stab).
-
Permutations on 42c points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- on the cosets of C4.
-
Permutations on 56 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 84 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 168 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- regular.
- All faithful irreducibles in characteristic 2.
-
Dimension 3a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- the natural representation as L3(2).
-
Dimension 3b over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- the dual of the above.
-
Dimension 8 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- the Steinberg representation for L3(2).
- All faithful irreducibles in characteristic 3.
-
Dimension 3a over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 3b over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- the dual of the above.
-
Dimension 6b over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- reducible over GF(9).
-
Dimension 6a over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- the deleted (7 point) permutation representation.
-
Dimension 7 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- All faithful irreducibles in characteristic 7.
-
Dimension 3 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- the natural representation as O3(7).
-
Dimension 5 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 7 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- the Steinberg representation for L2(7).
- All faithful irreducibles in characteristic 0.
- a and
b as 3 × 3 matrices over Z[b7].
- a and
b as 3 × 3 matrices over Z[b7] - the dual of the above.
- a and
b as 6 × 6 matrices over Z - reducible over Q(b7).
- a and
b as 6 × 6 matrices over Z - the deleted permutation representation.
- a and
b as 7 × 7 monomial matrices over Z.
- a and
b as 8 × 8 matrices over Z.
The representations of SL2(7) = 2.L2(7) = 2.L3(2) available are:
- All faithful transitive permutation representations.
-
Permutations on 16 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- pseudoprimitive.
-
Permutations on 48 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Permutations on 112 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Permutations on 336 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- regular.
- All faithful irreducibles in characteristic 3.
-
Dimension 4a over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 4b over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- the dual of the above.
-
Dimension 6a over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 6b over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- the dual of the above.
-
Dimension 8 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- reducible over GF(9).
-
Dimension 12 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- reducible over GF(9).
- All faithful irreducibles in characteristic 7.
-
Dimension 2 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- the natural representation as SL2(7).
-
Dimension 4 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 6 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Faithful irreducibles in characteristic 0.
- A and
B as 4 × 4 matrices over Z[b7].
- A and
B as 4 × 4 matrices over Z[b7] - the dual of the above.
- A and
B as 8 × 8 monomial matrices over Z - reducible over Q(b7).
- A and
B as
5 × 5 matrices over GF(2).
- A and
B as
5 × 5 matrices over GF(2).
The representations of PGL2(7) = L2(7):2 = L3(2):2 available are:
- All faithful transitive permutation representations.
-
Permutations on 8 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- primitive.
-
Permutations on 14 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- pseudoprimitive.
-
Permutations on 16 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 21 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- primitive.
-
Permutations on 24 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 28 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- on the cosets of D12; primitive.
-
Permutations on 28b points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- on the cosets of A4.
-
Permutations on 42a points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- on the cosets of inner D8.
-
Permutations on 42b points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- on the cosets of outer D8.
-
Permutations on 42c points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- on the cosets of C8.
-
Permutations on 48 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 56a points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- on the cosets of inner D6.
-
Permutations on 56b points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- on the cosets of outer D6.
-
Permutations on 56c points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- on the cosets of C6.
-
Permutations on 84a points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- on the cosets of inner 2^2.
-
Permutations on 84b points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- on the cosets of outer 2^2.
-
Permutations on 84c points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- on the cosets of C4.
-
Permutations on 112 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 168a points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- on the cosets of inner C2.
-
Permutations on 168b points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- on the cosets of outer C2.
-
Permutations on 336 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- regular.
- Both faithful irreducibles in characteristic 2.
- c and
d as
6 × 6 matrices over GF(2).
- c and
d as
8 × 8 matrices over GF(2).
- Faithful irreducibles in characteristic 3.
- c and
d as
6 × 6 matrices over GF(3).
- c and
d as
6 × 6 matrices over GF(9).
- c and
d as
12 × 12 matrices over GF(3) - reducible over GF(9).
- c and
d as
7 × 7 matrices over GF(3).
- Faithful irreducibles in characteristic 0.
- c and
d as 6 × 6 matrices over Z.
- c and
d as 6 × 6 matrices over Z[r2].
- c and
d as 12 × 12 matrices over Z - reducible over Q(r2).
- c and
d as 7 × 7 matrices over Z.
- c and
d as 8 × 8 matrices over Z.
The representations of 2.L2(7).2 (plus type, ATLAS version) available are:
- All faithful transitive permutation representations.
-
Permutations on 32 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- pseudoprimitive.
-
Permutations on 96 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Permutations on 224 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Permutations on 672 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- regular.
The representations of 2.L2(7):2 (minus type, non-ATLAS version) available are:
- All faithful transitive permutation representations.
-
Permutations on 16a points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- pseudoprimitive.
-
Permutations on 16b points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- pseudoprimitive.
-
Permutations on 32 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Permutations on 48a points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Permutations on 48b points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Permutations on 96 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Permutations on 112a points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- on the cosets of D6; pseudoprimitive.
-
Permutations on 112b points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- on the cosets of C6; not pseudoprimitive.
-
Permutations on 224 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Permutations on 336 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Permutations on 672 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- regular.
The maximal subgroups of L2(7) = L3(2) are as follows.
-
S4, with [standard?] generators
abbaba, b; or
aba, b; or
a, babb.
- the point stabiliser.
-
S4, with [standard?] generators
ababba, b; or
abba, b; or
a, bab.
- the line stabiliser.
-
7:3 = F21, with generators
abababba, b; or
babba, b; or
baba, b.
NB: Word programs in same line give conjugate subgroups, not necessarily
identical subgroups.
The maximal subgroups of L2(7):2 = L3(2):2 are as follows.
The following are conjugacy class representatives of L2(7) = L3(2).
- 1A: identity.
- 2A: a.
- 3A: b; B is class +3A.
- 4A: ababb or [a, b]; ABABB is class +4A.
- 7A: ab; AB is class +7A.
- 7B: abb; ABB is class -7B.
The following are conjugacy class representatives of L2(7):2 = L3(2):2.
- 1A: identity.
- 2A: cdcdcdcd = (cd)^4.
- 3A: d.
- 4A: cdcd = (cd)^2 or [c, d].
- 7AB: cdcdcddcdd or [c, dcd].
- 2B: c.
- 6A: cdcdcdd.
- 8A: cd.
- 8B: cdcdcd = (cd)^3.
Go to main ATLAS (version 2.0) page.
Go to linear groups page.
Go to old L2(7) = L3(2) page - ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 14th September 2004, from a version 1 file last updated on 11th February 1998.
Last updated 16.09.04 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.