ATLAS: Linear group L_{2}(7),
Linear group L_{3}(2)
Order = 168 = 2^{3}.3.7.
Mult = 2.
Out = 2.
See also the ATLAS of Finite Groups, page 3.
The page for the group 2^{3}.L_{3}(2) (nonsplit extension)
is available here
[but beware of incompatibilities with this page].
The following information is available for L_{2}(7) = L_{3}(2):
Standard generators of L_{2}(7) = L_{3}(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.L_{2}(7)
= SL_{2}(7) = 2.L_{3}(2) are preimages
A and B where
B has order 3 and AB has order 7.
Standard generators of L_{2}(7):2 = PGL_{2}(7) = L_{3}(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.PGL_{2}(7) are
preimages C and D where
D has order 3.
An outer automorphism, u say, of L_{2}(7) = L_{3}(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 SL_{2}(7) = 2.L_{3}(2) maps (A, B) to (A^{1}, B^{1}).
To obtain our standard generators for L_{2}(7):2 = L_{3}(2):2 we may take
c = u and d = b^{ababb}.
This forces a = [c, d]^{2} = (cddcd)^{2} = (ddcdc)^{2} and b = (dc)^{3}(ddc)^{3} (and u = c).
Alternatively, we can take
c = u^{babbab} and d = b, in which case we force
a = ((cd)^{4})^{dcdc} and b = d (and u = c^{dcd}cc^{dcd}).
To find standard generators for L_{2}(7) = L_{3}(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 L_{2}(7).2 = L_{3}(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 L_{2}(7) = L_{3}(2) and L_{2}(7):2 = L_{3}(2):2 in terms of their standard generators are given below.
< a, b  a^{2} = b^{3} = (ab)^{7} = [a, b]^{4} = 1 >.
< c, d  c^{2} = d^{3} = (cd)^{8} = [c, d]^{4} = 1 >.
These presentations, and those of the covering groups, are available in Magma format as follows:
L_{2}(7) = L_{3}(2) on a and b;
SL_{2}(7) = 2.L_{3}(2) on A and B;
PGL_{2}(7) = L_{3}(2):2 on c and d;
Representations are available for the following decorations of L_{2}(7) = L_{3}(2).
The representations of L_{2}(7) = L_{3}(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 SL_{2}(7) = 2.L_{2}(7) = 2.L_{3}(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 PGL_{2}(7) = L_{2}(7):2 = L_{3}(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.L_{2}(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.L_{2}(7):2 (minus type, nonATLAS 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 L_{2}(7) = L_{3}(2) are as follows.

S_{4}, with [standard?] generators
abbaba, b; or
a^{ba}, b; or
a, b^{abb}.
 the point stabiliser.

S_{4}, with [standard?] generators
ababba, b; or
a^{bba}, b; or
a, b^{ab}.
 the line stabiliser.

7:3 = F_{21}, with generators
abababba, b; or
b^{abba}, b; or
b^{aba}, b.
NB: Word programs in same line give conjugate subgroups, not necessarily
identical subgroups.
The maximal subgroups of L_{2}(7):2 = L_{3}(2):2 are as follows.
The following are conjugacy class representatives of L_{2}(7) = L_{3}(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 L_{2}(7):2 = L_{3}(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.