# 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

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.

### Automorphisms

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).

### Black box algorithms

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

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

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.
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.

### Maximal subgroups

The maximal subgroups of L2(7) = L3(2) are as follows.
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.

### Conjugacy classes

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.