# ATLAS: Linear group L2(7), Linear group L3(2)

Order = 168 = 23.3.7.
Mult = 2.
Out = 2.

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 of L2(7) = L3(2) of order 2 may be obtained by mapping (a, b) to (a, b-1).

To obtain our standard generators for L2(7):2 = L3(2):2 we may take c to be the above automorphism and d = bababb.
This forces a = [c, d]2 = (cddcd)2 = (ddcdc)2 and b = (dc)3(ddc)3.

### 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 >.

### Representations

Representations are available for the following decorations of L2(7) = L3(2).
The representations of L2(7) = L3(2) available are:
• All primitive permutation representations.
• a and b as permutations on 7 points - the action on points.
• a and b as permutations on 7 points - the action on lines.
• a and b as permutations on 8 points.
• All faithful irreducibles in characteristic 2.
• a and b as 3 × 3 matrices over GF(2) - the natural representation as L3(2).
• a and b as 3 × 3 matrices over GF(2) - the dual of the above.
• a and b as 8 × 8 matrices over GF(2) - the Steinberg representation for the group considered as L3(2).
• All faithful irreducibles in characteristic 3 and over GF(3).
• a and b as 3 × 3 matrices over GF(9).
• a and b as 3 × 3 matrices over GF(9) - the dual of the above.
• a and b as 6 × 6 matrices over GF(3) - reducible over GF(9).
• a and b as 6 × 6 matrices over GF(3) - the deleted permutation representation.
• a and b as 7 × 7 matrices over GF(3).
• All faithful irreducibles in characteristic 7.
• a and b as 3 × 3 matrices over GF(7) - the natural representation as O3(7).
• a and b as 5 × 5 matrices over GF(7).
• a and b as 7 × 7 matrices over GF(7) - the Steinberg representation for the group considered as 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:
• A and B as permutations on 16 points.
• Faithful irreducibles in characteristic 3 and over GF(3).
• A and B as 4 × 4 matrices over GF(9).
• A and B as 4 × 4 matrices over GF(9) - the dual of the above.
• A and B as 8 × 8 matrices over GF(3) - reducible over GF(9).
• Faithful irreducibles in characteristic 7.
• A and B as 2 × 2 matrices over GF(7) - the natural representation as SL2(7).
• A and B as 4 × 4 matrices over GF(7).
• 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:
The representations of 2.L2(7):2 (plus type, non-ATLAS version) available are:
The representations of 2.L2(7).2 (minus type, ATLAS version) available are:

### Maximal subgroups

The maximal subgroups of L2(7) = L3(2) are as follows.
The maximal subgroups of L2(7):2 = L3(2):2 are as follows.
• L2(7).
• 7:6.
• D16.
• D12.

### Conjugacy classes

The following are conjugacy class representatives of L2(7) = L3(2).
• 1A: identity.
• 2A: a.
• 3A: b.
• 4A: ababb or [a, b].
• 7A: ab.
• 7B: abb.
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.