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): To find standard generators for L2(7).2 = L3(2).2:

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: The representations of SL2(7) = 2.L2(7) = 2.L3(2) available are: 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.

Conjugacy classes

The following are conjugacy class representatives of L2(7) = L3(2). The following are conjugacy class representatives of L2(7):2 = L3(2):2.
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Last updated 11th February 1998,
R.A.Wilson, R.A.Parker and J.N.Bray