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.


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


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: 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, ATLAS version) available are: The representations of 2.L2(7):2 (minus type, non-ATLAS version) available are:

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). The following are conjugacy class representatives of L2(7):2 = L3(2):2.
Main ATLAS page Go to main ATLAS (version 2.0) page.
Linear groups page Go to linear groups page.
Old L2(7) = L3(2) page Go to old L2(7) = L3(2) page - ATLAS version 1.
ftp access 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.