ATLAS: Linear group L2(11)

Order = 660 = 22.3.5.11.
Mult = 2.
Out = 2.

The following information is available for L2(11):


Standard generators

Standard generators of L2(11) are a and b where a has order 2, b has order 3 and ab has order 11.
Standard generators of the double cover 2.L2(11) = SL2(11) are preimages A and B where B has order 3 and AB has order 11.

Standard generators of L2(11):2 = PGL2(11) are c and d where c is in class 2B, d has order 3, cd has order 10 and cdcdd has order 11. These conditions imply that cd is in class 10A.
Standard generators of either double cover 2.L2(11).2 = 2.PGL2(11) are preimages C and D where D has order 3.


Presentations

Presentations for L2(11) and L2(11):2 = PGL2(11) in terms of their standard generators are given below.

< a, b | a2 = b3 = (ab)11 = [a, babab]2 = 1 >.

< c, d | c2 = d3 = (cd)10 = (cdcdcd−1)4 = [c, dcdcd−1cd−1cd−1cdcd] = 1 >.

We have < c, d | c2 = d3 = (cd)10 = (cdcdcd−1)4 = 1 > = (3 × L2(11)):2 (with trivial centre). To quotient out the normal 3, we can add a relation such as [c, d]11 = 1, though the shortest such relation (when written in terms of c, d, c−1, d−1) is [c, dcdcd−1cd−1cd−1cdcd] = 1, which we have used above.


Representations

The representations L2(11) available are: The representations of 2.L2(11) = SL2(11) available are: The representations of L2(11):2 = PGL2(11) available are:

Maximal subgroups

The maximal subgroups of L2(11) are as follows. The maximal subgroups of L2(11):2 = PGL2(11) are as follows.

Conjugacy classes

A set of generators for the maximal cyclic subgroups of L2(11) can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

The 8 conjugacy classes of L2(11) are as follows.

A set of generators for the maximal cyclic subgroups of L2(11):2 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

The 13 conjugacy classes of L2(11):2 = PGL2(11) are as follows.


Main ATLAS page Go to main ATLAS (version 2.0) page.
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Old L2(11) page Go to old L2(11) page - ATLAS version 1.
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Version 2.0 file created on 21st January 2002, from Version 1 file last updated on 29.03.99.
Last updated 26.01.12 by JNB.
Information checked to Level 0 on 21.01.02 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.