An ATLAS of Group Presentations

Well, my research is in group theory which is a part of algebra. In particular, I study symmetric presentations. If you don't know what a group is then click here for a definition.

If the reader seeks a representation of a group, then he should refer to the ATLAS of group representations which is maintained by R.A.Wilson, S.J.Nickerson and me.

NB: This page was originally meant to focus on symmetric presentations, but the current page title seems to reflect the content more accurately. Also, these pages are somewhat neglected at present. Hopefully, this will not remain the case forever.

We shall give some presentations of groups (symmetric or otherwise). Firstly, choose a group. We have divided them into a number of classes to make location easier. Those groups that have just one (finite) non-abelian composition factor up to isomorphism are listed in the first seven sections according to the isomorphism type of this composition factor. Soluble groups and groups with at least two non-isomorphic non-abelian composition factors are listed under `other groups'.

Some of the presentations we list are standard Coxeter presentations. Click here for a list of these. Other sources of presentations include:

and, of course, me (unlabelled or labelled JNB). The `standard' Coxeter presentations and the odd relatives have been left unlabelled.

Each of the pages linked to below have a couple of sensitve images. Those like this:
return you to my homepage and those like this:
return you to this page.

Alternating groups

A5 A6 A7 A8 A9 A10 A11 A12 A13

Sporadic groups

M11 M12 M22 M23 M24 J1 J2 J3 J4
HS McL Suz Co3 Co2 Co1 Fi22 Fi23 Fi24'
He HN Th B = BM M = F1 Ru O'N Ly T = 2F4(2)'

Linear groups

L2(4) = L2(5) = A5 L3(2) = L2(7) L2(9) = A6 = S4(2)' = M10' L4(2) = A8
L2(8) L2(11) L2(13) L2(16) L2(17) L2(19) L2(23) L2(25) L2(27) L2(29) L2(31) L2(32)
L3(3) L3(4) = M21 L3(5) L3(7) L3(8) L3(9) L3(11) L4(3) L5(2)

Unitary groups

U3(3) U3(4) U3(5) U3(7) U3(8) U3(9) U3(11)
U4(2) U4(3) U5(2) U6(2) = Fi21

Symplectic groups

S4(2) = S6 = A6:21 S4(3) = U4(2) S4(4) S4(5) S6(2) S6(3) S8(2)

Orthogonal groups

O8+(2) O8-(2) O10+(2) O10-(2) O7(3) O8+(3) O8-(3)

Exceptional groups of Lie type

G2(2) = U3(3):2 G2(3) G2(4) G2(5) F4(2) E6(2) E7(2) E8(2) 3D4(2) 2E6(2)
Sz(8) = 2B2(8) Sz(32) = 2B2(32) T.2 = 2F4(2) R(3) = 2G2(3) = L2(8):3. R(27) = 2G2(27)

Other groups

Soluble groups (A4, S4, M9 etc.)
A5 × L3(2)

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Last updated 28th August 1997 (or 19th July 2007)
John N. Bray