Finite Group DatabasebetaFinite Group Databasebeta| order | 43008 |
| isoclinism | isoclinic class of #72227823 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #72209659 |
| commutator, hypocenter | G | C1 |
| Frattini, Fitting, cosocle | C26×C4 | GL(3,𝔽2) |
| socle | C2 | #72209871 |
| LCS | G | |
| UCS | C1 ◁ C4 | |
1958 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 1958 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,6,5,7,12,13)(3,8,10,14,11,15,19,20)(9,16,17,22,18,23,24,29)(21,25,27,30,28,31,32,26);(1,3,9)(4,10,17)(5,11,18)(12,19,24)(16,21,26)(22,27,25)(23,28,30)(29,32,31)56 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,6,5,7,12,13)(3,8,10,14,11,15,19,20)(9,16,17,22,18,23,24,29)(21,25,27,30,28,31,32,26), (1,3,9)(4,10,17)(5,11,18)(12,19,24)(16,21,26)(22,27,25)(23,28,30)(29,32,31));