Finite Group DatabasebetaFinite Group Databasebeta| order | 40320 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C4 |
| projective special semilinear | PΣL(3,𝔽4) |
| socle, derived | PSL(3,𝔽4) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | PSL(3,𝔽4) | C2 |
| LCS | G ▷ PSL(3,𝔽4) | |
| UCS | C1 | |
152 conjugacy classes of subgroups; only the 3 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 3 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 21 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,3,20,14,16,6,5)(2,17,11,4,13,15,9)(7,10,18,19,21,8,12);(1,4,11,14,3,12,17,5)(2,18,19,15,7,10,9,8)(13,21,20,16)14 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((1,3,20,14,16,6,5)(2,17,11,4,13,15,9)(7,10,18,19,21,8,12), (1,4,11,14,3,12,17,5)(2,18,19,15,7,10,9,8)(13,21,20,16));