Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
| direct product | C23×#5934 |
| Fitting | C23×H(5,𝔽2) ⋊ D3 |
| derived | #997 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C24 | #1020 |
| commutator, cosocle, hypocenter | #997 | C24 |
| Frattini | C2 | C23×#1020 |
| Fitting | C23×H(5,𝔽2) | D3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C24 | |
23800 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 23800 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 14 points:
(7,12,11,9)(8,14,13,10);(5,6)(8,11,9)(12,14,13);(3,4);(1,2)104 conjugacy classes — view the full character table.
| maximal subgroup of | #72241375, C2×D4×#5934, C22×D3×#5934, … |
| maximal normal subgroup of | #72241375, C2×D4×#5934, C22×#11715839, … |
| maximal quotient of | C22×#72209653, C2×D4×#5934, C22×D3×#5934, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((7,12,11,9)(8,14,13,10), (5,6)(8,11,9)(12,14,13), (3,4), (1,2));