Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
| direct product | C23×#73026 |
| Fitting | C23×B(4,𝔽2) ⋊ D3 |
| derived | #997 ⋊ C25 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C24 | #1020⋊C2 |
| commutator, cosocle, hypocenter | #997 | C25 |
| Frattini | C23 | C24×S4 |
| Fitting | C23×B(4,𝔽2) | D3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C24 | |
68902 conjugacy classes of subgroups; only the 575 normal subgroups are shown.
This group has 68902 conjugacy classes of subgroups — too many to draw. Only its 575 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 — 5 generators on 14 points:
(7,13,12,9,8,14,10,11);(5,6)(7,9,13)(8,11,14);(3,4)(7,8)(9,11)(13,14);(7,8)(9,11)(13,14);(1,2)160 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#72217027, C2×D4×#73026, C2×#72246533 |
| maximal normal subgroup of | C22×#72217027, C2×D4×#73026, C2×#72246533 |
| maximal quotient of | C2×D4×#73026 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((7,13,12,9,8,14,10,11), (5,6)(7,9,13)(8,11,14), (3,4)(7,8)(9,11)(13,14), (7,8)(9,11)(13,14), (1,2));