Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C217 |
| direct product | C24×#1020⋊C2 |
| Fitting | C24×#120 ⋊ D3 |
| derived | C22⋊A4 ⋊ C26 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | #1020⋊C2 |
| commutator, cosocle, hypocenter | C22⋊A4 | C26 |
| Frattini | C22 | C25×S4 |
| Fitting | C24×#120 | D3 |
| socle | C26 | C2×S4 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C24 | |
349358 conjugacy classes of subgroups; only the 3640 normal subgroups are shown.
This group has 349358 conjugacy classes of subgroups — too many to draw. Only its 3640 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 6 generators on 16 points:
(7,8)(10,16,14)(11,12,15);(5,6);(1,2)(9,10)(11,13)(12,14)(15,16);(11,15)(13,16);(3,4);(9,10)(11,13)(12,14)(15,16)224 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72222918, C22×D4×#1020⋊C2 |
| maximal normal subgroup of | C2×#72222918, C22×D4×#1020⋊C2 |
| maximal quotient of | C22×D4×#1020⋊C2, C22×#72217027, C2×#72220683, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((7,8)(10,16,14)(11,12,15), (5,6), (1,2)(9,10)(11,13)(12,14)(15,16), (11,15)(13,16), (3,4), (9,10)(11,13)(12,14)(15,16));