Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C212 |
| direct product | C24×#120⋊C3 |
| Fitting, Sylow | C24×#120 ⋊ C3 |
| derived | C24 ⋊ C25×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | #120⋊C3 |
| commutator, cosocle, hypocenter | C24 | C25×C3 |
| Frattini | C22 | C25×A4 |
| Fitting | C24×#120 | C3 |
| socle | C26 | C2×A4 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C24 | |
96030 conjugacy classes of subgroups; only the 1189 normal subgroups are shown.
This group has 96030 conjugacy classes of subgroups — too many to draw. Only its 1189 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 — 5 generators on 16 points:
(7,8)(11,15,12)(13,16,14);(5,6)(10,16)(13,14);(9,10)(11,13)(12,14)(15,16);(3,4);(1,2)160 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72220591, C24×#1020⋊C2, C2×#72220575, … |
| maximal normal subgroup of | C2×#72220591, C24×#1020⋊C2, C2×#72220575, … |
| maximal quotient of | C22×D4×#120⋊C3, C22×#60399⋊C3, C2×#72220478, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((7,8)(11,15,12)(13,16,14), (5,6)(10,16)(13,14), (9,10)(11,13)(12,14)(15,16), (3,4), (1,2));