Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C22×C4×#857 |
| Fitting | C22×C43 ⋊ D3 |
| derived | C42⋊C3 ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | #857 |
| commutator, hypocenter | C42⋊C3 | C23×C4 |
| Frattini | C23 | C23×S4 |
| Fitting | C22×C43 | D3 |
| socle | C25 | C2×S4 |
| prime radical | C23×#857 | C2 |
| cosocle | C2×C42⋊C3 | C24 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C22×C4 | |
5482 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 5482 conjugacy classes of subgroups — too many to draw. Only its 199 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) = 48 |
A minimal faithful permutation representation — 4 generators on 16 points:
(5,6,7,9)(8,15,13,14,11,12,10,16);(3,4)(5,8,12)(6,10,14)(7,11,15)(9,13,16);(1,2)(5,7)(6,9)(12,15)(14,16);(5,6,7,9)(8,10,11,13)(12,14,15,16)160 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72240898, C22×#11713628, C2×C4×#11715754 |
| maximal normal subgroup of | C22×#11713628 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6,7,9)(8,15,13,14,11,12,10,16), (3,4)(5,8,12)(6,10,14)(7,11,15)(9,13,16), (1,2)(5,7)(6,9)(12,15)(14,16), (5,6,7,9)(8,10,11,13)(12,14,15,16));