Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C22×C4×#120⋊C3 |
| Fitting, Sylow | C22×C4×#120 ⋊ C3 |
| derived | C24 ⋊ C23×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | #120⋊C3 |
| commutator, hypocenter | C24 | C23×C3×C4 |
| Frattini | C23 | C24×A4 |
| Fitting | C22×C4×#120 | C3 |
| socle | C25 | C22×A4 |
| prime radical | C23×#120⋊C3 | C2 |
| cosocle | C25 | C24×C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C22×C4 | |
14382 conjugacy classes of subgroups; only the 381 normal subgroups are shown.
This group has 14382 conjugacy classes of subgroups — too many to draw. Only its 381 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 — 4 generators on 16 points:
(1,2)(3,4)(5,6,7,8)(11,12,16)(13,15,14);(3,4)(10,15)(13,14);(1,2)(3,4)(9,10)(11,13)(12,15)(14,16);(1,2)160 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4)(5,6,7,8)(11,12,16)(13,15,14), (3,4)(10,15)(13,14), (1,2)(3,4)(9,10)(11,13)(12,15)(14,16), (1,2));