Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C23×C32 |
| direct product | C2×C32×A4⋊S4 |
| Fitting, socle | C25×C32 ⋊ C3⋊D3 |
| derived | A42 ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | A4⋊S4 |
| commutator, cosocle, hypocenter | A42 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C25×C32 | C3⋊D3 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C2×C32 | |
2886 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 2886 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 162 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,6)(4,5)(7,8,10)(14,15,16);(4,6)(8,9)(11,12,13);(4,5,6)(8,9,10)252 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)(3,6)(4,5)(7,8,10)(14,15,16), (4,6)(8,9)(11,12,13), (4,5,6)(8,9,10));