Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×S42 |
| Fitting, socle | C24×C32 ⋊ D32 |
| derived | A42 ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | S42 |
| commutator, cosocle, hypocenter | A42 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C24×C32 | D32 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C32 | |
2116 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 2116 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,4,2,3)(5,7,8,6)(12,13,14);(2,3,4)(7,8)(9,10,11)225 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×S42, C3×D3×S42 |
| maximal normal subgroup of | C3⋊D3×S42, C3×D3×S42 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,2,3)(5,7,8,6)(12,13,14), (2,3,4)(7,8)(9,10,11));