Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C32 |
| direct product | C22×C3⋊D3×PSU(3,𝔽2) |
| Fitting, socle | C22×C34 ⋊ C2×Q8 |
| Sylow | C34 ⋊ C23×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊D3×PSU(3,𝔽2) |
| commutator, cosocle | C32×C3⋊D3 | C25 |
| Frattini | C1 | G |
| Fitting, socle | C22×C34 | C2×Q8 |
| prime radical | C22×C3⋊D32 | C22 |
| hypocenter | C34 | C23×Q8 |
| LCS | G ▷ C32×C3⋊D3 ▷ C34 | |
| UCS | C1 ◁ C22 | |
5854 conjugacy classes of subgroups; only the 856 normal subgroups are shown.
This group has 5854 conjugacy classes of subgroups — too many to draw. Only its 856 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 5 generators on 19 points:
(4,5,6)(7,15,8,9)(10,14,13,12);(1,2,3)(8,9,11,14)(10,15,12,13);(18,19);(2,3)(5,6);(16,17)144 conjugacy classes — view the full character table.
| maximal subgroup of | C22×C3⋊D3×PSU(3,𝔽2)⋊C3 |
| maximal normal subgroup of | C22×C3⋊D3×PSU(3,𝔽2)⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5,6)(7,15,8,9)(10,14,13,12), (1,2,3)(8,9,11,14)(10,15,12,13), (18,19), (2,3)(5,6), (16,17));