Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C32 |
| direct product | C3⋊D3×#6900 |
| Fitting, Sylow | C32×#6900 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C33×C3⋊D3 |
| commutator, socle, cosocle | C34 | C2×C33 |
| Fitting | C32×#6900 | C2 |
| prime radical | C33×C3⋊D3 | C32 |
| hypercenter | #6900 | C3⋊D3 |
| hypocenter | C32 | C2×#6900 |
| LCS | G ▷ C34 ▷ C32 | |
| UCS | C1 ◁ C32 ◁ #6900 | |
4000 conjugacy classes of subgroups; only the 315 normal subgroups are shown.
This group has 4000 conjugacy classes of subgroups — too many to draw. Only its 315 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(5,6)(16,22,23,21,19,20,18,17,24);(4,5,6)(7,13,14,12,10,11,9,8,15);(1,2,3)(8,10,13)(11,15,14)(17,19,22)(20,24,23)306 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3⋊D3×#92314, C3⋊D3×#92265, #72243705, … |
| maximal normal subgroup of | C3⋊D3×#92314, C3⋊D3×#92265, #72243705, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(5,6)(16,22,23,21,19,20,18,17,24), (4,5,6)(7,13,14,12,10,11,9,8,15), (1,2,3)(8,10,13)(11,15,14)(17,19,22)(20,24,23));