Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C34 |
| direct product | C3×C3⋊#92314 |
| Fitting, Sylow | C32×#6900 ⋊ C2 |
| derived | C3×C92 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊#92314 |
| commutator, cosocle, hypocenter | C3×C92 | C2×C32 |
| Frattini | C32 | C32×C3⋊(C3⋊D3) |
| Fitting | C32×#6900 | C2 |
| socle | C34 | C3×C3⋊D3 |
| LCS | G ▷ C3×C92 | |
| UCS | C1 ◁ C3 | |
4504 conjugacy classes of subgroups; only the 370 normal subgroups are shown.
This group has 4504 conjugacy classes of subgroups — too many to draw. Only its 370 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) = 486 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3)(5,6)(8,10)(9,14)(11,12)(13,15)(17,19)(18,23)(20,21)(22,24);(4,5,6)(8,11,13)(10,12,15)(17,20,22)(19,21,24);(16,22,19,18,20,21,23,17,24);(7,13,10,9,11,12,14,8,15)243 conjugacy classes — view the full character table.
| maximal subgroup of | #72243235, C3×D3×#92314, C3×#72214284, … |
| maximal normal subgroup of | C3×D3×#92314, C3×#72214284, D3×C3⋊#92314, … |
| maximal quotient of | #72243235 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(5,6)(8,10)(9,14)(11,12)(13,15)(17,19)(18,23)(20,21)(22,24), (4,5,6)(8,11,13)(10,12,15)(17,20,22)(19,21,24), (16,22,19,18,20,21,23,17,24), (7,13,10,9,11,12,14,8,15));