Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C37 |
| direct product | C32×#92301 |
| Fitting, Sylow | C32×C32⋊H(3,𝔽3) ⋊ C2 |
| derived | C34 ⋊ C2×C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | #92301 |
| commutator, cosocle, hypocenter | C34 | C2×C33 |
| Frattini | C32 | C33×C3⋊D3 |
| Fitting | C32×C32⋊H(3,𝔽3) | C2 |
| socle | C34 | C3×C3⋊D3 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C32 | |
17980 conjugacy classes of subgroups; only the 298 normal subgroups are shown.
This group has 17980 conjugacy classes of subgroups — too many to draw. Only its 298 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) = 54 |
A minimal faithful permutation representation — 3 generators on 24 points:
(4,5,6)(8,10)(9,12)(11,13)(14,15)(17,19)(18,21)(20,22)(23,24);(8,11,15)(10,13,14)(16,17,23)(18,20,22)(19,21,24);(1,2,3)(7,8,10)(9,11,14)(12,15,13)270 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #72243422, C3×#72214065, C32×#11757702 |
| maximal normal subgroup of | #72243422, C3×#72214065, C32×#11757702 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5,6)(8,10)(9,12)(11,13)(14,15)(17,19)(18,21)(20,22)(23,24), (8,11,15)(10,13,14)(16,17,23)(18,20,22)(19,21,24), (1,2,3)(7,8,10)(9,11,14)(12,15,13));