Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C35 |
| direct product | C3×C3⋊#92300 |
| Fitting, Sylow | C32×C32⋊H(3,𝔽3) ⋊ C2 |
| derived | C32×H(3,𝔽3) ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | C3⋊(C3⋊((C3⋊D3)⋊C3)) |
| commutator, cosocle, hypocenter | C32×H(3,𝔽3) | C2×C32 |
| Frattini | C32 | C32×C3⋊(C3⋊D3) |
| Fitting | C32×C32⋊H(3,𝔽3) | C2 |
| socle | C34 | C3×C3⋊D3 |
| LCS | G ▷ C32×H(3,𝔽3) | |
| UCS | C1 ◁ C32 | |
18487 conjugacy classes of subgroups; only the 278 normal subgroups are shown.
This group has 18487 conjugacy classes of subgroups — too many to draw. Only its 278 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 — 4 generators on 24 points:
(5,6)(8,13,15,10,11,14)(9,12)(17,19)(20,24)(21,23);(1,2,3)(4,5,6);(17,20,21)(19,23,24);(7,8,10)(9,11,14)(12,15,13)(16,17,19)(18,21,23)(20,24,22)270 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #72253773, C3×#72224516, C3×#72224543, … |
| maximal normal subgroup of | C3×#72224516, C3×#72224543, C3×#72224542 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(8,13,15,10,11,14)(9,12)(17,19)(20,24)(21,23), (1,2,3)(4,5,6), (17,20,21)(19,23,24), (7,8,10)(9,11,14)(12,15,13)(16,17,19)(18,21,23)(20,24,22));