Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C33 |
| direct product | C3×C3⋊#92302 |
| Fitting, Sylow | C32×#6891 ⋊ C2 |
| derived | C3×C32⋊C9 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | C3⋊(C3⋊AGL(1,ℤ/9)) |
| commutator, cosocle, hypocenter | C3×C32⋊C9 | C2×C32 |
| Frattini | C32 | C32×C3⋊(C3⋊D3) |
| Fitting | C32×#6891 | C2 |
| socle | C34 | C3×C3⋊D3 |
| LCS | G ▷ C3×C32⋊C9 | |
| UCS | C1 ◁ C32 | |
13924 conjugacy classes of subgroups; only the 278 normal subgroups are shown.
This group has 13924 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:
(1,2,3)(5,6)(8,10)(9,13)(11,14)(12,15)(17,19)(20,24)(21,23);(7,11,12,9,15,14,13,8,10)(16,17,19)(18,20,24)(21,23,22);(17,21,20)(19,24,23);(4,5,6)(17,20,21)(19,23,24)270 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #72243459, #72253773, #72253800, … |
| maximal normal subgroup of | #72253800, C3×#72253805 |
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)(9,13)(11,14)(12,15)(17,19)(20,24)(21,23), (7,11,12,9,15,14,13,8,10)(16,17,19)(18,20,24)(21,23,22), (17,21,20)(19,24,23), (4,5,6)(17,20,21)(19,23,24));