Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C32 |
| direct product | C3×D3×#92314 |
| Fitting, Sylow | C32×#6900 ⋊ C22 |
| derived | C3×C92 ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D3×#92314 |
| commutator, cosocle, hypocenter | C3×C92 | C22×C32 |
| Frattini | C32 | C32×D3×C3⋊D3 |
| Fitting | C32×#6900 | C22 |
| socle | C34 | C2×C3×C3⋊D3 |
| LCS | G ▷ C3×C92 | |
| UCS | C1 ◁ C3 | |
4232 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 4232 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 324 |
A minimal faithful permutation representation — 3 generators on 24 points:
(4,5,6)(8,14,15,10,11,12)(9,13)(17,23,24,19,20,21)(18,22);(1,2,3)(5,6)(16,20,19,18,24,21,22,17,23);(7,11,10,9,15,12,13,8,14)270 conjugacy classes. The character table is too large to display.
| maximal subgroup of | AGL(1,ℤ/9)×#92314, C3×D3×#92265⋊C2, D32×#92314, … |
| maximal normal subgroup of | C3×D3×#92265⋊C2, D32×#92314, D3×#72214299 |
| maximal quotient of | AGL(1,ℤ/9)×#92314 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5,6)(8,14,15,10,11,12)(9,13)(17,23,24,19,20,21)(18,22), (1,2,3)(5,6)(16,20,19,18,24,21,22,17,23), (7,11,10,9,15,12,13,8,14));