Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210×C3 |
| direct product | C23×D9×C3⋊D3 |
| Fitting | C23×C32×C9 ⋊ C22 |
| derived, Sylow | C32×C9 ⋊ C25 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D9×C3⋊D3 |
| commutator, cosocle, hypocenter | C32×C9 | C25 |
| Frattini | C3 | C23×D3×C3⋊D3 |
| Fitting | C23×C32×C9 | C22 |
| socle | C23×C33 | C2×D3 |
| LCS | G ▷ C32×C9 | |
| UCS | C1 ◁ C23 | |
7804 conjugacy classes of subgroups; only the 1003 normal subgroups are shown.
This group has 7804 conjugacy classes of subgroups — too many to draw. Only its 1003 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 5 generators on 21 points:
(5,6)(7,8,9)(13,18,16,15,20,17,19,14,21);(1,2)(3,4)(5,6)(8,9)(11,12);(10,11,12)(14,16)(15,19)(17,20)(18,21);(3,4);(1,2)288 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×D4×D9×C3⋊D3 |
| maximal normal subgroup of | C2×D4×D9×C3⋊D3 |
| maximal quotient of | C2×D4×D9×C3⋊D3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(7,8,9)(13,18,16,15,20,17,19,14,21), (1,2)(3,4)(5,6)(8,9)(11,12), (10,11,12)(14,16)(15,19)(17,20)(18,21), (3,4), (1,2));