Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×C3×C9⋊#1020 |
| Fitting | C25×C3×C9 ⋊ D3 |
| derived | C9×C22⋊A4 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C9⋊#1020 |
| commutator, cosocle, hypocenter | C9×C22⋊A4 | C22×C3 |
| Frattini | C3 | C2×C3×C3⋊#1020 |
| Fitting | C25×C3×C9 | D3 |
| socle | C25×C32 | C3⋊D3 |
| LCS | G ▷ C9×C22⋊A4 | |
| UCS | C1 ◁ C2×C3 | |
2094 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 2094 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 22 points:
(3,6,5,4)(7,8,10,9)(11,12,13)(15,17)(16,20)(18,21)(19,22);(1,2)(14,19,17,16,21,18,20,15,22);(4,5,6)(8,10,9)252 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,6,5,4)(7,8,10,9)(11,12,13)(15,17)(16,20)(18,21)(19,22), (1,2)(14,19,17,16,21,18,20,15,22), (4,5,6)(8,10,9));