Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C3×#11709313 |
| Fitting | C3×#15947 ⋊ C3 |
| derived, Sylow | #15947 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | #5464 |
| commutator, cosocle, hypocenter | #15947 | C32 |
| Frattini | C24 | C3×C22⋊A4 |
| Fitting | C3×#15947 | C3 |
| socle | C24×C3 | C22⋊A4 |
| LCS | G ▷ #15947 | |
| UCS | C1 ◁ C22×C3 | |
2406 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 2406 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 23 points:
(4,5,8)(6,9,12)(7,10,13)(11,14,15)(17,20,18)(21,23,22);(1,2,3)(9,14)(12,15)(17,21)(18,22);(4,6)(5,9)(7,11)(10,14)(16,20)(17,18)(19,23)(21,22)90 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#72209884, C2×C3×#11709313 |
| maximal normal subgroup of | C3×#72209884, C2×C3×#11709313 |
| maximal quotient of | C2×C3×#11709313 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5,8)(6,9,12)(7,10,13)(11,14,15)(17,20,18)(21,23,22), (1,2,3)(9,14)(12,15)(17,21)(18,22), (4,6)(5,9)(7,11)(10,14)(16,20)(17,18)(19,23)(21,22));