Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C23×C3×C4 |
| direct product | C2×C3×#2679⋊C3 |
| Fitting | C2×C3×#2679 ⋊ C3 |
| derived | #2679 ⋊ C2×C32 |
| Sylow | C2×#2679 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | #5464 |
| commutator, cosocle, hypocenter | #2679 | C2×C32 |
| Frattini | C23 | C2×C3×C22⋊A4 |
| Fitting | C2×C3×#2679 | C3 |
| socle | C24×C3 | C22⋊A4 |
| LCS | G ▷ #2679 | |
| UCS | C1 ◁ C22×C3 | |
2586 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 2586 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2)(6,7,10)(8,11,14)(9,12,15)(13,16,17);(3,4,5)(11,16)(14,17);(6,8)(7,11)(9,13)(12,16)96 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#11714279 |
| maximal normal subgroup of | C2×C3×#11714279 |
| 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((1,2)(6,7,10)(8,11,14)(9,12,15)(13,16,17), (3,4,5)(11,16)(14,17), (6,8)(7,11)(9,13)(12,16));