Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#11714279 |
| Fitting | C2×C3×#2679 ⋊ D3 |
| derived | #2679⋊C3 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | #85559 |
| commutator, cosocle, hypocenter | #2679⋊C3 | C22×C3 |
| Frattini | C23 | C2×C3×#1020 |
| Fitting | C2×C3×#2679 | D3 |
| socle | C24×C3 | #1020 |
| LCS | G ▷ #2679⋊C3 | |
| UCS | C1 ◁ C22×C3 | |
5269 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 5269 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 17 points:
(1,2)(3,4,5)(6,12,15)(7,10,9)(8,16,14)(11,17,13);(6,8)(7,14,16,15)(9,13)(10,12,17,11)138 conjugacy classes — view the full character table.
| maximal quotient of | C2×C3×#72209884 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4,5)(6,12,15)(7,10,9)(8,16,14)(11,17,13), (6,8)(7,14,16,15)(9,13)(10,12,17,11));