Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#11715850 |
| Fitting | C22×C3×H(5,𝔽2) ⋊ D3 |
| derived | C22⋊#997 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #87588 |
| commutator, cosocle, hypocenter | C22⋊#997 | C2×C3 |
| Frattini | C2 | C3×#87588 |
| Fitting | C22×C3×H(5,𝔽2) | D3 |
| socle | C23×C3 | #1020 |
| LCS | G ▷ C22⋊#997 | |
| UCS | C1 ◁ C2×C3 | |
2439 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 2439 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,4,3,2)(6,7,10,11)(9,12)(13,14,15);(2,3,4)(6,7,9)(10,11,12);(5,6)(7,9)(8,10)(11,12)87 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#11715850, C3×#72209653, #72231618, … |
| maximal normal subgroup of | D3×#11715850, C3×#72209653, #72231618, … |
| maximal quotient of | C3×#72209641, C3×#72209772, C3×#72209645, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3,2)(6,7,10,11)(9,12)(13,14,15), (2,3,4)(6,7,9)(10,11,12), (5,6)(7,9)(8,10)(11,12));