Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C27 |
| direct product | D3×#11715850 |
| Fitting | C22×C3×H(5,𝔽2) ⋊ C2×D3 |
| derived | C3×C22⋊#997 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×#87588 |
| commutator, cosocle, hypocenter | C3×C22⋊#997 | C22 |
| Fitting | C22×C3×H(5,𝔽2) | C2×D3 |
| socle | C23×C3 | C2×#1020 |
| LCS | G ▷ C3×C22⋊#997 | |
| UCS | C1 ◁ C2 | |
18698 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 18698 conjugacy classes of subgroups — too many to draw. Only its 63 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)(6,7,9)(10,11,12)(14,15);(2,4)(6,11,10,7)(9,12)(13,14,15);(5,6)(7,9)(8,10)(11,12)87 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#72209653, C2×D3×#11715850 |
| maximal normal subgroup of | D3×#72209653, C2×D3×#11715850 |
| maximal quotient of | D3×#72209641, D3×#72209772, C2×D3×#11715850 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3)(6,7,9)(10,11,12)(14,15), (2,4)(6,11,10,7)(9,12)(13,14,15), (5,6)(7,9)(8,10)(11,12));