Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C27 |
| direct product | D3×#87588⋊C2 |
| Fitting | C22×C3×#120 ⋊ C2×D3 |
| derived | C3×C22⋊(C22⋊A4) ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C22⋊(C22⋊A4) | C23 |
| Frattini | C22 | C2×D3×#1020 |
| Fitting | C22×C3×#120 | C2×D3 |
| socle | C24×C3 | C22×S4 |
| LCS | G ▷ C3×C22⋊(C22⋊A4) | |
| UCS | C1 | |
16194 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 16194 conjugacy classes of subgroups — too many to draw. Only its 65 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) = 48 |
A minimal faithful permutation representation — 3 generators on 15 points:
(2,3,4)(6,9,11)(7,8,12)(14,15);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,15);(2,3)(7,12)(9,10)90 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×#87588⋊C2 |
| maximal normal subgroup of | C2×D3×#87588⋊C2 |
| maximal quotient of | D3×#72209653, C2×D3×#87588⋊C2 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4)(6,9,11)(7,8,12)(14,15), (1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,15), (2,3)(7,12)(9,10));