Finite Group DatabasebetaFinite Group Databasebeta| order | 6912 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C3×#87588⋊C3 |
| Fitting, socle | C27×C3 ⋊ C3×D3 |
| derived | C22⋊(C22⋊A4) ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #87588⋊C3 |
| commutator, cosocle, hypocenter | C22⋊(C22⋊A4) | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C27×C3 | C3×D3 |
| LCS | G ▷ C22⋊(C22⋊A4) | |
| UCS | C1 ◁ C2×C3 | |
5446 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 5446 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 54 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2)(3,6,5)(7,10)(8,14,12)(9,11);(4,6)(7,8)(9,12)(10,13)(11,14)(15,16,17)144 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#72215845 |
| maximal quotient of | C2×C3×#72215854 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,6,5)(7,10)(8,14,12)(9,11), (4,6)(7,8)(9,12)(10,13)(11,14)(15,16,17));