Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×D3×#5934 |
| Fitting | C32×H(5,𝔽2) ⋊ C2×D3 |
| derived | C3×#997 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D3×#1020 |
| commutator, cosocle, hypocenter | C3×#997 | C22×C3 |
| Frattini | C2 | C3×D3×#1020 |
| Fitting | C32×H(5,𝔽2) | C2×D3 |
| socle | C2×C32 | C2×#1020 |
| LCS | G ▷ C3×#997 | |
| UCS | C1 ◁ C2×C3 | |
1974 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 1974 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,7,5)(3,8,6)(9,10,11)(13,14);(1,2,5,3)(4,6,8,7)(12,13,14)117 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/9)×#5934 |
| maximal quotient of | AGL(1,ℤ/9)×#5934 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,7,5)(3,8,6)(9,10,11)(13,14), (1,2,5,3)(4,6,8,7)(12,13,14));