Finite Group DatabasebetaFinite Group Databasebeta| order | 2268 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | AGL(1,𝔽7)×AGL(1,ℤ/9) |
| Fitting | C7×3−1+2 ⋊ C22×C3 |
| derived | C7×C9 ⋊ C22×C32 |
| Hall | ΓL(1,𝔽8)×3−1+2 ⋊ C22 |
| Sylow | C7 ⋊ C2×C3×AGL(1,ℤ/9) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C7×C9 | C22×C32 |
| Frattini | C3 | C3×D3×AGL(1,𝔽7) |
| Fitting | C7×3−1+2 | C22×C3 |
| socle | C3×C7 | C2×C32×D3 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 | |
220 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,5,6,3,4,7)(8,14,11,10,12,13,15,9,16);(1,2,4,7,6,5,3)(9,13,14,11,12,16)(10,15)70 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,5,6,3,4,7)(8,14,11,10,12,13,15,9,16), (1,2,4,7,6,5,3)(9,13,14,11,12,16)(10,15));