Finite Group DatabasebetaFinite Group Databasebeta| order | 10368 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
| direct product | AGL(1,ℤ/9)×#5934 |
| Fitting | 3−1+2×H(5,𝔽2) ⋊ C2×D3 |
| derived | C9×#997 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | AGL(1,ℤ/9)×#1020 |
| commutator, cosocle, hypocenter | C9×#997 | C22×C3 |
| Frattini, socle | C2×C3 | C3×D3×#1020 |
| Fitting | 3−1+2×H(5,𝔽2) | C2×D3 |
| LCS | G ▷ C9×#997 | |
| UCS | C1 ◁ C2 | |
2924 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 2924 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,7,5)(3,8,6)(10,14,15,12,13,17)(11,16);(1,2,5,3)(4,6,8,7)(9,15,12,11,13,14,16,10,17)130 conjugacy classes — view the full character table.
| automorphism group of | C22×Dic9 |
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)(10,14,15,12,13,17)(11,16), (1,2,5,3)(4,6,8,7)(9,15,12,11,13,14,16,10,17));