Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×D9×#997 |
| Fitting | C2×C9×H(5,𝔽2) ⋊ C2×C3 |
| derived | C9×H(5,𝔽2) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D9×C22⋊A4 |
| commutator, cosocle, hypocenter | C9×H(5,𝔽2) | C22×C3 |
| Frattini | C2×C3 | C2×D3×C22⋊A4 |
| Fitting | C2×C9×H(5,𝔽2) | C2×C3 |
| socle | C22×C3 | D3×C22⋊A4 |
| LCS | G ▷ C9×H(5,𝔽2) | |
| UCS | C1 ◁ C22 | |
3727 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 3727 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 19 points:
(4,7,5)(8,9,10)(12,14)(13,17)(15,18)(16,19);(1,2)(4,8)(5,10)(11,18,19,13,12,14,17,16,15);(3,9,6,7)(4,5,8,10)132 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AGL(1,ℤ/9)×#997 |
| maximal normal subgroup of | C2×AGL(1,ℤ/9)×#997 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,7,5)(8,9,10)(12,14)(13,17)(15,18)(16,19), (1,2)(4,8)(5,10)(11,18,19,13,12,14,17,16,15), (3,9,6,7)(4,5,8,10));