Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×D3×A4×D7 |
| Fitting, socle | C23×C3×C7 ⋊ C22×C3 |
| derived | C22×C3×C7 ⋊ C23×C3 |
| Sylow | C7 ⋊ C22×D3×A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×A4×D7 |
| commutator, cosocle, hypocenter | C22×C3×C7 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3×C7 | C22×C3 |
| LCS | G ▷ C22×C3×C7 | |
| UCS | C1 ◁ C2 | |
732 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 732 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 126 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(4,6,5)(7,8,9)(10,11,13,15,16,14,12);(3,6)(4,5)(11,12)(13,14)(15,16);(8,9)120 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7×#997, C2×D3×A4×AGL(1,𝔽7) |
| maximal normal subgroup of | C2×D3×A4×AGL(1,𝔽7) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,6,5)(7,8,9)(10,11,13,15,16,14,12), (3,6)(4,5)(11,12)(13,14)(15,16), (8,9));