Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C3×C3⋊#5935 |
| Fitting | C2×C32×H(5,𝔽2) ⋊ D3 |
| derived | C3×#997 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | C3⋊#1020 |
| commutator, cosocle, hypocenter | C3×#997 | C22×C3 |
| Frattini | C2 | C2×C3×C3⋊#1020 |
| Fitting | C2×C32×H(5,𝔽2) | D3 |
| socle | C22×C32 | #1020 |
| LCS | G ▷ C3×#997 | |
| UCS | C1 ◁ C22×C3 | |
1889 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 1889 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 96 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,9)(4,6)(5,8)(7,10)(11,12,13)(14,15,16);(4,7)(5,9)(8,10)(15,16);(4,8,5)(7,9,10)144 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72220371, C2×#72219864, C2×#72220369, … |
| maximal normal subgroup of | C3×#72215523 |
| maximal quotient of | C2×#72252942 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,9)(4,6)(5,8)(7,10)(11,12,13)(14,15,16), (4,7)(5,9)(8,10)(15,16), (4,8,5)(7,9,10));