Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×(C3⋊D3)⋊#997 |
| Fitting | C2×C32×H(5,𝔽2) ⋊ C2×C3 |
| derived | C32×H(5,𝔽2) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | (C3⋊D3)⋊(C22⋊A4) |
| commutator, cosocle, hypocenter | C32×H(5,𝔽2) | C22×C3 |
| Frattini | C2×C3 | C2×D3×C22⋊A4 |
| Fitting | C2×C32×H(5,𝔽2) | C2×C3 |
| socle | C22×C3 | D3×C22⋊A4 |
| LCS | G ▷ C32×H(5,𝔽2) | |
| UCS | C1 ◁ C22 | |
7007 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 7007 conjugacy classes of subgroups — too many to draw. Only its 87 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:
(1,2)(4,5,8)(7,9,10)(11,12,18)(13,15,17)(14,16,19);(3,6)(4,9)(12,14)(13,16)(15,17)(18,19);(3,7,6,8)(4,10,9,5)100 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72220370 |
| maximal normal subgroup of | C2×#72220370 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,5,8)(7,9,10)(11,12,18)(13,15,17)(14,16,19), (3,6)(4,9)(12,14)(13,16)(15,17)(18,19), (3,7,6,8)(4,10,9,5));