Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C3⋊#10593110 |
| derived | C3×#63703 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊#64188 |
| commutator, cosocle, hypocenter | C3×#63703 | C22 |
| Frattini | C2×C3 | C2×C3⋊#1020 |
| Fitting | C2×C32×H(5,𝔽2) | D3 |
| socle | C22×C32 | #1020 |
| LCS | G ▷ C3×#63703 | |
| UCS | C1 ◁ C22 | |
1894 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 1894 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) = 22 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2)(3,4,5)(6,7)(8,12)(9,10)(11,13);(4,5)(7,10)(8,13)(9,12)(15,17)(16,20)(18,21)(19,22);(7,8,12)(9,13,10)(14,19,17,16,21,18,20,15,22)114 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72245426, C2×#72252942 |
| maximal normal subgroup of | C2×#72252942 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4,5)(6,7)(8,12)(9,10)(11,13), (4,5)(7,10)(8,13)(9,12)(15,17)(16,20)(18,21)(19,22), (7,8,12)(9,13,10)(14,19,17,16,21,18,20,15,22));