Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
| direct product | D32×#997 |
| Fitting, derived | C32×H(5,𝔽2) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32×C22⋊A4 |
| commutator, Fitting, cosocle, hypocenter | C32×H(5,𝔽2) | C22×C3 |
| socle | C2×C32 | C22×C22⋊A4 |
| LCS | G ▷ C32×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
5219 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 5219 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 14 points:
(2,3,7)(5,6,8)(10,11)(12,13);(2,6)(5,7)(9,10,11)(13,14);(1,5,4,7)(2,8,6,3)99 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,7)(5,6,8)(10,11)(12,13), (2,6)(5,7)(9,10,11)(13,14), (1,5,4,7)(2,8,6,3));