Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C3×#10596770 |
| Fitting, socle | C25×C3 ⋊ #157 |
| derived | A42 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #10596770 |
| commutator, hypocenter | A42 | C2×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C25×C3 | #157 |
| prime radical | C2×C3×A4⋊S4 | C2 |
| cosocle | A4⋊S4 | C22×C3 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C2×C3 | |
508 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 13 points:
(3,4,10,9,5,7,6,8)(11,12,13);(1,2)(5,10,6)(7,8,9)78 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#10596770⋊C2, C2×#72275397, C2×D3×#10596770 |
| maximal normal subgroup of | C2×C3×#10596770⋊C2, C2×#72275397, C2×D3×#10596770 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,10,9,5,7,6,8)(11,12,13), (1,2)(5,10,6)(7,8,9));