Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×(#1020⋊C3)⋊D3 |
| Fitting, socle | C25×C3 ⋊ D32 |
| derived | C3×A42 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (#1020⋊C3)⋊D3 |
| commutator, cosocle, hypocenter | C3×A42 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C25×C3 | D32 |
| LCS | G ▷ C3×A42 | |
| UCS | C1 ◁ C2 | |
1421 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 1421 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 3 generators on 13 points:
(3,5,9,6)(4,7)(12,13);(3,4)(5,10,6,8,9,7);(1,2)(11,12,13)60 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×(#1020⋊C3)⋊C2 |
| maximal normal subgroup of | C2×D3×(#1020⋊C3)⋊C2 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,5,9,6)(4,7)(12,13), (3,4)(5,10,6,8,9,7), (1,2)(11,12,13));