Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×S42 |
| Fitting, socle | C25×C3 ⋊ D32 |
| derived | A42 ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | S42 |
| commutator, cosocle, hypocenter | A42 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C25×C3 | D32 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C2×C3 | |
2892 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 2892 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 54 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,3)(4,5)(6,7,9)(10,13,11);(7,8)(12,13);(12,13)150 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×S42, C3×#72215293, S42×AGL(1,𝔽7), … |
| maximal normal subgroup of | C3×#72215293, C2×D3×S42, #72252996, … |
| maximal quotient of | C3×D3×S42, C3×#72215280, C3×#72215293, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,5)(6,7,9)(10,13,11), (7,8)(12,13), (12,13));