Finite Group DatabasebetaFinite Group Databasebeta| order | 5616 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×S4×D13⋊C3 |
| Fitting, socle | C22×C3×C13 ⋊ C2×C3×D3 |
| derived | A4×C13 ⋊ C22×C32 |
| Sylow | C13 ⋊ C2×C32×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | S4×D13⋊C3 |
| commutator, cosocle, hypocenter | A4×C13 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C13 | C2×C3×D3 |
| LCS | G ▷ A4×C13 | |
| UCS | C1 ◁ C3 | |
460 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 156 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,4,3)(5,6)(7,9)(8,14)(10,12)(11,13)(15,16)(18,19,20);(2,4)(6,8,11)(7,10,12)(9,14,16)(13,15,17)120 conjugacy classes — view the full character table.
| maximal subgroup of | C3×S4×AGL(1,𝔽13), D3×S4×D13⋊C3, S4×C3⋊AGL(1,𝔽13) |
| maximal normal subgroup of | C3×S4×AGL(1,𝔽13), D3×S4×D13⋊C3, S4×C3⋊AGL(1,𝔽13) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3)(5,6)(7,9)(8,14)(10,12)(11,13)(15,16)(18,19,20), (2,4)(6,8,11)(7,10,12)(9,14,16)(13,15,17));