Finite Group DatabasebetaFinite Group Databasebeta| order | 5616 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | C3⋊S4×D13⋊C3 |
| Fitting, socle | C22×C3×C13 ⋊ C2×C3×D3 |
| derived | C3×A4×C13 ⋊ C22×C3 |
| Sylow | C13 ⋊ C2×C3×C3⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×A4×C13 | C22×C3 |
| Fitting, socle | C22×C3×C13 | C2×C3×D3 |
| LCS | G ▷ C3×A4×C13 | |
| UCS | C1 | |
416 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 47 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 — 3 generators on 20 points:
(1,4)(2,3)(6,10,11,7,8,12)(9,15,16,13,14,17)(18,19,20);(2,4)(5,6,9,11,10,14,16,17,15,8,12,13,7)(19,20);(2,3,4)72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×S4×D13⋊C3, #72242121, C3⋊S4×AGL(1,𝔽13), … |
| maximal normal subgroup of | D3×S4×D13⋊C3, #72242121, C3⋊S4×AGL(1,𝔽13) |
| maximal quotient of | D13⋊C3×A4⋊S4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4)(2,3)(6,10,11,7,8,12)(9,15,16,13,14,17)(18,19,20), (2,4)(5,6,9,11,10,14,16,17,15,8,12,13,7)(19,20), (2,3,4));