Finite Group DatabasebetaFinite Group Databasebeta| order | 5616 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×A4⋊AGL(1,𝔽13) |
| Fitting, socle | C22×C3×C13 ⋊ C3×Dic3 |
| derived | A4×C13 ⋊ C32×C4 |
| Sylow | C13 ⋊ C32×SL(2,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | A4⋊AGL(1,𝔽13) |
| commutator, hypocenter | A4×C13 | C32×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C13 | C3×Dic3 |
| prime radical | C3×A4×D13⋊C3 | C2 |
| cosocle | A4×D13 | C2×C32 |
| LCS | G ▷ A4×C13 | |
| UCS | C1 ◁ C3 | |
292 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(1,4,3,2)(5,6,8,7)(9,17,15,13)(10,14,11,12)(18,19,20);(2,3,4)(6,8,13)(7,11,16)(9,14,15)(10,12,17)102 conjugacy classes — view the full character table.
| maximal subgroup of | C3×S4×AGL(1,𝔽13), D3×A4⋊AGL(1,𝔽13), #72242121 |
| maximal normal subgroup of | C3×S4×AGL(1,𝔽13), D3×A4⋊AGL(1,𝔽13), #72242121 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3,2)(5,6,8,7)(9,17,15,13)(10,14,11,12)(18,19,20), (2,3,4)(6,8,13)(7,11,16)(9,14,15)(10,12,17));