Finite Group DatabasebetaFinite Group Databasebeta| order | 7488 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×S4×AGL(1,𝔽13) |
| Fitting, socle | C23×C13 ⋊ C3×C4×D3 |
| derived | A4×C13 ⋊ C22×C3×C4 |
| Sylow | C13 ⋊ C2×C3×C4×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | S4×AGL(1,𝔽13) |
| commutator, hypocenter | A4×C13 | C22×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C23×C13 | C3×C4×D3 |
| prime radical | C2×S4×D13⋊C3 | C2 |
| cosocle | A4×D13 | C23×C3 |
| LCS | G ▷ A4×C13 | |
| UCS | C1 ◁ C2 | |
964 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 964 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 78 |
A minimal faithful permutation representation — 3 generators on 19 points:
(3,6,5)(8,13,17,14,10,18,11,19,15,9,16,12);(1,2)(7,8,12,15,16,14,19,13,9,10,17,18,11);(4,6)130 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,6,5)(8,13,17,14,10,18,11,19,15,9,16,12), (1,2)(7,8,12,15,16,14,19,13,9,10,17,18,11), (4,6));