Finite Group DatabasebetaFinite Group Databasebeta| order | 5760 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#11757163⋊C2 |
| Fitting, socle | C24×C3 ⋊ S5 |
| derived | #11757163 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #11757163⋊C2 |
| commutator, cosocle, hypocenter | #11757163 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3 | S5 |
| LCS | G ▷ #11757163 | |
| UCS | C1 ◁ C3 | |
415 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 415 conjugacy classes of subgroups — too many to draw. Only its 8 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) = 30 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,10,8)(2,7,5)(3,4)(6,9)(11,12,13);(1,7,4)(2,10,3)(5,6,8,9)54 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#11757163⋊C2, D3×#11757163⋊C2 |
| maximal normal subgroup of | C2×C3×#11757163⋊C2, D3×#11757163⋊C2 |
| maximal quotient of | C2×C3×#11757163⋊C2 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,10,8)(2,7,5)(3,4)(6,9)(11,12,13), (1,7,4)(2,10,3)(5,6,8,9));