Finite Group DatabasebetaFinite Group Databasebeta| order | 5760 |
| isoclinism | isoclinic class of #11757163 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×C3×#11757163 |
| Fitting, socle | C25×C3 ⋊ A5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #11757163 |
| commutator, hypocenter | #11757163 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C25×C3 | A5 |
| cosocle | C24 | C2×C3×A5 |
| LCS | G ▷ #11757163 | |
| UCS | C1 ◁ C2×C3 | |
546 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 546 conjugacy classes of subgroups — too many to draw. Only its 12 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:
(4,13,12,8,10,7,11,9,5,6);(1,2,3)(5,10,13,8,6,11)(9,12)72 conjugacy classes — view the full character table.
| maximal subgroup of | #72227477, C2×C3×#11757163⋊C2, C2×D3×#11757163, … |
| maximal normal subgroup of | C2×C3×#11757163⋊C2, C2×D3×#11757163, C2×#72275455 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,13,12,8,10,7,11,9,5,6), (1,2,3)(5,10,13,8,6,11)(9,12));