Finite Group DatabasebetaFinite Group Databasebeta| order | 960 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#5711 |
| Fitting | C3×C5×#126 ⋊ C2 |
| Hall | C3×C5 ⋊ #560 |
| Sylow | C3 ⋊ C5×#560 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5 | C23×D3 |
| commutator, cosocle | C22×C3 | C24×C5 |
| Frattini | C22 | C23×C5×D3 |
| Fitting | C3×C5×#126 | C2 |
| socle | C22×C3×C5 | C24 |
| hypercenter | C5×#126 | D3 |
| hypocenter | C3 | C5×#560 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#126 | |
504 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 182 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4,5)(6,7,8)(10,14)(13,16)(17,19,20,23)(18,22,21,24);(9,11,12,15)(10,13,14,16)(18,21)(22,24);(9,10)(11,16)(12,14)(13,15)(17,18)(19,22)(20,21)(23,24);(7,8)(11,15)(13,16)(19,23)(22,24)165 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D3×#560, C2×C5×#5711, D5×#5711, … |
| maximal normal subgroup of | C5×D3×#560, C2×C5×#5711, D5×#5711, … |
| maximal quotient of | C5×#75840, C5×#75312, C5×#75490, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(960, 10291);