Finite Group DatabasebetaFinite Group Databasebeta| order | 960 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C3×C5×#361 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C5 | #102 |
| commutator | C8 | C2×C3×C4×C5 |
| Frattini, cosocle | C2×C8 | C22×C3×C5 |
| prime radical | C2×C3×C5×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ C2×C3×C4×C5 ◁ C2×C3×C5×C8 ◁ G | |
124 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 2 generators on 24 points:
(1,2,3)(4,5,6,7,8)(9,10,12,16,13,18,23,17)(11,19,21,15,22,14,20,24);(10,14)(11,20)(12,23)(15,16)(17,24)(18,19)(21,22)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(960, 624);