Finite Group DatabasebetaFinite Group Databasebeta| order | 960 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42×C5 · Schur multiplier (H2) = C22 |
| direct product | C3×C5×#341 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C5 | D4 |
| commutator | C4 | C3×C42×C5 |
| Frattini, cosocle | C22×C4 | C22×C3×C5 |
| socle | C22×C3×C5 | C4⋊C4 |
| prime radical | C23×C3×C5 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C5 ◁ C22×C3×C4×C5 ◁ G | |
180 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 480 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3)(4,5,6,7,8)(9,10)(11,12)(13,14,16,21,18,22,17,20)(15,23,24,19,26,27,25,28);(10,12)(13,15,17,25,18,26,16,24)(14,19,21,27,22,28,20,23)420 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(960, 604);