Finite Group DatabasebetaFinite Group Databasebeta| order | 960 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#66050 |
| Fitting | C3×C5×#127 ⋊ C2 |
| Hall | C3×C5 ⋊ #532 |
| Sylow | C5 ⋊ C3×#532 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C23×D5 |
| commutator, cosocle | C22×C5 | C24×C3 |
| Frattini | C22 | C23×C3×D5 |
| Fitting | C3×C5×#127 | C2 |
| socle | C22×C3×C5 | C24 |
| hypercenter | C3×#127 | D5 |
| hypocenter | C5 | C3×#532 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#127 | |
620 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 480 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3)(4,5,7,8,6)(10,12)(14,18)(15,19);(13,15,16,19)(14,17,18,20);(9,10,11,12);(5,6)(7,8)(13,14)(15,17)(16,18)(19,20)168 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#10604604, #14281781, C3×#10604605, … |
| maximal normal subgroup of | C3×#10604604, #14281781, C3×#10604605, … |
| maximal quotient of | C3×#10606049, C3×#10608020, C3×#10608021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(960, 9922);