Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C42⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C5 · Schur multiplier (H2) = C3×C4 |
| Fitting | C3×C42×C5 ⋊ C3 |
| derived, Sylow | C42 ⋊ C32×C5 |
| Hall | C42×C5 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | C42⋊C3 |
| commutator, cosocle, hypocenter | C42 | C32×C5 |
| Frattini | C22 | C3×C5×A4 |
| Fitting | C3×C42×C5 | C3 |
| socle | C22×C3×C5 | A4 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C3×C5 | |
52 conjugacy classes of subgroups.
| transitive degree | μt(G) = 180 |
120 conjugacy classes — view the full character table.
| derived subgroup of | C3⋊(C5⋊#857) |
| maximal subgroup of | C3×C5×#857, C3×C5⋊#857, C5×C3⋊#857, … |
| maximal normal subgroup of | C3×C5×#857, C3×C5⋊#857, C5×C3⋊#857, … |
| maximal quotient of | C3×C5×#95⋊C3, C2×C3×C5×C42⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 302);