Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of #67138 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×#67138 |
| Hall | C32×C5 ⋊ C2×D4 |
| Sylow | C32 ⋊ C2×C22⋊D5 |
| Sylow | C5 ⋊ C2×#638 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D5×C3⋊D3 |
| commutator, cosocle | C2×C32×C5 | C23 |
| Frattini | C2 | C2×D5×C3⋊D3 |
| Fitting, socle | C22×C32×C5 | C22 |
| hypocenter | C32×C5 | C2×D4 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C22 | |
324 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | #13085436, #13085492, #13085523, … |
| maximal normal subgroup of | #13085436, #13085492, #13085523, … |
| maximal quotient of | #13086784, #13086791, #13086792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 636);