Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of Dic18 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C2×C5×#611 |
| Fitting | C23×C5×C9 ⋊ C2 |
| Hall | C5×C9 ⋊ C2×D4 |
| Sylow | C9 ⋊ C2×C5×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5 | C2×D9 |
| commutator, cosocle | C2×C9 | C23×C5 |
| Frattini | C2×C3 | C22×C5×D3 |
| Fitting | C23×C5×C9 | C2 |
| socle | C22×C3×C5 | C2×D3 |
| hypercenter | C23×C5 | D9 |
| hypocenter | C9 | C2×C5×D4 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C5 | |
162 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
210 conjugacy classes — view the full character table.
| maximal subgroup of | #13083753, C5×#63365, #13084295, … |
| maximal normal subgroup of | #13083753, C5×#63365, #13084295, … |
| maximal quotient of | C2×C5×Dic9⋊C4, C5×#63486, C2×C5×(C2×D9)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 174);