Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D5×D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×D5×Dic9 |
| derived, Hall | C5×C9 ⋊ C22×C4 |
| Sylow | C9 ⋊ C2×C4×D5 |
| Sylow | C5 ⋊ C22×Dic9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D5×D9 |
| commutator, hypocenter | C5×C9 | C22×C4 |
| Frattini | C2×C3 | C2×D3×D5 |
| Fitting | C22×C5×C9 | C22 |
| socle | C22×C3×C5 | C2×D3 |
| prime radical | C22×C3×D5 | D3 |
| cosocle | C2×C5×C9 | C23 |
| LCS | G ▷ C5×C9 | |
| UCS | C1 ◁ C22 | |
162 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
96 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×AGL(1,𝔽5))⋊Dic9, C4×D5×Dic9, D5×Dic9⋊C4, … |
| maximal normal subgroup of | (C2×AGL(1,𝔽5))⋊Dic9, C4×D5×Dic9, D5×Dic9⋊C4, … |
| maximal quotient of | C2×D5×C9⋊C8, D5×#3667, #13084061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 144);