Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | Dic9⋊Dic5 |
| Hall | C5×C9 ⋊ C4⋊C4 |
| Sylow | C9 ⋊ C4⋊Dic5 |
| Sylow | C5 ⋊ Dic9⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D5×D9 |
| commutator | C2×C5×C9 | C2×C4 |
| Frattini | C22×C3 | D3×D5 |
| Fitting, cosocle | C22×C5×C9 | C22 |
| socle, prime radical | C22×C3×C5 | C2×D3 |
| hypocenter | C5×C9 | C4⋊C4 |
| LCS | G ▷ C2×C5×C9 ▷ C5×C9 | |
| UCS | C1 ◁ C22 | |
78 conjugacy classes of subgroups.
| transitive degree | μt(G) = 720 |
Polycyclic presentation:
2^4,3,5,3|a^2=c,b^2=d,e^3=g,b^a=bd,f^a=f^4,e^b=e^2g^2,g^b=g^290 conjugacy classes — view the full character table.
| maximal subgroup of | Dic5×Dic18, #13084069, #13084070, … |
| maximal normal subgroup of | Dic5×Dic18, #13084069, #13084070, … |
| maximal quotient of | #13083714, #13083716, Dic9⋊#168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 27);