Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C5×C9 ⋊ D8 |
| Sylow | C9 ⋊ D40 |
| Sylow | C5 ⋊ #3673 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #67083 |
| commutator, Fitting, cosocle | C4×C5×C9 | C22 |
| Frattini | C3×C4 | D3×D5 |
| socle | C2×C3×C5 | #67 |
| hypercenter | C4 | D5×D9 |
| hypocenter | C5×C9 | D8 |
| LCS | G ▷ C4×C5×C9 ▷ C2×C5×C9 ▷ C5×C9 | |
| UCS | C1 ◁ C2 ◁ C4 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 360 |
Polycyclic presentation:
2^4,3,5,3|b^2=cd,c^2=d,e^3=g,b^a=bc,c^a=cd,f^a=f^4,e^b=e^2g^2,g^b=g^275 conjugacy classes — view the full character table.
| maximal subgroup of | #13084021, D9×D40, #13084023, … |
| maximal normal subgroup of | #13084021, D9×D40, #13084023, … |
| maximal quotient of | #13083701, #13083703, #13083705, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 18);