Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C5×D4×C9 ⋊ C2 |
| Hall | C5×C9 ⋊ D8 |
| Sylow | C9 ⋊ #686 |
| Sylow | C5 ⋊ #3673 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #67099 |
| commutator, cosocle | C4×C5×C9 | C22 |
| Frattini | C3×C4 | C2×D15 |
| Fitting | C5×D4×C9 | C2 |
| socle | C2×C3×C5 | #67 |
| hypercenter | D4 | D45 |
| hypocenter | C5×C9 | D8 |
| LCS | G ▷ C4×C5×C9 ▷ C2×C5×C9 ▷ C5×C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 360 |
Polycyclic presentation:
2^4,3,5,3|c^2=d,e^3=g,b^a=bc,c^a=cd,e^a=e^2g^2,f^a=f^4,g^a=g^2,c^b=cd117 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#3673, #13084239, #13084242, … |
| maximal normal subgroup of | D5×#3673, #13084239, #13084242, … |
| maximal quotient of | #13083850, #13083853, #13083869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 77);