Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C5×C9 ⋊ Q16 |
| Sylow | C9 ⋊ #689 |
| Sylow | C5 ⋊ #3674 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #67082 |
| commutator, Fitting, cosocle | C4×C5×C9 | C22 |
| Frattini | C3×C4 | D3×D5 |
| socle, prime radical | C2×C3×C5 | #67 |
| hypercenter | C4 | D5×D9 |
| hypocenter | C5×C9 | Q16 |
| LCS | G ▷ C4×C5×C9 ▷ C2×C5×C9 ▷ C5×C9 | |
| UCS | C1 ◁ C2 ◁ C4 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 720 |
Polycyclic presentation:
2^4,3,5,3|a^2=d,b^2=d,c^2=d,e^3=g,b^a=bc,c^a=cd,f^a=f^4,c^b=cd,e^b=e^2g^2,g^b=g^269 conjugacy classes — view the full character table.
| maximal subgroup of | #13084031, #13084032, #13084036, … |
| maximal normal subgroup of | #13084031, #13084032, #13084036, … |
| maximal quotient of | #13083713, #13083719, #13083748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 23);