Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×Q8×C9 ⋊ C2 |
| Hall | C5×C9 ⋊ #41 |
| Sylow | C9 ⋊ #713 |
| Sylow | C5 ⋊ #3701 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D45 |
| commutator, cosocle | C2×C5×C9 | C23 |
| Frattini | C2×C3 | C22×D15 |
| Fitting | C5×Q8×C9 | C2 |
| socle | C2×C3×C5 | C22×D3 |
| hypercenter | Q8 | D45 |
| hypocenter | C5×C9 | #41 |
| LCS | G ▷ C2×C5×C9 ▷ C5×C9 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
120 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
Polycyclic presentation:
2^4,3,5,3|b^2=d,c^2=d,e^3=g,c^a=cd,e^a=e^2g^2,f^a=f^4,g^a=g^2,c^b=cd120 conjugacy classes — view the full character table.
| maximal subgroup of | #13084271, #13084274, #13084279, … |
| maximal normal subgroup of | #13084271, #13084274, #13084279, … |
| maximal quotient of | #13084539, #13084541, #13084542, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 182);