Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C2×C32×C4×C5 ⋊ C2 |
| Hall | C32×C5 ⋊ #41 |
| Sylow | C32 ⋊ #709 |
| Sylow | C5 ⋊ #3828 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×C3⋊D15 |
| commutator, socle, cosocle | C2×C32×C5 | C23 |
| Frattini | C2 | C22×C3⋊D15 |
| Fitting | C2×C32×C4×C5 | C2 |
| hypercenter | C2×C4 | C3⋊D15 |
| hypocenter | C32×C5 | #41 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
240 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
Polycyclic presentation:
2^4,3^2,5|b^2=d,c^a=cd,e^a=e^2,f^a=f^2,g^a=g^4186 conjugacy classes — view the full character table.
| maximal subgroup of | #13084748, #13084749, #13084856, … |
| maximal normal subgroup of | #13084748, #13084749, #13084856, … |
| maximal quotient of | C4×#67180, #13088139, #13088141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 752);