Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Hall | C32×C5 ⋊ D8 |
| Sylow | C32 ⋊ #686 |
| Sylow | C5 ⋊ #3745 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #67140 |
| commutator, Fitting, cosocle | C32×C4×C5 | C22 |
| Frattini, hypercenter | C4 | D5×C3⋊D3 |
| socle | C2×C32×C5 | D4 |
| hypocenter | C32×C5 | D8 |
| LCS | G ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 ◁ C4 | |
132 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
Polycyclic presentation:
2^4,3^2,5|a^2=c,c^2=d,b^a=bc,g^a=g^4,c^b=cd,e^b=e^2,f^b=f^281 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#6654, #13085677, #13085749, … |
| maximal normal subgroup of | D3×#6654, #13085677, #13085749, … |
| maximal quotient of | #13084707, #13084712, #13084713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 228);