Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C32×C5 ⋊ Q16 |
| Sylow | C32 ⋊ Dic20 |
| Sylow | C5 ⋊ #3776 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #157⋊D5 |
| commutator, cosocle | C5×C3⋊Dic3 | C22 |
| Fitting, socle, prime radical | C2×C32×C5 | D4 |
| hypocenter | C32×C5 | Q16 |
| LCS | G ▷ C5×C3⋊Dic3 ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 240 |
Polycyclic presentation:
2^4,3^2,5|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=e^2f,g^a=g^4,e^b=e^2f^2,f^b=e^2f,e^c=e^2,f^c=f^239 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#3776, #13088465, #13088466, … |
| maximal normal subgroup of | D5×#3776, #13088465, #13088466, … |
| maximal quotient of | #13085615, #13085618, C2×#10624014 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 483);