Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Hall | C32×C5 ⋊ M16 |
| Sylow | C32 ⋊ #676 |
| Sylow | C5 ⋊ #3792 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×#157 |
| commutator, Fitting, socle | C2×C32×C5 | C2×C4 |
| prime radical | C2×C32×D5 | C4 |
| cosocle | C5×C3⋊Dic3 | C22 |
| hypocenter | C32×C5 | M16 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
72 conjugacy classes of subgroups.
| transitive degree | μt(G) = 120 |
Polycyclic presentation:
2^4,3^2,5|b^2=c,c^2=d,b^a=bd,g^a=g^4,e^b=f,f^b=e^2,e^c=e^2,f^c=f^242 conjugacy classes — view the full character table.
| maximal subgroup of | #13088472, #13088473, #13088476, … |
| maximal normal subgroup of | #13088472, #13088473, #13088476, … |
| maximal quotient of | #13085520, #13085521, #13085528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 458);