Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C32×C5 ⋊ C22⋊C4 |
| Sylow | C32 ⋊ C22⋊Dic5 |
| Sylow | C5 ⋊ #3722 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | (C3⋊D3)⋊D5 |
| commutator | C2×C32×C5 | C2×C4 |
| Fitting, socle, cosocle | C22×C32×C5 | C22 |
| prime radical | C22×C5×C3⋊D3 | C2 |
| hypocenter | C32×C5 | C22⋊C4 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C22 | |
168 conjugacy classes of subgroups.
| transitive degree | μt(G) = 120 |
Polycyclic presentation:
2^4,3^2,5|a^2=c,b^a=bd,e^a=e^2,g^a=g^4,e^b=e^2,f^b=f^278 conjugacy classes — view the full character table.
| maximal subgroup of | #13085592, #13085613, D5×#3722, … |
| maximal normal subgroup of | #13085592, #13085613, D5×#3722, … |
| maximal quotient of | #13084890, #13084892, #13084894, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 299);