Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊(#41⋊C9) |
| unambiguous semidirect product | #713⋊C9 |
| Hall | #713 ⋊ C9 |
| Sylow | C5 ⋊ #41⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | D5×A4 |
| commutator, hypocenter | C5×Q8 | C2×C9 |
| Fitting, cosocle | C3×C5×Q8 | C2×C3 |
| socle | C2×C3×C5 | C2×A4 |
| prime radical | C3×#713 | C3 |
| LCS | G ▷ C5×Q8 | |
| UCS | C1 ◁ C2×C3 | |
50 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
Polycyclic presentation:
2,3^2,2^2,5,2|a^2=g,b^3=c,d^2=g,e^2=g,f^a=f^4,d^b=de,e^b=d,e^d=eg84 conjugacy classes — view the full character table.
| maximal subgroup of | #13085140, #4133⋊C9, #13085146, … |
| maximal normal subgroup of | #13085140, #4133⋊C9, #13085146, … |
| maximal quotient of | Dic5×Q8⋊C9, C2×C5⋊(#41⋊C9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 108);