Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊#3689 |
| derived | C5×Q8⋊C9 ⋊ C2 |
| Sylow | C5 ⋊ #3689 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C5⋊((C22⋊C9)⋊C2) |
| commutator, cosocle, hypocenter | C5×Q8⋊C9 | C2 |
| Frattini | C2×C3 | C5⋊S4 |
| Fitting | C3×C5×Q8 | D3 |
| socle | C2×C3×C5 | S4 |
| LCS | G ▷ C5×Q8⋊C9 | |
| UCS | C1 ◁ C2 | |
50 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
Polycyclic presentation:
2,3,5,3,2^3|b^3=d,e^2=g,f^2=g,b^a=b^2d^2,c^a=c^4,d^a=d^2,e^a=fg,f^a=eg,e^b=f,f^b=efg,f^e=fg57 conjugacy classes — view the full character table.
| maximal subgroup of | #13085148, #13085149, #13085152, … |
| maximal normal subgroup of | #13085148, #13085149, #13085152, … |
| maximal quotient of | #13083943, C2×C5⋊#3689 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 106);