Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D45 |
| automorphism | ? |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C9⋊#672 |
| derived, Hall | C5×C9 ⋊ C16 |
| Sylow | C9 ⋊ #672 |
| Sylow | C5 ⋊ C9⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | D45 |
| commutator, hypocenter | C5×C9 | C16 |
| Frattini | C3×C8 | D15 |
| Fitting, cosocle | C5×C8×C9 | C2 |
| socle, prime radical | C2×C3×C5 | C3⋊C8 |
| LCS | G ▷ C5×C9 | |
| UCS | C1 ◁ C8 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 720 |
Polycyclic presentation:
2^4,3,5,3|a^2=b,b^2=c,c^2=d,e^3=g,e^a=e^2g^2,f^a=f^4,g^a=g^2192 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#672, #13083692, D5×C9⋊C16, … |
| maximal normal subgroup of | D9×#672, #13083692, D5×C9⋊C16, … |
| maximal quotient of | C9⋊#3926, C2×C9⋊#672 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 3);