Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C3⋊D15 |
| automorphism | ? |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C3⋊#672) |
| derived, Hall | C32×C5 ⋊ C16 |
| Sylow | C32 ⋊ #672 |
| Sylow | C5 ⋊ C3⋊(C3⋊C16) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | C3⋊D15 |
| commutator, hypocenter | C32×C5 | C16 |
| Fitting, cosocle | C32×C5×C8 | C2 |
| socle, prime radical | C2×C32×C5 | C8 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C8 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 720 |
Polycyclic presentation:
2^4,3^2,5|a^2=b,b^2=c,c^2=d,e^a=e^2,f^a=f^2,g^a=g^4192 conjugacy classes — view the full character table.
| maximal subgroup of | #13084631, D5×C3⋊(C3⋊C16), C3⋊D3×#672, … |
| maximal normal subgroup of | #13084631, D5×C3⋊(C3⋊C16), C3⋊D3×#672, … |
| maximal quotient of | C3⋊(C3⋊#3926), C2×C3⋊(C3⋊#672) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 93);