Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of (C3⋊D3)⋊D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Hall | C32×C5 ⋊ C2×C8 |
| Sylow | C32 ⋊ C2×#168 |
| Sylow | C5 ⋊ #3710 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | (C3⋊D3)⋊D5 |
| commutator, hypocenter | C32×C5 | C2×C8 |
| Fitting, cosocle | C32×C4×C5 | C22 |
| socle | C2×C32×C5 | C2×C4 |
| prime radical | C2×C5×C3⋊D3 | C4 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C4 | |
104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 120 |
Polycyclic presentation:
2^4,3^2,5|a^2=c,c^2=d,e^a=e^2,g^a=g^4,e^b=e^2,f^b=f^284 conjugacy classes — view the full character table.
| maximal subgroup of | D32⋊#168, #13085579, #13085581, … |
| maximal normal subgroup of | D32⋊#168, #13085579, #13085581, … |
| maximal quotient of | #13084875, #13084877, #13084886, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 287);