Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of (C3⋊D3)⋊D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#67155 |
| derived, Hall | C32×C5 ⋊ C22×C4 |
| Sylow | C32 ⋊ C22×Dic5 |
| Sylow | C5 ⋊ C2×#624 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | (C3⋊D3)⋊D5 |
| commutator, hypocenter | C32×C5 | C22×C4 |
| Frattini | C2 | C2×(C3⋊D3)⋊D5 |
| Fitting, socle | C22×C32×C5 | C22 |
| prime radical | C22×C5×C3⋊D3 | C2 |
| cosocle | C2×C32×C5 | C23 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C22 | |
248 conjugacy classes of subgroups.
| transitive degree | μt(G) = 120 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | #13085590, #13085612, Dic5×#624, … |
| maximal normal subgroup of | #13085590, #13085612, Dic5×#624, … |
| maximal quotient of | #13087616, #13087619, #13087633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 700);