Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D3×D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C22×C3×D3×D5 |
| unambiguous semidirect product | (C5×D3)⋊(C23×C3) |
| unambiguous semidirect product | (C3×D5)⋊(C23×C3) |
| unambiguous semidirect product | D15⋊(C23×C3) |
| Fitting, socle | C22×C32×C5 ⋊ C22 |
| derived | C3×C5 ⋊ C24×C3 |
| Hall | C32×C5 ⋊ C24 |
| Sylow | C32 ⋊ C23×D5 |
| Sylow | C5 ⋊ C23×C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | D3×D5 |
| commutator, cosocle, hypocenter | C3×C5 | C24×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32×C5 | C22 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C22×C3 | |
568 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 120 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | D5×(C2×D3)⋊Dic3, D3×#6665, #13085856, … |
| maximal normal subgroup of | D5×(C2×D3)⋊Dic3, D3×#6665, #13085856, … |
| maximal quotient of | C2×C3×D5×Dic6, C2×C3×#6765, C2×C3×#6766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 828);