Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D3×S5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | D3×S5 |
| unambiguous semidirect product | (C3×S5)⋊C2 |
| unambiguous semidirect product | ΓL(2,𝔽4)⋊C2 |
| unambiguous semidirect product | (D3×A5)⋊C2 |
| Fitting | C3 ⋊ C2×S5 |
| socle, derived | C3×A5 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | C3×A5 | C22 |
| Fitting | C3 | C2×S5 |
| LCS | G ▷ C3×A5 | |
| UCS | C1 | |
121 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 121 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 15 |
21 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 767);