Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D3×A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×Dic3×A4 |
| unambiguous semidirect product | C3⋊(C4×C5×A4) |
| unambiguous semidirect product | (C22×Dic3)⋊(C3×C5) |
| unambiguous semidirect product | (C22×C5×Dic3)⋊C3 |
| derived | C22×C3 ⋊ C3×C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D3×A4 |
| commutator, hypocenter | C22×C3 | C3×C4×C5 |
| Frattini | C2 | C5×D3×A4 |
| Fitting, socle | C23×C3×C5 | C2×C3 |
| prime radical | C2×C3×C5×A4 | C2 |
| cosocle | C23×C3 | C2×C3×C5 |
| LCS | G ▷ C22×C3 | |
| UCS | C1 ◁ C2×C5 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 180 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | C5×#64204, C5×Dic3×S4, C5×(C3⋊S4)⋊C4, … |
| maximal normal subgroup of | C5×#64204, C5×Dic3×S4, C5×(C3⋊S4)⋊C4, … |
| maximal quotient of | C5×A4×C3⋊C8, C5×Dic3×SL(2,𝔽3), C5×C3⋊(#131⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 543);