Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of A4×D15 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| direct product | A4×Dic15 |
| unambiguous semidirect product | C3⋊(A4×Dic5) |
| unambiguous semidirect product | (C22×Dic15)⋊C3 |
| derived | C22×C3×C5 ⋊ C3×C4 |
| Sylow | C5 ⋊ Dic3×A4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×D15 |
| commutator, hypocenter | C22×C3×C5 | C3×C4 |
| Fitting, socle, cosocle | C23×C3×C5 | C2×C3 |
| prime radical | C2×C3×C5×A4 | C2 |
| LCS | G ▷ C22×C3×C5 | |
| UCS | C1 ◁ C2 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 180 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | #13088554, S4×Dic15, #13088556, … |
| maximal normal subgroup of | #13088554, S4×Dic15, #13088556, … |
| maximal quotient of | A4×C3⋊#168, SL(2,𝔽3)×Dic15, C3⋊(#4094⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 546);