Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of A4⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×A4⋊AGL(1,𝔽5) |
| unambiguous semidirect product | A4⋊(C3×AGL(1,𝔽5)) |
| Fitting, socle | C22×C3×C5 ⋊ Dic3 |
| derived | C5×A4 ⋊ C3×C4 |
| Sylow | C5 ⋊ C3×SL(2,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | A4⋊AGL(1,𝔽5) |
| commutator, hypocenter | C5×A4 | C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C5 | Dic3 |
| prime radical | C3×D5×A4 | C2 |
| cosocle | D5×A4 | C2×C3 |
| LCS | G ▷ C5×A4 | |
| UCS | C1 ◁ C3 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 60 |
42 conjugacy classes — view the full character table.
| maximal subgroup of | C3×AGL(1,𝔽5)×S4, ΓL(1,𝔽16)⋊S4, D3×A4⋊AGL(1,𝔽5), … |
| maximal normal subgroup of | C3×AGL(1,𝔽5)×S4, ΓL(1,𝔽16)⋊S4, D3×A4⋊AGL(1,𝔽5), … |
| maximal quotient of | C3×#91885, C3×#91886, C3×A4⋊#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 787);