Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of A4×ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| direct product | A4×ΓL(1,𝔽16) |
| unambiguous semidirect product | C3⋊(A4×AGL(1,𝔽5)) |
| unambiguous semidirect product | (C22×ΓL(1,𝔽16))⋊C3 |
| Fitting, socle, derived | C22×C3×C5 ⋊ C3×C4 |
| Sylow | C5 ⋊ Dic3×A4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C22×C3×C5 | C3×C4 |
| prime radical | C3×D5×A4 | C2 |
| cosocle | C22×C3×D5 | C2×C3 |
| LCS | G ▷ C22×C3×C5 | |
| UCS | C1 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 60 |
36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 790);