Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of A4×C7⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| direct product | A4×C7⋊AGL(1,𝔽5) |
| Fitting, socle, derived | C22×C5×C7 ⋊ C3×C4 |
| Hall | C22×C7⋊AGL(1,𝔽5) ⋊ C3 |
| Hall | C5×C7 ⋊ C4×A4 |
| Sylow | C7 ⋊ A4×AGL(1,𝔽5) |
| Sylow | C5 ⋊ A4×Dic7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C22×C5×C7 | C3×C4 |
| prime radical | C7×D5×A4 | C2 |
| cosocle | C22×C7×D5 | C2×C3 |
| LCS | G ▷ C22×C5×C7 | |
| UCS | C1 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 140 |
68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 948);