Finite Group DatabasebetaFinite Group Databasebeta| order | 420 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | Aut(G) = C2×C4×AGL(1,𝔽7) (order 336) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C4×C5×ΓL(1,𝔽8) |
| unambiguous semidirect product | (C4×C7)⋊(C3×C5) |
| unambiguous semidirect product | (C4×C5×C7)⋊C3 |
| Fitting, Hall | C4×C5×C7 ⋊ C3 |
| derived, Sylow | C7 ⋊ C3×C4×C5 |
| Hall | C5×C7 ⋊ C3×C4 |
| Hall | C4×C7 ⋊ C3×C5 |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 140 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,3,4)(5,7,6)(8,9,10,11,12)(13,14,15,16);(1,2,3,5,4,6,7)100 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(420, 4);