Finite Group DatabasebetaFinite Group Databasebeta| order | 420 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | Aut(G) = C2×C4×AGL(1,𝔽7) (order 336) · Inn(G) = AGL(1,𝔽7) · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2 |
| Fitting, socle | C2×C5×C7 ⋊ C2×C3 |
| derived, Sylow | C7 ⋊ C22×C3×C5 |
| Hall | C2×C5×D7 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C22 |
| Hall | C5×C7 ⋊ C22×C3 |
| Hall | C2×D7 ⋊ C3×C5 |
| Hall | ΓL(1,𝔽8) ⋊ C22×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C7 | C22×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C2×C5×C7 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C5 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 70 |
A minimal faithful permutation representation — 2 generators on 14 points:
(3,4,5,6,7)(9,12,13,10,11,14);(1,2)(8,9,11,14,13,12,10)70 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(420, 17);