Finite Group DatabasebetaFinite Group Databasebeta| order | 420 |
| isoclinism | isoclinic class of C5⋊AGL(1,𝔽7) |
| automorphism | Aut(G) = C2×AGL(1,𝔽5)×AGL(1,𝔽7) (order 1680) · Inn(G) = C5⋊AGL(1,𝔽7) · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊Dic5 |
| unambiguous semidirect product | Dic35⋊C3 |
| derived, Hall | C5×C7 ⋊ C3×C4 |
| Hall | Dic35 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C4 |
| Hall | ΓL(1,𝔽8) ⋊ Dic5 |
| Sylow | C7 ⋊ C3×Dic5 |
| Sylow | C5 ⋊ ΓL(1,𝔽8)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C5⋊AGL(1,𝔽7) |
| commutator, hypocenter | C5×C7 | C3×C4 |
| Fitting, socle, cosocle | C2×C5×C7 | C2×C3 |
| prime radical | C2×C5×ΓL(1,𝔽8) | C2 |
| LCS | G ▷ C5×C7 | |
| UCS | C1 ◁ C2 | |
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:
(2,5,6,3,4,7)(9,10)(11,12)(13,14,15,16);(1,2,4,7,6,5,3)(8,9,11,12,10)Polycyclic presentation:
2,3,2,5,7|a^2=c,d^a=d^4,e^a=e^6,e^b=e^234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(420, 3);