Finite Group DatabasebetaFinite Group Databasebeta| order | 420 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊AGL(1,𝔽5) |
| automorphism | Aut(G) = AGL(1,𝔽5)×AGL(1,𝔽7) (order 840) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊AGL(1,𝔽5) |
| unambiguous semidirect product | (C7⋊AGL(1,𝔽5))⋊C3 |
| Fitting, socle, derived, Hall | C5×C7 ⋊ C3×C4 |
| Hall | C7⋊AGL(1,𝔽5) ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C4 |
| Hall | ΓL(1,𝔽8) ⋊ AGL(1,𝔽5) |
| Sylow | C7 ⋊ C3×AGL(1,𝔽5) |
| Sylow | C5 ⋊ ΓL(1,𝔽8)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C5×C7 | C3×C4 |
| prime radical | D5×ΓL(1,𝔽8) | C2 |
| cosocle | C7×D5 | C2×C3 |
| LCS | G ▷ C5×C7 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 35 |
A minimal faithful permutation representation — 2 generators on 12 points:
(2,3,4,5)(7,10,11,8,9,12);(1,2,5,3,4)(6,7,9,12,11,10,8)Polycyclic presentation:
2,3,2,5,7|a^2=c,d^a=d^3,e^a=e^6,e^b=e^2,d^c=d^419 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(420, 15);