Finite Group DatabasebetaFinite Group Databasebeta| order | 420 |
| isoclinism | isoclinic class of C3⋊(C7⋊AGL(1,𝔽5)) |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(C7⋊AGL(1,𝔽5)) |
| unambiguous semidirect product | C7⋊ΓL(1,𝔽16) |
| Fitting, socle, derived, Hall | C3×C5×C7 ⋊ C4 |
| Hall | C5×C7 ⋊ Dic3 |
| Hall | C3×C7 ⋊ AGL(1,𝔽5) |
| Hall | C3×C5 ⋊ Dic7 |
| Sylow | C7 ⋊ ΓL(1,𝔽16) |
| Sylow | C5 ⋊ Dic21 |
| Sylow | C3 ⋊ C7⋊AGL(1,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C5×C7 | C4 |
| prime radical, cosocle | C3×C7×D5 | C2 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 105 |
A minimal faithful permutation representation — 2 generators on 15 points:
(2,3)(4,5)(6,7)(9,10,11,12)(14,15);(1,2,4,6,7,5,3)(8,9,10,12,11)(13,14,15)Polycyclic presentation:
2^2,3,5,7|a^2=b,c^a=c^2,d^a=d^2,e^a=e^6,d^b=d^445 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(420, 23);