Finite Group DatabasebetaFinite Group Databasebeta| order | 1260 |
| isoclinism | isoclinic class of C3⋊(C7⋊AGL(1,𝔽5)) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C3×C3⋊(C7⋊AGL(1,𝔽5)) |
| unambiguous semidirect product | C3⋊(C3×C7⋊AGL(1,𝔽5)) |
| Fitting, socle, Hall | C32×C5×C7 ⋊ C4 |
| derived | C3×C5×C7 ⋊ C3×C4 |
| Hall | C32×C7 ⋊ AGL(1,𝔽5) |
| Hall | C32×C5 ⋊ Dic7 |
| Hall | C5×C7 ⋊ C3×Dic3 |
| Sylow | C32 ⋊ C7⋊AGL(1,𝔽5) |
| Sylow | C7 ⋊ C3×ΓL(1,𝔽16) |
| Sylow | C5 ⋊ C3×Dic21 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊(C7⋊AGL(1,𝔽5)) |
| commutator, hypocenter | C3×C5×C7 | C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C32×C5×C7 | C4 |
| prime radical | C32×C7×D5 | C2 |
| cosocle | C3×C7×D5 | C2×C3 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 ◁ C3 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 210 |
135 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1260, 85);