Finite Group DatabasebetaFinite Group Databasebeta| order | 1260 |
| isoclinism | isoclinic class of C5⋊AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C7⋊C9)⋊Dic5 |
| unambiguous semidirect product | Dic35⋊C9 |
| derived, Hall | C5×C7 ⋊ C4×C9 |
| Hall | C5×C7⋊C9 ⋊ C4 |
| Hall | Dic35 ⋊ C9 |
| Hall | C7⋊C9 ⋊ Dic5 |
| Sylow | C7 ⋊ C9×Dic5 |
| Sylow | C5 ⋊ Dic7⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | C5⋊AGL(1,𝔽7) |
| commutator, hypocenter | C5×C7 | C4×C9 |
| Fitting, socle, prime radical, cosocle | C2×C3×C5×C7 | C2×C3 |
| LCS | G ▷ C5×C7 | |
| UCS | C1 ◁ C2×C3 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 1260 |
Polycyclic presentation:
2,3,2,3,5,7|a^2=c,b^3=d,e^a=e^4,f^a=f^6,f^b=f^2102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1260, 3);