Finite Group DatabasebetaFinite Group Databasebeta| order | 1764 |
| isoclinism | isoclinic class of D49⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | Dic49⋊C9 |
| unambiguous semidirect product | (C49⋊C9)⋊C4 |
| derived, Sylow | C49 ⋊ C4×C9 |
| Hall | C49⋊C9 ⋊ C4 |
| Hall | Dic49 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D49⋊C3 |
| commutator, hypocenter | C49 | C4×C9 |
| Frattini, socle, prime radical | C2×C3×C7 | AGL(1,𝔽7) |
| Fitting, cosocle | C2×C3×C49 | C2×C3 |
| LCS | G ▷ C49 | |
| UCS | C1 ◁ C2×C3 | |
27 conjugacy classes of subgroups.
| minimal degree | μ(G) = 62 |
| transitive degree | μt(G) = 1764 |
Polycyclic presentation:
2,3,2,3,7^2|a^2=c,b^3=d,e^7=f,e^a=e^6f^6,f^a=f^6,e^b=e^4f^2,f^b=f^484 conjugacy classes — view the full character table.
| maximal subgroup of | C11×Dic49⋊C9, C11⋊(Dic49⋊C9), C13×Dic49⋊C9, … |
| maximal normal subgroup of | C11×Dic49⋊C9, C13×Dic49⋊C9, C17×Dic49⋊C9, … |
| maximal quotient of | C11×Dic49⋊C9, C11⋊(Dic49⋊C9), C13×Dic49⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1764, 1);