Finite Group DatabasebetaFinite Group Databasebeta| order | 1764 |
| isoclinism | isoclinic class of #5997⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #5997⋊C9 |
| derived, Sylow | C72 ⋊ C4×C9 |
| Hall | #89204 ⋊ C4 |
| Hall | #5997 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | #5997⋊C3 |
| commutator, hypocenter | C72 | C4×C9 |
| Fitting, socle | C3×C72 | C3×C4 |
| prime radical, cosocle | C3×C7⋊D7 | C2×C3 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C3 | |
42 conjugacy classes of subgroups.
| transitive degree | μt(G) = 126 |
Polycyclic presentation:
2,3,2,3,7^2|a^2=c,b^3=d,e^a=e^5f^3,f^a=e^3f^2,e^b=e^2,f^b=f^2,e^c=e^6,f^c=f^648 conjugacy classes — view the full character table.
| maximal subgroup of | C11×#5997⋊C9, C11⋊(#5997⋊C9), C13×#5997⋊C9, … |
| maximal normal subgroup of | C11×#5997⋊C9, C13×#5997⋊C9, C17×#5997⋊C9, … |
| maximal quotient of | C11×#5997⋊C9, C11⋊(#5997⋊C9), C13×#5997⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1764, 65);