Finite Group DatabasebetaFinite Group Databasebeta| order | 1134 |
| isoclinism | isoclinic class of C3⋊D189 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊D189 |
| unambiguous semidirect product | C7⋊(C3⋊D27) |
| unambiguous semidirect product | C27⋊D21 |
| Fitting, derived, Hall | C3×C7×C27 ⋊ C2 |
| Sylow | C3×C27 ⋊ D7 |
| Sylow | C7 ⋊ C3⋊D27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×C7×C27 | C2 |
| Frattini | C9 | C3⋊D21 |
| socle | C32×C7 | D9 |
| LCS | G ▷ C3×C7×C27 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 567 |
Polycyclic presentation:
2,3^2,7,3^2|c^3=e^2,e^3=f^2,b^a=b^2,c^a=c^2ef,d^a=d^6,e^a=e^2f,f^a=f^2285 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1134, 123);