Finite Group DatabasebetaFinite Group Databasebeta| order | 1620 |
| isoclinism | isoclinic class of C3⋊D135 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊Dic135 |
| unambiguous semidirect product | C27⋊Dic15 |
| derived, Hall | C3×C5×C27 ⋊ C4 |
| Sylow | C3×C27 ⋊ Dic5 |
| Sylow | C5 ⋊ C3⋊Dic27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊D135 |
| commutator, hypocenter | C3×C5×C27 | C4 |
| Frattini | C2×C9 | C3⋊D15 |
| Fitting, cosocle | C2×C3×C5×C27 | C2 |
| socle, prime radical | C2×C32×C5 | D9 |
| LCS | G ▷ C3×C5×C27 | |
| UCS | C1 ◁ C2 | |
84 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1620 |
Polycyclic presentation:
2^2,3^2,5,3^2|a^2=b,d^3=f,f^3=g,c^a=c^2,d^a=d^2f^2g^2,e^a=e^4,f^a=f^2g^2,g^a=g^2408 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1620, 57);