Finite Group DatabasebetaFinite Group Databasebeta| order | 1620 |
| isoclinism | isoclinic class of C3⋊(C3⋊#4380) |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C3⋊(C3⋊#4380) |
| Fitting, socle, derived, Hall | C34×C5 ⋊ C4 |
| Sylow | C34 ⋊ AGL(1,𝔽5) |
| Sylow | C5 ⋊ C3⋊(C3⋊#157) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C34×C5 | C4 |
| prime radical, cosocle | C32×C3⋊D15 | C2 |
| LCS | G ▷ C34×C5 | |
| UCS | C1 | |
208 conjugacy classes of subgroups.
| transitive degree | μt(G) = 180 |
Polycyclic presentation:
2^2,3^4,5|a^2=b,c^a=cd^2,d^a=c^2d^2,e^a=e^2,f^a=f^2,g^a=g^2,c^b=c^2,d^b=d^2,g^b=g^4111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1620, 439);