Finite Group DatabasebetaFinite Group Databasebeta| order | 1620 |
| isoclinism | isoclinic class of C3⋊(C27⋊AGL(1,𝔽5)) |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C27⋊AGL(1,𝔽5)) |
| unambiguous semidirect product | C27⋊ΓL(1,𝔽16) |
| Fitting, derived, Hall | C3×C5×C27 ⋊ C4 |
| Sylow | C3×C27 ⋊ AGL(1,𝔽5) |
| Sylow | C5 ⋊ C3⋊Dic27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C3×C5×C27 | C4 |
| Frattini | C9 | C3⋊ΓL(1,𝔽16) |
| socle | C32×C5 | Dic9 |
| prime radical | C32×D5 | D9 |
| cosocle | C3×D5×C27 | C2 |
| LCS | G ▷ C3×C5×C27 | |
| UCS | C1 | |
84 conjugacy classes of subgroups.
| transitive degree | μt(G) = 405 |
Polycyclic presentation:
2^2,3^2,5,3^2|a^2=b,d^3=f^2,f^3=g^2,c^a=c^2,d^a=d^2fg,e^a=e^2,f^a=f^2g,g^a=g^2,e^b=e^4165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1620, 96);