Finite Group DatabasebetaFinite Group Databasebeta| order | 1620 |
| isoclinism | isoclinic class of C81⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C81⋊AGL(1,𝔽5) |
| Fitting, derived, Hall | C5×C81 ⋊ C4 |
| Sylow | C81 ⋊ AGL(1,𝔽5) |
| Sylow | C5 ⋊ Dic81 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C5×C81 | C4 |
| Frattini | C27 | ΓL(1,𝔽16) |
| socle | C3×C5 | Dic27 |
| prime radical | C3×D5 | D27 |
| cosocle | D5×C81 | C2 |
| LCS | G ▷ C5×C81 | |
| UCS | C1 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 86 |
| transitive degree | μt(G) = 405 |
Polycyclic presentation:
2^2,3,5,3^3|a^2=b,c^3=e^2,e^3=f^2,f^3=g^2,c^a=c^2efg,d^a=d^2,e^a=e^2fg,f^a=f^2g,g^a=g^2,d^b=d^4165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1620, 6);