Finite Group DatabasebetaFinite Group Databasebeta| order | 1620 |
| isoclinism | isoclinic class of (C3⋊ΓL(1,𝔽16))⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C3⋊ΓL(1,𝔽16))⋊C9 |
| Fitting, Hall | C5×C32⋊C9 ⋊ C4 |
| derived | C32×C5 ⋊ C4×C9 |
| Sylow | C32⋊C9 ⋊ AGL(1,𝔽5) |
| Sylow | C5 ⋊ (C3⋊Dic3)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | (C3⋊ΓL(1,𝔽16))⋊C3 |
| commutator, hypocenter | C32×C5 | C4×C9 |
| Frattini | C32 | C3×ΓL(1,𝔽16) |
| Fitting | C5×C32⋊C9 | C4 |
| socle | C32×C5 | C3×Dic3 |
| prime radical, cosocle | C33×D5 | C2×C3 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C3 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 90 |
Polycyclic presentation:
2,3,2,3^3,5|a^2=c,b^3=d,e^a=e^2,f^a=f^2,g^a=g^2,e^b=ef,g^c=g^493 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×(C3⋊D3)⋊C9, #72247383, #72247386, … |
| maximal normal subgroup of | AGL(1,𝔽5)×(C3⋊D3)⋊C9, #72247383, #72247386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1620, 82);