Finite Group DatabasebetaFinite Group Databasebeta| order | 1620 |
| isoclinism | isoclinic class of (C27⋊C3)⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C27⋊C3)⋊AGL(1,𝔽5) |
| unambiguous semidirect product | (C27⋊AGL(1,𝔽5))⋊C3 |
| Fitting, Hall | C5×C27⋊C3 ⋊ C4 |
| derived | C5×C27 ⋊ C3×C4 |
| Sylow | C27⋊C3 ⋊ AGL(1,𝔽5) |
| Sylow | C5 ⋊ (C27⋊C3)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C5×C27 | C3×C4 |
| Frattini | C9 | C3×ΓL(1,𝔽16) |
| Fitting | C5×C27⋊C3 | C4 |
| socle | C3×C5 | C3×Dic9 |
| prime radical | C32×D5 | D9 |
| cosocle | D5×C27 | C2×C3 |
| LCS | G ▷ C5×C27 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 135 |
Polycyclic presentation:
2,3,2,3,5,3^2|a^2=c,d^3=f,f^3=g,d^a=d^2f^2g^2,e^a=e^2,f^a=f^2g^2,g^a=g^2,d^b=dg,e^c=e^475 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×D27⋊C3, #72247132 |
| maximal normal subgroup of | AGL(1,𝔽5)×D27⋊C3, #72247132 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1620, 87);