Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of C3⋊(C7⋊AGL(1,𝔽9)) |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(C7⋊AGL(1,𝔽9)) |
| unambiguous semidirect product | C7⋊(C3⋊AGL(1,𝔽9)) |
| Fitting, socle, derived, Hall | C33×C7 ⋊ C8 |
| Sylow | C33 ⋊ C7⋊C8 |
| Sylow | C7 ⋊ C3⋊AGL(1,𝔽9) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C33×C7 | C8 |
| prime radical | C3×C7×C3⋊D3 | C4 |
| cosocle | C3×C7×#157 | C2 |
| LCS | G ▷ C33×C7 | |
| UCS | C1 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2^3,3^3,7|a^2=b,b^2=c,d^a=e,e^a=de^2,f^a=f^2,g^a=g^6,d^b=de^2,e^b=d^2e^2,d^c=d^2,e^c=e^269 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 812);