Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of C3⋊(C3⋊AGL(1,𝔽7)) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C3⋊(ΓL(1,𝔽8)⋊C8)) |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C3⋊(C3⋊C8)) |
| derived | C32×C7 ⋊ C3×C8 |
| Hall | C32×ΓL(1,𝔽8) ⋊ C8 |
| Sylow | C7 ⋊ C3×C3⋊(C3⋊C8) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | C3⋊(C3⋊AGL(1,𝔽7)) |
| commutator, hypocenter | C32×C7 | C3×C8 |
| Fitting, cosocle | C32×C4×C7 | C2×C3 |
| socle | C2×C32×C7 | C3×C4 |
| prime radical | C2×C32×ΓL(1,𝔽8) | C4 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C4 | |
144 conjugacy classes of subgroups.
| transitive degree | μt(G) = 504 |
Polycyclic presentation:
2,3,2^2,3^2,7|a^2=c,c^2=d,e^a=e^2,f^a=f^2,g^a=g^6,g^b=g^2108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 427);