Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of #67356⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #67366⋊C4 |
| derived | C7×C9 ⋊ C2×C3×C4 |
| Hall | #4427 ⋊ C2×C4 |
| Sylow | C7 ⋊ C4×AGL(1,ℤ/9) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #67356⋊C2 |
| commutator, hypocenter | C7×C9 | C2×C3×C4 |
| Frattini | C2×C3 | D3×AGL(1,𝔽7) |
| Fitting, cosocle | C2×C7×C9 | C22×C3 |
| socle | C2×C3×C7 | C2×C3×D3 |
| prime radical | C2×#67366 | C2 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 ◁ C2 | |
102 conjugacy classes of subgroups.
| transitive degree | μt(G) = 252 |
Polycyclic presentation:
2^2,3,2,3,7,3|a^2=d,b^2=d,e^3=g,f^a=f^6,e^b=e^2g^2,g^b=g^2,e^c=eg,f^c=f^252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 105);