Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of D7×D27 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D189⋊C4 |
| derived, Hall | C7×C27 ⋊ C2×C4 |
| Sylow | C27 ⋊ C4×D7 |
| Sylow | C7 ⋊ C4×D27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D7×D27 |
| commutator, hypocenter | C7×C27 | C2×C4 |
| Frattini | C2×C9 | D3×D7 |
| Fitting, cosocle | C2×C7×C27 | C22 |
| socle | C2×C3×C7 | C2×D9 |
| prime radical | C2×D189 | C2 |
| LCS | G ▷ C7×C27 | |
| UCS | C1 ◁ C2 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 756 |
Polycyclic presentation:
2^3,3,7,3^2|a^2=c,b^2=c,d^3=f,f^3=g,e^a=e^6,d^b=d^2f^2g^2,f^b=f^2g^2,g^b=g^2150 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 14);