Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #4246⋊C9 |
| Hall | C3×C7⋊C9 ⋊ D4 |
| Sylow | C7 ⋊ C9×#67 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | D3×AGL(1,𝔽7) |
| commutator | C2×C3×C7 | C22×C9 |
| Fitting, socle, cosocle | C2×C32×C7 | C22×C3 |
| prime radical | C3×#4246 | C3 |
| hypocenter | C3×C7 | D4×C9 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2×C3 | |
116 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 252 |
Polycyclic presentation:
2^2,3,2,3^2,7|b^2=d,c^3=e,b^a=bd,g^a=g^6,f^b=f^2,g^c=g^299 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 122);