Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C3 |
| direct product | ΓL(1,𝔽8)×Q8⋊C9 |
| unambiguous semidirect product | (Q8×ΓL(1,𝔽8))⋊C9 |
| derived, Hall | C7×Q8 ⋊ C3×C9 |
| Sylow | Q8 ⋊ C9×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×Q8⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | A4×ΓL(1,𝔽8) |
| commutator, hypocenter | C7×Q8 | C3×C9 |
| Fitting, cosocle | C3×C7×Q8 | C32 |
| socle | C2×C3×C7 | C3×A4 |
| prime radical | C2×C3×ΓL(1,𝔽8) | A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C2×C3 | |
72 conjugacy classes of subgroups.
| transitive degree | μt(G) = 504 |
105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 132);