Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C27 · Schur multiplier (H2) = C1 |
| direct product | D3×Q8⋊C27 |
| unambiguous semidirect product | C3⋊(C2×Q8⋊C27) |
| unambiguous semidirect product | (D3×Q8)⋊C27 |
| derived | C3×Q8 ⋊ C2×C27 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C9 | D3×A4 |
| commutator, hypocenter | C3×Q8 | C2×C27 |
| Fitting, cosocle | C3×Q8×C9 | C2×C3 |
| socle | C2×C32 | C2×C22⋊C9 |
| prime radical | C2×C3×D3 | C22⋊C9 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C2×C9 | |
91 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
189 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 752);