Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C33 |
| direct product | SL(2,𝔽3)×C3⋊(C3⋊D3) |
| unambiguous semidirect product | C3⋊(C3⋊D3×SL(2,𝔽3)) |
| Fitting, derived | C33×Q8 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×C3⋊(C3⋊D3) |
| commutator, Fitting, cosocle, hypocenter | C33×Q8 | C2×C3 |
| socle | C2×C33 | C2×A4 |
| LCS | G ▷ C33×Q8 | |
| UCS | C1 ◁ C2 | |
738 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3292);