Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C32 |
| direct product | C23×C3⋊#272 |
| Fitting | C23×C3×H(3,𝔽3) ⋊ C2 |
| derived, Sylow | C3×H(3,𝔽3) ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | C3⋊(C3⋊D3) |
| commutator, cosocle, hypocenter | C3×H(3,𝔽3) | C24 |
| Frattini | C3 | C23×C3⋊(C3⋊D3) |
| Fitting | C23×C3×H(3,𝔽3) | C2 |
| socle | C23×C32 | C3⋊D3 |
| LCS | G ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C23×C3 | |
3162 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3076);