Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C24×C34 |
| direct product | H(3,𝔽3)×C22⋊A4 |
| Fitting | C24×H(3,𝔽3) ⋊ C3 |
| Sylow | C24 ⋊ C3×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C32×C22⋊A4 |
| commutator, socle, cosocle | C24×C3 | C33 |
| Fitting | C24×H(3,𝔽3) | C3 |
| hypercenter | H(3,𝔽3) | C22⋊A4 |
| hypocenter | C24 | C3×H(3,𝔽3) |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C3 ◁ H(3,𝔽3) | |
612 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
88 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3056);