Finite Group DatabasebetaFinite Group Databasebeta| order | 1323 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C32⋊(C49⋊C3) |
| unambiguous semidirect product | C49⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×C49⋊C3)⋊C3 |
| Fitting | C32×C49 ⋊ C3 |
| Sylow | C49 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center | C3 | C3×C49⋊C3 |
| commutator, cosocle | C3×C49 | C32 |
| Frattini, socle | C3×C7 | C3×ΓL(1,𝔽8) |
| Fitting | C32×C49 | C3 |
| hypercenter | C32 | C49⋊C3 |
| hypocenter | C49 | H(3,𝔽3) |
| LCS | G ▷ C3×C49 ▷ C49 | |
| UCS | C1 ◁ C3 ◁ C32 | |
33 conjugacy classes of subgroups.
| transitive degree | μt(G) = 441 |
Polycyclic presentation:
3^3,7^2|d^7=e,b^a=bc,d^a=d^4e^2,e^a=e^4155 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1323, 8);