Finite Group DatabasebetaFinite Group Databasebeta| order | 1323 |
| isoclinism | isoclinic class of C49⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C49⋊C27 |
| derived, Sylow | C49 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | C49⋊C3 |
| commutator, hypocenter | C49 | C27 |
| Frattini | C7×C9 | ΓL(1,𝔽8) |
| Fitting, cosocle | C9×C49 | C3 |
| socle, prime radical | C3×C7 | C7⋊C9 |
| LCS | G ▷ C49 | |
| UCS | C1 ◁ C9 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 76 |
| transitive degree | μt(G) = 1323 |
Polycyclic presentation:
3^3,7^2|a^3=b,b^3=c,d^7=e,d^a=d^4e^2,e^a=e^4171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1323, 1);