Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #12902⋊C3 |
| Fitting, Sylow | #12902 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | #127⋊C3 |
| commutator, hypocenter | #338 | C3×C4 |
| Frattini | C22×C8 | C2×A4 |
| Fitting | #12902 | C3 |
| socle | C2 | #1843⋊C3 |
| prime radical | #338⋊C3 | C4 |
| cosocle | #1747 | C2×C3 |
| LCS | G ▷ #338 | |
| UCS | C1 ◁ C8 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|b^2=g,c^2=e,d^2=f,g^2=h,h^2=i,c^a=cdegh,d^a=ce,e^a=efi,f^a=e,c^b=ce,d^b=df,d^c=dh,f^c=fi,e^d=ei60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083572);