Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #12542⋊C3 |
| Fitting, Sylow | #12542 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | C42⋊C3 |
| commutator, hypocenter | #338 | C3×C4 |
| Frattini | C22×C8 | C2×A4 |
| Fitting | #12542 | C3 |
| socle | C2 | #1726⋊C3 |
| prime radical | #338⋊C3 | C4 |
| cosocle | #1747 | C2×C3 |
| LCS | G ▷ #338 | |
| UCS | C1 ◁ C16 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|b^2=e,c^2=f,d^2=g,e^2=h,h^2=i,c^a=dfi,d^a=cdehi,f^a=gi,g^a=fg,d^c=dh,g^c=gi,f^d=fi72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083560);