Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #12539⋊C3 |
| Fitting, Sylow | #12539 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | C42⋊C3 |
| commutator, hypocenter | #95 | C3×C8 |
| Frattini | C22×C8 | C2×A4 |
| Fitting | #12539 | C3 |
| socle | C23 | C8×A4 |
| prime radical | #95⋊C3 | C8 |
| cosocle | #1726 | C2×C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C16 | |
103 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
3,2^8|b^2=e,c^2=f,d^2=g,e^2=h,h^2=i,b^a=bgi,c^a=cdf,d^a=cfi,f^a=fgi,g^a=f,d^c=di96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083559);