Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of C162⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C16 |
| unambiguous semidirect product | C162⋊C3 |
| Fitting, derived, Sylow | C162 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C162 | C3 |
| Frattini | C82 | A4 |
| socle | C22 | C82⋊C3 |
| LCS | G ▷ C162 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=d,c^2=e,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bcdefgi,c^a=bhi,d^a=defghi,e^a=d,f^a=fghi,g^a=f,h^a=hi,i^a=h88 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083477);