Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #560⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #13830⋊C3 |
| Fitting, Sylow | #13830 ⋊ C3 |
| derived | C42 ⋊ C3×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #560⋊C3 |
| commutator, hypocenter | C42 | C3×C42 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | #13830 | C3 |
| prime radical | C22×C42⋊C3 | C22 |
| cosocle | C22×C42 | C22×C3 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C22 | |
296 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=i,d^2=fg,e^2=f,b^a=bg,c^a=ceg,d^a=eh,e^a=degh,f^a=g,g^a=fg,c^b=cf,d^b=dg,e^b=efg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083605);