Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×#1313 ⋊ C2 |
| Sylow | C3 ⋊ #11405 |
282 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^8,3|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,b^a=bf,e^a=eh,c^b=ch,d^b=dh,i^c=i^2132 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#11405, #72255440, #72255444, … |
| maximal normal subgroup of | D3×#11405, #72255440, #72255444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 78571);