Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #12929 |
458 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 135 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:
2^8,3|a^2=f,b^2=fg,d^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,f^b=fh,g^b=gh,i^b=i^290 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#12929, #72219844, #72219845 |
| maximal normal subgroup of | D3×#12929, #72219844, #72219845 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 86124);