Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| Sylow | C3 ⋊ #12517 |
354 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 103 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|b^2=dfh,c^2=eg,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,i^b=i^2108 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#12517, #72225918, #72225920, … |
| maximal normal subgroup of | D3×#12517, #72225918, #72225920, … |
| maximal quotient of | #72226164, #72226165 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 84285);