Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#1335 ⋊ C2 |
| Sylow | C3 ⋊ #12310 |
406 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^8,3|d^2=f,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,g^a=gh,i^a=i^2,d^b=df,e^b=eg,g^b=gh,i^b=i^2,d^c=dg,e^c=eg,f^c=fh,g^c=gh78 conjugacy classes — view the full character table.
| maximal subgroup of | #72214541, D3×#12310, #72215228, … |
| maximal normal subgroup of | D3×#12310, #72215228, #72215232, … |
| maximal quotient of | #72214541, C2×#10708582 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 82963);