Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C3×#361 ⋊ C2 |
| Sylow | C3 ⋊ #13310 |
552 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 93 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|d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gh,i^a=i^2,c^b=cf,d^b=dg,e^b=eh,g^b=gh,e^c=eh,i^c=i^266 conjugacy classes — view the full character table.
| maximal subgroup of | #72270233, #72270252, #72270254, … |
| maximal normal subgroup of | #72270233, #72270252, #72270254, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 88302);