Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #60087⋊C3 |
| Fitting, Sylow | #60087 ⋊ C3 |
1094 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 1094 conjugacy classes of subgroups — too many to draw. Only its 117 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:
3,2^8|b^2=h,g^2=i,b^a=bdghi,c^a=cefg,d^a=efhi,e^a=degh,c^b=cg,f^b=fi,g^b=gi,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72225973, #72226110, #72226173, … |
| maximal normal subgroup of | #72226110, #72226173, #72226271, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084602);