Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #60109⋊C3 |
| Fitting, Sylow | #60109 ⋊ C3 |
624 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 624 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) = 128 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=g,g^2=i,b^a=bdgh,c^a=cefgi,d^a=defgi,e^a=dfhi,c^b=cg,g^b=gi,f^c=fi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72253048 |
| maximal normal subgroup of | #72253048 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084622);