Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #15210⋊C3 |
| Fitting, Sylow | #15210 ⋊ C3 |
231 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 29 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:
3,2^8|b^2=fgi,c^2=fg,d^2=f,e^2=hi,b^a=bcg,c^a=dg,d^a=cdfg,f^a=gh,g^a=fgh,c^b=cf,d^b=dg,e^b=ei,d^c=dh48 conjugacy classes — view the full character table.
| maximal subgroup of | #72222701, #72222837, #72222847, … |
| maximal normal subgroup of | #72222701, #72222837, #72222847 |
| maximal quotient of | #72222699, #72299986 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083688);