Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #14693⋊C3 |
| Fitting, Sylow | #14693 ⋊ C3 |
1174 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 39 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=i,c^2=i,b^a=bgh,c^a=ch,d^a=defhi,e^a=dfgi,g^a=gh,h^a=g,c^b=cf,d^b=dg,e^b=eh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72243123, C4×AΣL(1,𝔽64), #72243707, … |
| maximal normal subgroup of | C4×AΣL(1,𝔽64), #72243707, #72243948, … |
| maximal quotient of | #72243123, #72248076, #72341331, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083655);