Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #20579⋊C3 |
| Fitting, Sylow | #20579 ⋊ C3 |
1075 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 1075 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
3,2^8|b^2=h,h^2=i,b^a=bfgi,c^a=cdi,d^a=cef,e^a=eg,f^a=fg,g^a=fi,c^b=cf,d^b=dg,e^b=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72233415, #72233784, #72252249, … |
| maximal normal subgroup of | #72233784, #72252249, #72252250, … |
| maximal quotient of | C2×#20579⋊C3, #72341931, #72342113 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083836);