Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #20609⋊C3 |
| Fitting, Sylow | #20609 ⋊ C3 |
792 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 792 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
3,2^8|b^2=h,b^a=bcdfgh,c^a=cdeghi,d^a=cgh,f^a=fgi,g^a=fi,h^a=fghi,c^b=cf,d^b=dg,e^b=ei,f^b=fi,g^c=gi,h^c=hi,f^d=fi46 conjugacy classes — view the full character table.
| maximal subgroup of | #72241384, #72241398, #72244982, … |
| maximal normal subgroup of | #72241384, #72241398, #72245412, … |
| maximal quotient of | #72253796, C2×#20609⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083840);