Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | C4⋊(#379⋊C3) |
| unambiguous semidirect product | #14680⋊C3 |
| Fitting, Sylow | #14680 ⋊ C3 |
1174 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 41 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=f,b^a=bh,d^a=deh,e^a=dgi,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 | #72341260, #72341265, #72341266, … |
| maximal normal subgroup of | #72341260, #72341265, #72341266, … |
| maximal quotient of | #72341331, #72341657, #72341661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083650);