Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #21481⋊C3 |
| Fitting, Sylow | #21481 ⋊ C3 |
324 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|b^2=h,e^2=h,b^a=bf,c^a=dh,d^a=cdhi,f^a=gi,g^a=fg,c^b=cf,d^b=dg,e^b=ei,g^c=gi,f^d=fi46 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#3026⋊C3, #72253616, #72253620, … |
| maximal normal subgroup of | C4×#3026⋊C3, #72253616, #72253620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083858);