Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #11391⋊C3 |
| Fitting, Sylow | #11391 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #120⋊C3 |
| commutator, hypocenter | #543 | C3×C4 |
| Frattini | C23×C4 | C2×A4 |
| Fitting | #11391 | C3 |
| socle | C22 | #379⋊C3 |
| prime radical | #543⋊C3 | C4 |
| cosocle | #3458 | C2×C3 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C2×C4 | |
192 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 22 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,c^2=g,d^2=g,h^2=i,b^a=bcdefgh,c^a=cdefh,d^a=ceghi,e^a=ef,f^a=eg,h^a=efgh,c^b=ce,d^b=df,e^b=ei,d^c=dg,f^c=fi,h^c=hi,e^d=ei60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083555);