Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #11322⋊C3 |
| Fitting, Sylow | #11322 ⋊ C3 |
| derived | #95 ⋊ C3×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #127⋊C3 |
| commutator, hypocenter | #95 | C3×C8 |
| Frattini | C23×C4 | C2×A4 |
| Fitting | #11322 | C3 |
| socle | C24 | C4×A4 |
| prime radical | C2×#95⋊C3 | C4 |
| cosocle | C4×#95 | C2×C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C2×C4 | |
203 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 203 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=e,d^2=f,h^2=i,b^a=bdefg,c^a=defi,d^a=cdegi,e^a=f,f^a=efg,c^b=ce,d^b=df,d^c=dg72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083550);