Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #7822⋊C3 |
| Fitting, Sylow | #7822 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #1655⋊C3 |
| commutator, cosocle, hypocenter | #2679 | C2×C3 |
| Frattini | C2×C42 | C2×A4 |
| Fitting | #7822 | C3 |
| socle | C23 | #120⋊C3 |
| prime radical | #2679⋊C3 | C2 |
| LCS | G ▷ #2679 | |
| UCS | C1 ◁ C2 | |
153 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
3,2^8|b^2=ghi,e^2=g,f^2=i,b^a=bg,c^a=dh,d^a=cdgh,e^a=f,f^a=efh,g^a=i,h^a=hi,i^a=gi,c^b=ce,d^b=df,e^b=eg,f^b=fi,e^c=eg,f^c=fh,e^d=eh,f^d=fi23 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083486);