Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #10391⋊C3 |
| Fitting, Sylow | #10391 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #393⋊C3 |
| commutator, cosocle, hypocenter | #2840 | C2×C3 |
| Frattini | C25 | C2×A4 |
| Fitting | #10391 | C3 |
| prime radical | #73293 | C2 |
| hypercenter | C23 | #120⋊C3 |
| LCS | G ▷ #2840 | |
| UCS | C1 ◁ C22 ◁ C23 | |
167 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 167 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
3,2^8|b^2=e,c^2=e,d^2=i,b^a=ci,c^a=bc,f^a=g,g^a=fgi,c^b=ce,d^b=df,f^b=fh,g^b=ghi,d^c=dg,f^c=fi,g^c=gh,e^d=eh36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083532);