Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #30236⋊C3 |
| Fitting, Sylow | #30236 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2644⋊C3 |
| commutator, cosocle, hypocenter | #338 | C22×C3 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | #30236 | C3 |
| prime radical | #2069⋊C3 | C2 |
| hypercenter | C8 | #126⋊C3 |
| LCS | G ▷ #338 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
116 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
3,2^8|b^2=gh,c^2=g,e^2=f,f^2=i,h^2=i,b^a=ce,c^a=bcfhi,d^a=dfgh,g^a=fh,h^a=ghi,c^b=cf,d^b=dg,g^b=gi,h^b=hi,d^c=dh,h^c=hi,e^d=ef,f^d=fi,h^d=hi30 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083939);