Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #32874⋊C3 |
| Fitting, Sylow | #32874 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #73210 |
| commutator, cosocle, hypocenter | #543 | C22×C3 |
| Frattini | C24 | C22×A4 |
| Fitting | #32874 | C3 |
| hypercenter | C22 | #5450 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C2 ◁ C22 | |
193 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 193 conjugacy classes of subgroups — too many to draw. Only its 29 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|c^2=h,d^2=h,e^2=i,b^a=bg,c^a=di,d^a=cdhi,f^a=gi,g^a=fgi,c^b=cf,d^b=dg,h^b=hi,d^c=dh,e^c=efg,f^c=fi,g^c=gi,e^d=ef,g^d=gi36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083968);