Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #47196⋊C3 |
| Fitting, Sylow | #47196 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #3568⋊C3 |
| commutator, cosocle, hypocenter | #2840 | C2×C3 |
| Frattini, socle | C23 | C2×C22⋊A4 |
| Fitting | #47196 | C3 |
| prime radical | #73292 | C2 |
| LCS | G ▷ #2840 | |
| UCS | C1 ◁ C2 | |
292 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 292 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.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=gi,c^2=ghi,d^2=g,b^a=dfgi,c^a=ceg,d^a=bcdef,e^a=fh,f^a=efgi,g^a=i,h^a=ghi,i^a=gi,c^b=cg,d^b=dh,f^b=fg,d^c=di,e^c=eg,f^c=fi,e^d=ei28 conjugacy classes — view the full character table.
| maximal subgroup of | #72284908, #72285368 |
| maximal normal subgroup of | #72284908, #72285368 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084464);