Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #55768⋊C3 |
| Fitting, Sylow | #55768 ⋊ C3 |
| derived | #2810 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #3556⋊C3 |
| commutator, cosocle, hypocenter | #2810 | C2×C3 |
| Frattini, socle | C23 | C2×C22⋊A4 |
| Fitting | #55768 | C3 |
| LCS | G ▷ #2810 | |
| UCS | C1 ◁ C2 | |
275 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 275 conjugacy classes of subgroups — too many to draw. Only its 21 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=hi,c^2=gi,e^2=h,f^2=g,b^a=bdegh,c^a=efg,d^a=dh,e^a=bdgi,f^a=bcdefghi,g^a=gi,h^a=i,i^a=hi,c^b=cg,d^b=dh,e^b=eh,f^b=fg,d^c=di,e^c=eg,f^c=fhi,e^d=ehi26 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084538);