Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, Sylow | #60308 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | C22⋊A4 |
| commutator, hypocenter | Q82 | C3×C4 |
| Frattini | C2×C4 | C2×C22⋊A4 |
| Fitting | #60308 | C3 |
| socle | C22 | C4×C22⋊A4 |
| prime radical | #85654 | C2 |
| cosocle | #3483 | C2×C3 |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C2×C8 | |
404 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 256 |
Polycyclic presentation:
3,2^8|b^2=h,f^2=h,h^2=i,b^a=cefh,c^a=bcdfhi,d^a=ei,e^a=deghi,c^b=cg,d^b=di,e^b=egi,d^c=dgi,e^c=ei,e^d=ei76 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084687);