Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #17316⋊C3 |
| Fitting, Sylow | #17316 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C22×A4 |
| commutator, hypocenter | Q8 | C3×C4×C8 |
| Fitting | #17316 | C3 |
| socle | C22 | C42×A4 |
| prime radical | C2×#41⋊C3 | C2×C4 |
| cosocle | C4×#41 | C22×C3 |
| hypercenter | #322 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C42 ◁ #322 | |
240 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 73 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=f,c^2=g,f^2=h,g^2=i,d^a=defh,e^a=dh,c^b=ch,e^d=eh152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083729);