Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #33974⋊C3 |
| Fitting, Sylow | #33974 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C16 | C22×A4 |
| commutator, hypocenter | Q8 | C2×C3×C16 |
| Fitting | #33974 | C3 |
| socle | C2 | C2×A4×C16 |
| prime radical | C2×#41⋊C3 | C8 |
| cosocle | #504 | C22×C3 |
| hypercenter | M64 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C16 ◁ M64 | |
127 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 127 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
3,2^8|b^2=f,f^2=g,g^2=h,h^2=i,b^a=bcdei,c^a=dei,d^a=cdhi,e^b=ei,d^c=di152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084104);