Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #33496⋊C3 |
| Fitting, Sylow | #33496 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C22×A4 |
| commutator, hypocenter | Q8 | C2×C3×C16 |
| Fitting | #33496 | C3 |
| socle | C22 | C2×C8×A4 |
| prime radical | C22×SL(2,𝔽3) | C8 |
| cosocle | C8×Q8 | C22×C3 |
| hypercenter | C22⋊C16 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C8 ◁ C22⋊C16 | |
166 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
3,2^8|b^2=g,c^2=f,d^2=f,g^2=h,h^2=i,b^a=bcdi,c^a=dfi,d^a=cde,c^b=cf,e^b=ef,d^c=df152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084083);