Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #12680⋊C3 |
| Fitting, Sylow | #12680 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | #120⋊C3 |
| commutator, hypocenter | C22×Q8 | C3×C8 |
| Frattini | C22×C8 | C2×A4 |
| Fitting | #12680 | C3 |
| socle | C23 | C8×A4 |
| prime radical | #5948 | C4 |
| cosocle | C22×#131 | C2×C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C8 | |
193 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 193 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
3,2^8|b^2=g,g^2=h,h^2=i,b^a=be,c^a=cdfhi,d^a=ce,e^a=ef,f^a=e,c^b=ce,d^b=df,d^c=di72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083562);