Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #34197⋊C3 |
| Fitting, Sylow | #34197 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C23 | C23×A4 |
| commutator | C2×Q8 | C22×C3×C4 |
| Fitting | #34197 | C3 |
| prime radical | C22×#41⋊C3 | C22 |
| cosocle | C22×Q8 | C23×C3 |
| hypercenter | #383 | A4 |
| hypocenter | Q8 | C3×#117 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C23 ◁ #383 | |
564 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 101 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=i,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bcgh,c^a=e,e^a=ceh,c^b=cg,d^b=dh,f^b=fg,e^c=eg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084189);