Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #21442⋊C3 |
| Fitting, Sylow | #21442 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | #120⋊C3 |
| commutator, hypocenter | C22×Q8 | C2×C3×C4 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | #21442 | C3 |
| socle | C23 | C2×C4×A4 |
| prime radical | #2388⋊C3 | C2 |
| cosocle | C22×#41 | C22×C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C8 | |
311 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=h,e^2=h,h^2=i,b^a=bfg,c^a=df,d^a=cdfghi,e^a=efg,f^a=g,g^a=fg,c^b=cf,d^b=dg,d^c=di72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083854);