Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #13856⋊C3 |
| Fitting, Sylow | #13856 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #127⋊C3 |
| commutator, hypocenter | #95 | C2×C3×C4 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | #13856 | C3 |
| prime radical | C2×#95⋊C3 | C22 |
| cosocle | C2×#95 | C22×C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C2×C4 | |
359 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 39 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=i,c^2=f,d^2=g,e^2=h,b^a=bcdeg,c^a=defg,d^a=cde,e^a=egh,f^a=gh,g^a=fg,c^b=cf,d^b=dg,d^c=dh72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083612);