Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #62686⋊C3 |
| Fitting, Sylow | #62686 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | C22⋊A4 |
| commutator, cosocle, hypocenter | #543 | C22×C3 |
| Frattini, socle | C22 | C22×C22⋊A4 |
| Fitting | #62686 | C3 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C42 | |
1316 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 1316 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|f^2=i,g^2=h,b^a=ceh,c^a=bcdefgh,d^a=dei,e^a=d,c^b=ch,d^b=di,e^b=ei,d^c=di72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085022);