Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #29686⋊C3 |
| Fitting, Sylow | #29686 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120⋊C3 |
| commutator, cosocle, hypocenter | #543 | C22×C3 |
| Frattini | C24 | C22×A4 |
| Fitting | #29686 | C3 |
| hypercenter | C2×C4 | #120⋊C3 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
297 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
3,2^8|e^2=fi,b^a=cfhi,c^a=bceg,g^a=h,h^a=fghi,c^b=cf,d^b=dg,h^b=hi,d^c=dh,g^c=gi,e^d=ef39 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083935);