Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #32884⋊C3 |
| Fitting, Sylow | #32884 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | AΣL(1,𝔽64) |
| commutator, cosocle | C22×#41 | C22×C3 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | #32884 | C3 |
| socle | C23 | D4×A4 |
| hypercenter | Q8 | #120⋊C3 |
| hypocenter | C22×Q8 | C3×D4 |
| LCS | G ▷ C22×#41 ▷ C22×Q8 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
729 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 729 conjugacy classes of subgroups — too many to draw. Only its 23 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|f^2=i,b^a=bgi,c^a=cdfi,d^a=defgh,e^a=di,f^a=fgi,g^a=ghi,h^a=g,c^b=cf,d^b=dg,e^b=eh,f^b=fi,f^c=fi,e^d=ei36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083976);