Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #12773⋊C3 |
| Fitting, Sylow | #12773 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | B(4,𝔽2)⋊C3 |
| commutator, hypocenter | Q82 | C3×C4 |
| Frattini | C2×#41 | C2×A4 |
| Fitting | #12773 | C3 |
| socle | C2 | #1790⋊C3 |
| prime radical | #85654 | C2 |
| cosocle | #3483 | C2×C3 |
| hypercenter | C2×C4 | #120⋊C3 |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
187 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 187 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
3,2^8|b^2=g,e^2=i,f^2=i,b^a=bcegh,c^a=dh,d^a=cdfghi,e^a=fi,f^a=efi,g^a=egi,c^b=ce,d^b=df,e^b=eh,h^b=hi,d^c=di,e^c=ei,f^c=fhi,g^c=ghi,e^d=eh,f^d=fi,g^d=gi,f^e=fi,g^e=gi40 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083565);