Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #11709160 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, Sylow | #10691 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #1304 | C2×C3 |
| Frattini | C25 | C2×A4 |
| Fitting | #10691 | C3 |
| socle | C22 | #401⋊C3 |
| prime radical | #1304⋊C3 | C2 |
| LCS | G ▷ #1304 | |
| UCS | C1 | |
87 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
3,2^8|b^2=efgh,c^2=ef,d^2=e,b^a=cdefghi,c^a=cehi,d^a=bdef,e^a=ghi,f^a=efgh,g^a=egh,h^a=hi,i^a=h,c^b=ce,d^b=df,e^b=eh,f^b=fhi,g^b=gi,d^c=dg,e^c=ei,f^c=fi,g^c=gh,f^d=fh,g^d=gi19 conjugacy classes — view the full character table.
| maximal subgroup of | #72284969 |
| maximal normal subgroup of | #72284969 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083541);