Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C2×#1020 |
| commutator, cosocle, hypocenter | #543⋊C3 | C22 |
| Fitting | C2×#543 | D3 |
| prime radical | C2×#543⋊C3 | C2 |
| hypercenter | C23 | #1020 |
| LCS | G ▷ #543⋊C3 | |
| UCS | C1 ◁ C22 ◁ C23 | |
391 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 391 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:
2^2,3,2^6|a^2=i,f^2=h,g^2=h,b^a=bhi,c^a=c^2,d^a=de,f^a=fg,g^a=gh,d^c=e,e^c=de,f^c=g,g^c=fgh,g^d=gi,f^e=fi,g^f=gh33 conjugacy classes — view the full character table.
| maximal subgroup of | #72227610, #72227715, #72227717, … |
| maximal normal subgroup of | #72227715, #72227726, #72227735, … |
| maximal quotient of | #72227601, #72271316 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1089985);