Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C32×#1419 ⋊ C2 |
| Sylow | C32 ⋊ #13658 |
582 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 582 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^8,3^2|a^2=d,c^2=eg,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bdeh,c^a=cgh,d^b=df,d^c=dh,e^a=eh,e^b=eh,f^b=fg,g^b=gh,i^c=i^2,j^c=j^2138 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#13658 |
| maximal normal subgroup of | C3⋊D3×#13658 |
| maximal quotient of | #72269676 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 3, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.5*F.7);;
gap> SetPower(coll, 4, F.6*F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.7*F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 3, F.9^2);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);