Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | #93255 ⋊ D3 |
| derived | #72210255 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #72210224 |
| commutator, cosocle, hypocenter | #72210255 | C2 |
| Frattini | #1726 | S4 |
| Fitting | #93255 | D3 |
| hypercenter | C8 | #67992 |
| LCS | G ▷ #72210255 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
197 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2,3,2^9|a^2=hjk,c^2=e,d^2=f,e^2=h,f^2=i,g^2=jk,h^2=k,i^2=k,j^2=k,b^a=b^2hi,c^a=cgk,c^b=deghijk,d^a=cdefhij,d^b=cdg,d^c=dg,e^a=ejk,e^b=fh,e^d=ej,f^a=efghi,f^b=efgjk,f^c=fjk,f^e=fk,g^a=gj,h^a=hk,h^b=ik,h^d=hk,i^a=hij,i^b=hijk,i^c=ik,j^a=jk44 conjugacy classes — view the full character table.
| maximal quotient of | #72210807, #72210816 |
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", "k");;
gap> coll := SingleCollector(F, [2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8*F.10*F.11);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10*F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.8*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.11);;
gap> SetConjugate(coll, 3, 2, F.4*F.5*F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.3*F.4*F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.3*F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.6*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.5*F.6*F.7*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.9*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);