Finite Group DatabasebetaFinite Group Databasebeta| order | 13122 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C81 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, Sylow | #72277430 ⋊ C2 |
| derived | C81 ⋊ C2×C81 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #72221799 |
| commutator, hypocenter | C81 | C2×C81 |
| Frattini | #10624443 | C3×D3 |
| Fitting | #72277430 | C2 |
| socle | C32 | #10624462⋊C2 |
| prime radical | C3×D81 | C27 |
| cosocle | #14546223 | C2×C3 |
| LCS | G ▷ C81 | |
| UCS | C1 ◁ C3 | |
70 conjugacy classes of subgroups.
| transitive degree | μt(G) = 162 |
Polycyclic presentation:
2,3^8|b^3=c,c^3=d,d^3=e,f^3=g,g^3=h,h^3=i,f^a=f^2g^2h^2i^2,f^b=fg^2h,f^c=fh^2i,f^d=fi^2,g^a=g^2h^2i^2,g^b=gh^2i,g^c=gi^2,h^a=h^2i^2,h^b=hi^2,i^a=i^2282 conjugacy classes — view the full character table.
| maximal subgroup of | #72277488 |
| maximal quotient of | #72277488 |
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");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7^2*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8^2*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9^2);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 7, 2, F.7*F.8^2*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2);;
gap> SetConjugate(coll, 8, 2, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);