Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C32×#518 ⋊ C22 |
| Sylow | C32 ⋊ C2×#1913 |
5116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3)(4,5,6)(7,10)(8,13)(9,15)(11,17)(12,18)(14,20)(16,22)(19,21);(2,3)(5,6);(7,9)(8,12,14,19)(10,15)(11,16)(13,18,20,21)(17,22);(5,6)(7,8)(9,12,16,19)(10,13)(11,14)(15,21,22,18)(17,20)Polycyclic presentation:
2^8,3^2|a^2=g,b^2=ef,b^a=be,d^a=dg,e^a=eh,e^b=eh,e^d=eh,f^b=fh,f^d=fh,g^b=gh,i^a=i^2,i^c=i^2,j^a=i^2j,j^c=j^2150 conjugacy classes — view the full character table.
| maximal subgroup of | #72252356, #72252376 |
| maximal normal subgroup of | #72252356, #72252376 |
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.7);;
gap> SetPower(coll, 2, F.5*F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 3, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.9^2*F.10);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);