Finite Group DatabasebetaFinite Group Databasebeta| order | 64800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D32400 |
| Fitting | C16×C25×C81 ⋊ C2 |
| Hall | C25×C81 ⋊ D16 |
| Sylow | C81 ⋊ D400 |
| Sylow | C25 ⋊ D1296 |
210 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 122 |
| transitive degree | μt(G) = 32400 |
A minimal faithful permutation representation — 2 generators on 122 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97)(98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,97)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(29,86)(30,85)(31,84)(32,83)(33,82)(34,81)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(99,122)(100,121)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)Polycyclic presentation:
2^5,3,5,3,5,3^2|b^2=cd,c^2=de,d^2=e,f^3=h,g^5=i,h^3=j,j^3=k,b^a=bc,c^a=cd,d^a=de,f^a=f^2h^2j^2k^2,g^a=g^4i^4,h^a=h^2j^2k^2,i^a=i^4,j^a=j^2k^2,k^a=k^216203 conjugacy classes. The character table is too large to display.
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, 2, 2, 2, 2, 3, 5, 3, 5, 3, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.8^2*F.10^2*F.11^2);;
gap> SetConjugate(coll, 7, 1, F.7^4*F.9^4);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.10^2*F.11^2);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.10^2*F.11^2);;
gap> SetConjugate(coll, 11, 1, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);