Relator,Name,Number of Generators,Torsion,Abelianisation,Small Cancellation,Center,Free-by-cyclic,Geometric,Manifold,Knot,Link Exterior,Further Information,References ,Trivial group,0,False,Z^0,True,,False,True,,,,, aa,"C_2",1,a,Z^0+Z/2Z,True,a,False,True,,,,, aaa,"C_3",1,a,Z^0+Z/3Z,True,a,False,True,,,,, aaaa,"C_4",1,a,Z^0+Z/4Z,True,a,False,True,,,,, aabb,"T(2,2)",2,False,Z^1+Z/2Z,False,bb,True,True,,,,, abAB,"BS(1,1)",2,False,Z^2,False,a,True,True,,,,, aaaaa,"C_5",1,a,Z^0+Z/5Z,True,a,False,True,,,,, aaabb,"T(3,2)",2,False,Z^1,False,bb,True,True,,,3_1,, aabaB,"BS(1,-2)",2,False,Z^1+Z/3Z,False,,False,False,,,,, aabAB,"BS(1,2)",2,False,Z^1,False,,False,False,,,,, aaaaaa,"C_5",1,a,Z^0+Z/6Z,True,a,False,True,,,,, aaaabb,"T(4,2)",2,False,Z^1+Z/2Z,False,bb,True,True,,,,, aaabaB,"BS(1,-3)",2,False,Z^1+Z/4Z,False,,False,False,,,,, aaabbb,"T(3,3)",2,False,Z^1+Z/3Z,False,bbb,True,True,,,,, aaabAB,"B(1,3)",2,False,Z^1+Z/2Z,False,,False,False,,,,, aabaaB,"BS(2,-2)",2,False,Z^1+Z/4Z,False,,False,,,,,, aabaBB,,2,False,Z^1,False,,True,False,,,,, aabbaB,,2,False,Z^1,False,,True,False,,,,, aabbcc,"\pi_1(N_3)",3,False,Z^2+Z/2Z,False,,False,True,,,,, aabbAB,,2,False,Z^1,False,,True,False,,,,, aabAAB,"BS(2,2)",2,False,Z^2,False,aa,True,,,,9^2_43,, aaaaaaa,"C_7",1,a,Z^0+Z/7Z,True,a,False,True,,,,, aaaaabb,"T(5,2)",2,False,Z^1,False,bb,True,True,,,5_1,, aaaabaB,"BS(1,-4)",2,False,Z^1+Z/5Z,False,,False,False,,,,, aaaabbb,"T(4,3)",2,False,Z^1,False,bbb,True,True,,,8_19,, aaaabAB,"BS(1,4)",2,False,Z^1+Z/3Z,False,,False,,,,,, aaabaaB,"BS(2,-3)",2,False,Z^1+Z/5Z,False,,False,False,,,,, aaabaBB,,2,False,Z^1,False,,True,False,,,,, aaabbaB,,2,False,Z^1,False,,True,False,,,,, aaabbcc,,3,False,Z^2,False,,False,True,,,,, aaabbAb,,2,False,Z^1,False,,True,False,,,,, aaabbAB,,2,False,Z^1,False,,True,False,,,,, aaabcBC,,3,False,Z^2+Z/3Z,False,,False,True,,,,, aaabAbb,,2,False,Z^1,False,,True,False,,,,, aaabAAB,"BS(2,3)",2,False,Z^1,False,,False,False,,,,, aaabABB,,2,False,Z^1,False,,True,False,,,,, aabaaBB,,2,False,Z^1,False,,True,True,,,,, aabacbc,,3,False,Z^2,True,,False,False,,,,, aabbaBB,,2,False,Z^1+Z/3Z,False,,False,False,,,,, aabbAAB,,2,False,Z^1,False,,False,False,,,,, abacaCB,,3,False,Z^2+Z/3Z,True,,False,True,,,,, abacACB,,3,False,Z^2,True,,False,True,,,,, aaaaaaaa,"C_8",1,a,Z^0+Z/8Z,True,a,False,True,,,,, aaaaaabb,"T(6,2)",2,False,Z^1+Z/2Z,False,bb,True,True,,,,, aaaaabaB,"BS(1,-5)",2,False,Z^1+Z/6Z,False,,False,False,,,,, aaaaabbb,"T(5,3)",2,False,Z^1,False,bbb,True,True,,,10_124,, aaaaabAB,"BS(1,5)",2,False,Z^1+Z/4Z,False,,False,False,,,,, aaaabaaB,"BS(2,-4)",2,False,Z^1+Z/6Z,False,,False,False,,,,, aaaabaBB,,2,False,Z^1,False,,True,False,,,,, aaaabbaB,,2,False,Z^1,False,,True,False,,,,, aaaabbbb,"T(4,4)",2,False,Z^1+Z/4Z,False,bbbb,True,True,,,,, aaaabbAb,,2,False,Z^1+Z/3Z,False,,True,False,,,,, aaaabbAB,,2,False,Z^1,False,,True,False,,,,, aaaabAbb,,2,False,Z^1+Z/3Z,False,,True,False,,,,, aaaabAAB,"BS(2,4)",2,False,Z^1+Z/2Z,False,,False,False,,,,, aaaabABB,,2,False,Z^1,False,,True,False,,,,, aaabaaaB,"BS(3,-3)",2,False,Z^1+Z/6Z,False,,False,,,,,, aaabaaBB,,2,False,Z^1,False,,True,False,,,,, aaababbb,,2,False,Z^1+Z/4Z,False,,False,False,,,,, aaabaBBB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaabbaaB,,2,False,Z^1,False,,True,False,,,,, aaabbabb,,2,False,Z^1+Z/4Z,False,,True,True,,,,, aaabbaBB,,2,False,Z^1+Z/4Z,False,,False,False,,,,, aaabbbaB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaabbbAB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaabbAbb,,2,False,Z^1+Z/2Z,False,,True,True,,,,, aaabbAAb,,2,False,Z^1,False,,True,False,,,,, aaabbAAB,,2,False,Z^1,False,,True,False,,,,, aaabbABB,,2,False,Z^1+Z/2Z,False,,False,False,,,,, aaabAAbb,,2,False,Z^1,False,,True,False,,,,, aaabAAAB,"BS(3,3)",2,False,Z^2,False,aaa,True,,,,9^2_49,, aaabAABB,,2,False,Z^1,False,,True,False,,,,, aababbaB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aababbAB,,2,False,Z^1+Z/2Z,False,,False,False,,,,, aababAbb,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aabaBabb,,2,False,Z^1+Z/2Z,False,,False,False,,,,, aabaBaBB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aabaBAbb,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aabbaabb,,2,aabb,Z^1+Z/4Z,True,,False,False,,,,, aabbaaBB,,2,False,Z^1+Z/4Z,False,,False,False,,,,, aabbaBAb,,2,False,Z^1+Z/2Z,False,,False,False,,,,, aabbAABB,,2,False,Z^2,False,,False,True,,,,, aabbABAB,,2,False,Z^2,False,,False,False,,,,, aabABBAb,,2,False,Z^2,False,,True,True,,,,, abABabAB,,2,abAB,Z^2,True,,False,False,,,,, aaaabbcc,,3,False,Z^2+Z/2Z,False,,False,True,,,,, aaaabcBC,,3,False,Z^2+Z/4Z,False,,False,True,,,,, aaabacbc,,3,False,Z^2+Z/2Z,False,,False,,,,,, aaabbbcc,,3,False,Z^2,False,,False,True,,,,, aaabbcbC,,3,False,Z^2+Z/3Z,False,,False,False,,,,, aaabbcBC,,3,False,Z^2,False,,False,False,,,,, aabaacbc,,3,False,Z^2+Z/2Z,False,,False,,,,,, aabacaCB,,3,False,Z^2+Z/4Z,True,,False,,,,,, aabacbcc,,3,False,Z^2,True,,False,False,,,,, aabacbCB,,3,False,Z^2,True,,False,False,,,,, aabaccBC,,3,False,Z^2,True,,False,True,,,,, aabacACB,,3,False,Z^2+Z/2Z,True,,False,,,,,, aabacBCB,,3,False,Z^2,True,,False,False,,,,, aabacBCC,,3,False,Z^2,True,,False,True,,,,, aabcbCAb,,3,False,Z^2,True,,False,True,,,,, aabcbCAB,,3,False,Z^2,True,,False,False,,,,, aabcBCAb,,3,False,Z^2,True,,False,True,,,,, aabcBCAc,,3,False,Z^2,True,,False,False,,,,, aabbccdd,"\pi_1(N_4)",4,False,Z^3+Z/2Z,True,,False,True,,,,, abcdABCD,"\pi_1(S_2)",4,False,Z^4,True,,False,True,,,,, aaaaaaaaa,,1,a,Z^0+Z/9Z,True,a,False,True,,,,, aaaaaaabb,,2,False,Z^1,False,bb,True,True,,,7_1,, aaaaaabaB,,2,False,Z^1+Z/7Z,False,,False,False,,,,, aaaaaabbb,,2,False,Z^1+Z/3Z,False,bbb,True,True,,,,, aaaaaabAB,,2,False,Z^1+Z/5Z,False,,False,False,,,,, aaaaabaaB,,2,False,Z^1+Z/7Z,False,,False,False,,,,, aaaaabaBB,,2,False,Z^1,False,,True,False,,,,, aaaaabbaB,,2,False,Z^1,False,,True,False,,,,, aaaaabbbb,,2,False,Z^1,False,bbbb,True,True,,,,, aaaaabbAb,,2,False,Z^1,False,,True,False,,,,, aaaaabbAB,,2,False,Z^1,False,,True,False,,,,, aaaaabAbb,,2,False,Z^1,False,,True,False,,,,, aaaaabAAB,,2,False,Z^1+Z/3Z,False,,False,False,,,,, aaaaabABB,,2,False,Z^1,False,,True,False,,,,, aaaabaaaB,,2,False,Z^1+Z/7Z,False,,False,False,,,,, aaaabaaBB,,2,False,Z^1,False,,True,False,,,,, aaaababbb,,2,False,Z^1,False,,True,False,,,,, aaaabaBBB,,2,False,Z^1,False,,True,False,,,,, aaaabbaaB,,2,False,Z^1,False,,True,False,,,,, aaaabbabb,,2,False,Z^1,False,,True,True,,,,, aaaabbaBB,,2,False,Z^1+Z/5Z,False,,False,False,,,,, aaaabbbab,,2,False,Z^1,False,,True,False,,,,, aaaabbbaB,,2,False,Z^1,False,,True,False,,,,, aaaabbbAb,,2,False,Z^1,False,,True,False,,,,, aaaabbbAB,,2,False,Z^1,False,,True,False,,,,, aaaabbAbb,,2,False,Z^1,False,,True,True,,,,, aaaabbAAb,,2,False,Z^1,False,,True,False,,,,, aaaabbAAB,,2,False,Z^1,False,,True,False,,,,, aaaabbABB,,2,False,Z^1+Z/3Z,False,,False,False,,,,, aaaabAbbb,,2,False,Z^1,False,,True,False,,,,, aaaabAAbb,,2,False,Z^1,False,,True,False,,,,, aaaabAAAB,,2,False,Z^1,False,,False,False,,,,, aaaabAABB,,2,False,Z^1,False,,True,False,,,,, aaaabABBB,,2,False,Z^1,False,,True,False,,,,, aaabaaaBB,,2,False,Z^1,False,,True,True,,,,, aaabaabbb,,2,False,Z^1,False,,True,False,,,,, aaabaaBBB,,2,False,Z^1,False,,True,False,,,,, aaababaBB,,2,False,Z^1+Z/5Z,False,,False,False,,,,, aaababbaB,,2,False,Z^1,False,,True,False,,,,, aaababbAb,,2,False,Z^1,False,,True,False,,,,, aaababbAB,,2,False,Z^1,False,,True,False,,,,, aaababAbb,,2,False,Z^1,False,,True,False,,,,, aaababABB,,2,False,Z^1+Z/3Z,False,,False,False,,,,, aaabaBabb,,2,False,Z^1,False,,True,False,,,,, aaabaBaBB,,2,False,Z^1,False,,True,False,,,,, aaabaBAbb,,2,False,Z^1,False,,True,False,,,,, aaabaBABB,,2,False,Z^1,False,,True,False,,,,, aaabaBBab,,2,False,Z^1+Z/5Z,False,,True,True,m003,,,, aaabaBBAb,,2,False,Z^1+Z/3Z,False,,True,False,,,,, aaabbaabb,,2,False,Z^1,False,bb,True,True,,,,, aaabbaaBB,,2,False,Z^1+Z/5Z,False,,False,False,,,,, aaabbabaB,,2,False,Z^1,False,,True,False,,,,, aaabbabbb,,2,False,Z^1,False,,True,False,,,,, aaabbabAb,,2,False,Z^1,False,,True,False,,,,, aaabbabAB,,2,False,Z^1,False,,True,False,,,,, aaabbaBab,,2,False,Z^1,False,,True,False,,,,, aaabbaBAb,,2,False,Z^1,False,,False,False,,,,, aaabbaBAB,,2,False,Z^1+Z/3Z,False,,False,False,,,,, aaabbaBBB,,2,False,Z^1,False,,True,False,,,,, aaabbbaaB,,2,False,Z^1,False,,True,False,,,,, aaabbbaBB,,2,False,Z^1,False,,True,False,,,,, aaabbbAAB,,2,False,Z^1,False,,True,False,,,,, aaabbbABB,,2,False,Z^1,False,,True,False,,,,, aaabbAbab,,2,False,Z^1,False,,True,False,,,,, aaabbAAbb,,2,False,Z^1,False,,True,True,,,,, aaabbAAAb,,2,False,Z^1+Z/3Z,False,,False,False,,,,, aaabbAAAB,,2,False,Z^1,False,,False,False,,,,, aaabbAABB,,2,False,Z^1,False,,False,False,,,,, aaabbABAB,,2,False,Z^1,False,,False,False,,,,, aaabAbabb,,2,False,Z^1,False,,True,False,,,,, aaabAbbab,,2,False,Z^1,False,,True,False,,,,, aaabABaBB,,2,False,Z^1,False,,True,False,,,,, aaabABABB,,2,False,Z^1,False,,True,False,,,,, aaabABBAb,,2,False,Z^1,False,,True,True,m004,K2_1,4_1,, aabaabaBB,,2,False,Z^1+Z/5Z,False,,False,False,,,,, aabaabbaB,,2,False,Z^1,False,,True,False,,,,, aabaabbAB,,2,False,Z^1,False,,True,False,,,,, aabaabABB,,2,False,Z^1+Z/3Z,False,,False,False,,,,, aabaaBabb,,2,False,Z^1,False,,True,False,,,,, aabaaBaBB,,2,False,Z^1,False,,True,False,,,,, aabaaBAbb,,2,False,Z^1,False,,True,False,,,,, aabaaBABB,,2,False,Z^1,False,,True,False,,,,, aababaaBB,,2,False,Z^1+Z/5Z,False,,False,True,m006,,,, aababbAAB,,2,False,Z^1,False,,True,False,,,,, aababAABB,,2,False,Z^1,False,,False,False,,,,, aabaBaabb,,2,False,Z^1,False,,True,False,,,,, aabaBaaBB,,2,False,Z^1,False,,True,False,,,,, aabaBabaB,,2,False,Z^1+Z/5Z,False,,False,False,,,,, aabaBabAB,,2,False,Z^1+Z/3Z,False,,False,False,,,,, aabaBAbaB,,2,False,Z^1+Z/3Z,False,,False,False,,,,, aabaBAbAB,,2,False,Z^1,False,,False,False,,,,, aabaBBAbb,,2,False,Z^1,False,,False,False,,,,, aabaBBAAb,,2,False,Z^1,False,,False,False,,,,, aabbaabAB,,2,False,Z^1,False,,True,False,,,,, aabbaaBAb,,2,False,Z^1,False,,True,False,,,,, aabbaaBAB,,2,False,Z^1+Z/3Z,False,,False,True,m007,,,, aabbaBAAB,,2,False,Z^1,False,,False,False,,,,, aabABabaB,,2,False,Z^1+Z/3Z,False,,False,False,,,,, aabABabAB,,2,False,Z^1,False,,False,False,,,,, aabABAbaB,,2,False,Z^1,False,,False,False,,,,, aabABAbAB,,2,False,Z^1,False,,False,False,,,,, aaaaaaaaaa,"C_10",1,a,Z^0+Z/10Z,True,a,False,True,,,,, aaaaaaaabb,"T(8,2)",2,False,Z^1+Z/2Z,False,bb,True,True,,,,, aaaaaaabaB,"BS(7,-1)",2,False,Z^1+Z/8Z,False,,False,False,,,,, aaaaaaabbb,"T(7,3)",2,False,Z^1,False,bbb,True,True,,,,, aaaaaaabAB,"BS(7,1)",2,False,Z^1+Z/6Z,False,,False,False,,,,, aaaaaabaaB,"BS(6,-2)",2,False,Z^1+Z/8Z,False,,False,False,,,,, aaaaaabaBB,,2,False,Z^1,False,,True,False,,,,, aaaaaabbaB,,2,False,Z^1,False,,True,False,,,,, aaaaaabbbb,"T(6,4)",2,False,Z^1+Z/2Z,False,bbbb,True,True,,,,, aaaaaabbAb,,2,False,Z^1,False,,True,False,,,,, aaaaaabbAB,,2,False,Z^1,False,,True,False,,,,, aaaaaabAbb,,2,False,Z^1,False,,True,False,,,,, aaaaaabAAB,"BS(6,2)",2,False,Z^1+Z/4Z,False,,False,False,,,,, aaaaaabABB,,2,False,Z^1,False,,True,False,,,,, aaaaabaaaB,"BS(5,-3)",2,False,Z^1+Z/8Z,False,,False,False,,,,, aaaaabaaBB,,2,False,Z^1,False,,True,False,,,,, aaaaababbb,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaaabaBBB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaaabbaaB,,2,False,Z^1,False,,True,False,,,,, aaaaabbabb,,2,False,Z^1+Z/2Z,False,,True,True,,,,, aaaaabbaBB,,2,False,Z^1+Z/6Z,False,,False,False,,,,, aaaaabbbab,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaaabbbaB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaaabbbbb,"T(5,5)",2,False,Z^1+Z/5Z,False,bbbbb,True,True,,,,, aaaaabbbAb,,2,False,Z^1+Z/4Z,False,,True,False,,,,, aaaaabbbAB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaaabbAbb,,2,False,Z^1+Z/4Z,False,,True,True,,,,, aaaaabbAAb,,2,False,Z^1+Z/3Z,False,,True,False,,,,, aaaaabbAAB,,2,False,Z^1,False,,True,False,,,,, aaaaabbABB,,2,False,Z^1+Z/4Z,False,,False,False,,,,, aaaaabAbbb,,2,False,Z^1+Z/4Z,False,,True,False,,,,, aaaaabAAbb,,2,False,Z^1+Z/3Z,False,,True,False,,,,, aaaaabAAAB,"BS(5,3)",2,False,Z^1+Z/2Z,False,,False,False,,,,, aaaaabAABB,,2,False,Z^1,False,,True,False,,,,, aaaaabABBB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaabaaaaB,"BS(4,-4)",2,False,Z^1+Z/8Z,False,,False,,,,,, aaaabaaaBB,,2,False,Z^1,False,,True,False,,,,, aaaabaabbb,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaabaaBBB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaababaBB,,2,False,Z^1+Z/6Z,False,,False,False,,,,, aaaababbaB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaababbbb,,2,False,Z^1+Z/5Z,False,,False,False,,,,, aaaababbAb,,2,False,Z^1+Z/4Z,False,,False,False,,,,, aaaababbAB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaababAbb,,2,False,Z^1+Z/4Z,False,,False,False,,,,, aaaababABB,,2,False,Z^1+Z/4Z,False,,False,False,,,,, aaaabaBabb,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaabaBaBB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaabaBAbb,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaabaBABB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaabaBBab,,2,False,Z^1+Z/6Z,False,,True,True,m010,,,, aaaabaBBAb,,2,False,Z^1+Z/4Z,False,,True,False,,,,, aaaabaBBBB,,2,False,Z^1,False,,True,False,,,,, aaaabbaaaB,,2,False,Z^1,False,,True,False,,,,, aaaabbaabb,,2,False,Z^1+Z/2Z,False,bbbb,True,False,,,,, aaaabbaaBB,,2,False,Z^1+Z/6Z,False,,False,False,,,,, aaaabbabaB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaabbabbb,,2,False,Z^1+Z/5Z,False,,True,False,,,,, aaaabbabAb,,2,False,Z^1+Z/4Z,False,,True,False,,,,, aaaabbabAB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaabbaBab,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaabbaBAb,,2,False,Z^1+Z/2Z,False,,False,False,,,,, aaaabbaBAB,,2,False,Z^1+Z/4Z,False,,False,False,,,,, aaaabbaBBB,,2,False,Z^1,False,,True,False,,,,, aaaabbbaab,,2,False,Z^1+Z/2Z,False,,True,False,,,,, aaaabbbaaB,,2,False,Z^1+Z/2Z,False,,True,False,,,,, 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