================================================= Generating symmetries 8 elements in Aut 16 elements in Aut x Z2 27 Conjugacy classes Group( () ) 1 [ 1 .. 4 ] Group( () ) [ 1, 1 ] Group( [ (2,4) ] ) 2 [ 1, 2, 3, 2 ] Group( [ (2,4) ] ) [ 2, 1 ] Group( [ (1,3)(2,4) ] ) Group( [ (2,4)(5,6) ] ) Group( [ (5,6) ] ) Group( [ (1,3)(2,4)(5,6) ] ) 3 [ 1, 2, -1, -2 ] Group( [ (1,3)(2,4)(5,6) ] ) [ 2, 1 ] Group( [ (1,2)(3,4) ] ) 4 [ 1, 1, 3, 3 ] Group( [ (1,2)(3,4) ] ) [ 2, 1 ] Group( [ (1,2)(3,4)(5,6) ] ) 5 [ 1, -1, 3, -3 ] Group( [ (1,2)(3,4)(5,6) ] ) [ 2, 1 ] Group( [ (2,4), (5,6) ] ) Group( [ (1,3), (2,4) ] ) 6 [ 1, 2, 1, 2 ] Group( [ (1,3), (2,4) ] ) [ 4, 2 ] Group( [ (1,3)(2,4), (5,6) ] ) Group( [ (1,3)(2,4), (2,4)(5,6) ] ) Group( [ (1,3)(2,4), (1,2)(3,4) ] ) Group( [ (1,3)(2,4), (1,4,3,2) ] ) Group( [ (2,4), (1,3)(2,4)(5,6) ] ) 7 [ 1, 0, -1, 0 ] Group( [ (2,4), (1,3)(2,4)(5,6) ] ) [ 4, 2 ] Group( [ (1,3)(2,4), (1,2)(3,4)(5,6) ] ) Group( [ (1,3)(2,4), (1,4,3,2)(5,6) ] ) Group( [ (5,6), (1,2)(3,4) ] ) Group( [ (1,3)(2,4)(5,6), (1,2)(3,4) ] ) 8 [ 1, 1, -1, -1 ] Group( [ (1,3)(2,4)(5,6), (1,2)(3,4) ] ) [ 4, 2 ] Group( [ (2,4), (1,3), (5,6) ] ) Group( [ (2,4), (1,3), (1,2)(3,4) ] ) 9 [ 1, 1, 1, 1 ] Group( [ (2,4), (1,3), (1,2)(3,4) ] ) [ 8, 3 ] Group( [ (2,4), (1,3), (1,2)(3,4)(5,6) ] ) 10 [ 1, -1, 1, -1 ] Group( [ (2,4), (1,3), (1,2)(3,4)(5,6) ] ) [ 8, 3 ] Group( [ (1,3)(2,4), (5,6), (1,4,3,2) ] ) Group( [ (1,3)(2,4), (5,6), (1,2)(3,4) ] ) Group( [ (1,3)(2,4), (2,4)(5,6), (1,2)(3,4) ] ) Group( [ (1,3)(2,4), (2,4)(5,6), (1,4,3,2) ] ) Group( [ (2,4), (1,3), (5,6), (1,2)(3,4) ] ) 11 [ 0, 0, 0, 0 ] Group( [ (2,4), (1,3), (5,6), (1,2)(3,4) ] ) [ 16, 11 ] Subgroup structure [ [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ], [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], [ 2, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1 ], [ 3, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1 ], [ 4, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1 ], [ 5, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1 ], [ 6, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1 ], [ 7, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 ], [ 8, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1 ], [ 9, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1 ], [ 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ], [ 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ] ================================================= 4 11 1 2 1 2 2 1 2 2 1 1 1 1 2 2 2 2 4 4 4 8 8 16 1 2 3 4 1 2 3 4 1 4 3 2 1 2 3 4 3 2 1 4 1 2 3 4 -3 -4 -1 -2 1 2 3 4 2 1 4 3 1 2 3 4 4 3 2 1 1 2 3 4 -2 -1 -4 -3 1 2 3 4 -4 -3 -2 -1 1 2 3 4 3 2 1 4 1 4 3 2 3 4 1 2 1 2 3 4 -3 -2 -1 -4 1 4 3 2 -3 -4 -1 -2 1 2 3 4 -1 -4 -3 -2 3 2 1 4 -3 -4 -1 -2 1 2 3 4 -3 -4 -1 -2 2 1 4 3 -4 -3 -2 -1 1 2 3 4 -3 -4 -1 -2 4 3 2 1 -2 -1 -4 -3 1 2 3 4 3 2 1 4 4 3 2 1 4 1 2 3 1 4 3 2 3 4 1 2 2 3 4 1 2 1 4 3 1 2 3 4 -4 -3 -2 -1 3 2 1 4 -4 -1 -2 -3 1 4 3 2 -2 -3 -4 -1 3 4 1 2 -2 -1 -4 -3 1 2 3 4 -1 -2 -3 -4 3 2 1 4 -3 -2 -1 -4 4 3 2 1 -4 -3 -2 -1 4 1 2 3 -4 -1 -2 -3 1 4 3 2 -1 -4 -3 -2 3 4 1 2 -3 -4 -1 -2 2 3 4 1 -2 -3 -4 -1 2 1 4 3 -2 -1 -4 -3 mem limits rec( min := 24576, max := 1536000, kill := 0 ) mem stat before subtypes rec( partial := rec( livebags := 2072, livekb := 60, deadbags := 32005, deadkb := 660, freekb := 1348, totalkb := 24576 ) ) mem stat after subtypes rec( partial := rec( livebags := 2379, livekb := 61, deadbags := 32105, deadkb := 680, freekb := 1259, totalkb := 24576 ) ) 16=symmetries Symmetry H=H1, Symmetry type S1, Id(H)=Z1 Representation 1, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 4, dim Fix_U_i K = 1, N_H(K)/K=Z1 isotropy subgroups H[ 1 ], maximals H[ ], H/H'=Z1 Jumps from H1 to H[ ] Typejumps from S1 to S[ ] Symmetry H=H2, Symmetry type S2, Id(H)=Z2 Representation 1, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H2, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 1, 2 ], maximals H[ 1 ], H/H'=Z2 Representation 2, dimension of irred subspace U_i = 1 K=H2, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, N_H(K)/K=Z1 isotropy subgroups H[ 2 ], maximals H[ ], H/H'=Z1 Jumps from H2 to H[ 1 ] Typejumps from S2 to S[ 1 ] Symmetry H=H3, Symmetry type S2, Id(H)=Z2 Representation 1, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H3, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 1, 3 ], maximals H[ 1 ], H/H'=Z2 Representation 2, dimension of irred subspace U_i = 1 K=H3, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, N_H(K)/K=Z1 isotropy subgroups H[ 3 ], maximals H[ ], H/H'=Z1 Jumps from H3 to H[ 1 ] Typejumps from S2 to S[ 1 ] Symmetry H=H4, Symmetry type S3, Id(H)=Z2 Representation 1, dimension of irred subspace U_i = 1 K=H4, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K=Z1 isotropy subgroups H[ 4 ], maximals H[ ], H/H'=Z1 Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H4, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 1, 4 ], maximals H[ 1 ], H/H'=Z2 Jumps from H4 to H[ 1 ] Typejumps from S3 to S[ 1 ] Symmetry H=H5, Symmetry type S4, Id(H)=Z2 Representation 1, dimension of irred subspace U_i = 1 K=H5, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K=Z1 isotropy subgroups H[ 5 ], maximals H[ ], H/H'=Z1 Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H5, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 1, 5 ], maximals H[ 1 ], H/H'=Z2 Jumps from H5 to H[ 1 ] Typejumps from S4 to S[ 1 ] Symmetry H=H6, Symmetry type S4, Id(H)=Z2 Representation 1, dimension of irred subspace U_i = 1 K=H6, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K=Z1 isotropy subgroups H[ 6 ], maximals H[ ], H/H'=Z1 Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H6, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 1, 6 ], maximals H[ 1 ], H/H'=Z2 Jumps from H6 to H[ 1 ] Typejumps from S4 to S[ 1 ] Symmetry H=H7, Symmetry type S5, Id(H)=Z2 Representation 1, dimension of irred subspace U_i = 1 K=H7, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K=Z1 isotropy subgroups H[ 7 ], maximals H[ ], H/H'=Z1 Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H7, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 1, 7 ], maximals H[ 1 ], H/H'=Z2 Jumps from H7 to H[ 1 ] Typejumps from S5 to S[ 1 ] Symmetry H=H8, Symmetry type S5, Id(H)=Z2 Representation 1, dimension of irred subspace U_i = 1 K=H8, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K=Z1 isotropy subgroups H[ 8 ], maximals H[ ], H/H'=Z1 Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H8, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 1, 8 ], maximals H[ 1 ], H/H'=Z2 Jumps from H8 to H[ 1 ] Typejumps from S5 to S[ 1 ] Symmetry H=H9, Symmetry type S6, Id(H)=Z2xZ2 Representation 1, dimension of irred subspace U_i = 1 K=H9, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K=Z1 isotropy subgroups H[ 9 ], maximals H[ ], H/H'=Z1 Representation 2, dimension of irred subspace U_i = 1 K=H3, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H9, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 3, 9 ], maximals H[ 3 ], H/H'=Z2 Representation 3, dimension of irred subspace U_i = 1 K=H2, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H9, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 2, 9 ], maximals H[ 2 ], H/H'=Z2 Jumps from H9 to H[ 2, 3 ] Typejumps from S6 to S[ 2, 2 ] Symmetry H=H10, Symmetry type S7, Id(H)=Z2xZ2 Representation 1, dimension of irred subspace U_i = 1 K=H10, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z1 isotropy subgroups H[ 10 ], maximals H[ ], H/H'=Z1 Representation 2, dimension of irred subspace U_i = 1 K=H2, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H10, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 2, 10 ], maximals H[ 2 ], H/H'=Z2 Representation 4, dimension of irred subspace U_i = 1 K=H4, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H10, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 4, 10 ], maximals H[ 4 ], H/H'=Z2 Jumps from H10 to H[ 2, 4 ] Typejumps from S7 to S[ 2, 3 ] Symmetry H=H11, Symmetry type S7, Id(H)=Z2xZ2 Representation 1, dimension of irred subspace U_i = 1 K=H11, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z1 isotropy subgroups H[ 11 ], maximals H[ ], H/H'=Z1 Representation 2, dimension of irred subspace U_i = 1 K=H3, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H11, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 3, 11 ], maximals H[ 3 ], H/H'=Z2 Representation 4, dimension of irred subspace U_i = 1 K=H4, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H11, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 4, 11 ], maximals H[ 4 ], H/H'=Z2 Jumps from H11 to H[ 3, 4 ] Typejumps from S7 to S[ 2, 3 ] Symmetry H=H12, Symmetry type S8, Id(H)=Z2xZ2 Representation 1, dimension of irred subspace U_i = 1 K=H12, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z1 isotropy subgroups H[ 12 ], maximals H[ ], H/H'=Z1 Representation 2, dimension of irred subspace U_i = 1 K=H4, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H12, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 4, 12 ], maximals H[ 4 ], H/H'=Z2 Representation 3, dimension of irred subspace U_i = 1 K=H5, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H12, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 5, 12 ], maximals H[ 5 ], H/H'=Z2 Representation 4, dimension of irred subspace U_i = 1 K=H8, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H12, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 8, 12 ], maximals H[ 8 ], H/H'=Z2 Jumps from H12 to H[ 4, 5, 8 ] Typejumps from S8 to S[ 3, 4, 5 ] Symmetry H=H13, Symmetry type S8, Id(H)=Z2xZ2 Representation 1, dimension of irred subspace U_i = 1 K=H13, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z1 isotropy subgroups H[ 13 ], maximals H[ ], H/H'=Z1 Representation 2, dimension of irred subspace U_i = 1 K=H4, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H13, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 4, 13 ], maximals H[ 4 ], H/H'=Z2 Representation 3, dimension of irred subspace U_i = 1 K=H6, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H13, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 6, 13 ], maximals H[ 6 ], H/H'=Z2 Representation 4, dimension of irred subspace U_i = 1 K=H7, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H13, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 7, 13 ], maximals H[ 7 ], H/H'=Z2 Jumps from H13 to H[ 4, 6, 7 ] Typejumps from S8 to S[ 3, 4, 5 ] Symmetry H=H14, Symmetry type S9, Id(H)=D4 Representation 1, dimension of irred subspace U_i = 1 K=H14, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z1 isotropy subgroups H[ 14 ], maximals H[ ], H/H'=Z1 Representation 4, dimension of irred subspace U_i = 1 K=H9, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H14, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 9, 14 ], maximals H[ 9 ], H/H'=Z2 Representation 5, dimension of irred subspace U_i = 2 K=H1, dim Fix_P(U) K = 2, dim Fix_U_i K = 2, N_H(K)/K=D4 K=H2, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H5, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H14, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 1, 2, 5, 14 ], maximals H[ 2 5 ], H/H'=D4 Jumps from H14 to H[ 2, 5, 9 ] Typejumps from S9 to S[ 2, 4, 6 ] Symmetry H=H15, Symmetry type S10, Id(H)=D4 Representation 1, dimension of irred subspace U_i = 1 K=H15, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z1 isotropy subgroups H[ 15 ], maximals H[ ], H/H'=Z1 Representation 4, dimension of irred subspace U_i = 1 K=H9, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H15, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 9, 15 ], maximals H[ 9 ], H/H'=Z2 Representation 5, dimension of irred subspace U_i = 2 K=H1, dim Fix_P(U) K = 2, dim Fix_U_i K = 2, N_H(K)/K=D4 K=H2, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H7, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H15, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 1, 2, 7, 15 ], maximals H[ 2 7 ], H/H'=D4 Jumps from H15 to H[ 2, 7, 9 ] Typejumps from S10 to S[ 2, 5, 6 ] Symmetry H=H16, Symmetry type S11, Id(H)=[ 16, 11 ] Representation 4, dimension of irred subspace U_i = 1 K=H15, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H16, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 15, 16 ], maximals H[ 15 ], H/H'=Z2 Representation 5, dimension of irred subspace U_i = 1 K=H14, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H16, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 14, 16 ], maximals H[ 14 ], H/H'=Z2 Representation 10, dimension of irred subspace U_i = 2 K=H4, dim Fix_P(U) K = 2, dim Fix_U_i K = 2, N_H(K)/K=D4 K=H10, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H12, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K=Z2 K=H16, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K=Z1 isotropy subgroups H[ 4, 10, 12, 16 ], maximals H[ 10 12 ], H/H'=D4 Jumps from H16 to H[ 10, 12, 14, 15 ] Typejumps from S11 to S[ 7, 8, 9, 10 ] This next table is written to bifs.txt, and used by Mathematica. H=fromSym fromType {K=toSym irrepNum size(N_H(K)/K)} 1 1 2 2 1 1 2 3 2 1 1 2 4 3 1 2 2 5 4 1 2 2 6 4 1 2 2 7 5 1 2 2 8 5 1 2 2 9 6 2 3 2 3 2 2 10 7 2 2 2 4 4 2 11 7 3 2 2 4 4 2 12 8 4 2 2 5 3 2 8 4 2 13 8 4 2 2 6 3 2 7 4 2 14 9 2 5 2 5 5 2 9 4 2 15 10 2 5 2 7 5 2 9 4 2 16 11 10 10 2 12 10 2 14 5 2 15 4 2 This next table is written to digraph.txt and used for the bifurcation digraph. fromType toType 2 1 2 1 3 1 4 1 4 1 5 1 5 1 6 2 6 2 7 2 7 3 7 2 7 3 8 3 8 4 8 5 8 3 8 4 8 5 9 2 9 4 9 6 10 2 10 5 10 6 11 7 11 8 11 9 11 10 This next table is just for humans. It includes H/H', the symmetry group of the bifurcation. fromType: {toType K=toSym irrepNum size(N_H(K)/K) id(H/H')} 1 : 2 : 1 1 1 2 Z2 3 : 1 1 2 2 Z2 4 : 1 1 2 2 Z2 5 : 1 1 2 2 Z2 6 : 2 2 3 2 Z2 7 : 2 3 2 2 Z2 3 4 4 2 Z2 8 : 3 4 2 2 Z2 4 6 3 2 Z2 5 7 4 2 Z2 9 : 2 2 5 2 D4 4 5 5 2 D4 6 9 4 2 Z2 10 : 2 2 5 2 D4 5 7 5 2 D4 6 9 4 2 Z2 11 : 7 10 10 2 D4 8 12 10 2 D4 9 14 5 2 Z2 10 15 4 2 Z2 This next table is just for humans using C++ numbering of symmetries. 0 [ 16, 11 ] 0 :1 1 Z2 2 2 Z2 4 3 D4 6 4 D4 1 D4 1 :7 5 Z2 9 6 D4 14 9 D4 2 D4 2 :7 5 Z2 11 7 D4 14 9 D4 3 Z2xZ2 3 :9 6 Z2 10 7 Z2 12 8 Z2 4 Z2xZ2 3 :8 6 Z2 11 7 Z2 12 8 Z2 5 Z2xZ2 4 :12 8 Z2 13 9 Z2 6 Z2xZ2 4 :12 8 Z2 14 9 Z2 7 Z2xZ2 5 :13 9 Z2 14 9 Z2 8 Z2 6 :15 10 Z2 9 Z2 6 :15 10 Z2 10 Z2 7 :15 10 Z2 11 Z2 7 :15 10 Z2 12 Z2 8 :15 10 Z2 13 Z2 9 :15 10 Z2 14 Z2 9 :15 10 Z2 15 Z1 10 : 0 Group( [ (2,4), (1,3), (5,6), (1,2)(3,4) ] ) 1 Group( [ (2,4), (1,3), (1,2)(3,4)(5,6) ] ) 2 Group( [ (2,4), (1,3), (1,2)(3,4) ] ) 3 Group( [ (1,3)(2,4)(5,6), (1,4)(2,3) ] ) 4 Group( [ (1,3)(2,4)(5,6), (1,2)(3,4) ] ) 5 Group( [ (1,3), (1,3)(2,4)(5,6) ] ) 6 Group( [ (2,4), (1,3)(2,4)(5,6) ] ) 7 Group( [ (1,3), (2,4) ] ) 8 Group( [ (1,4)(2,3)(5,6) ] ) 9 Group( [ (1,2)(3,4)(5,6) ] ) 10 Group( [ (1,4)(2,3) ] ) 11 Group( [ (1,2)(3,4) ] ) 12 Group( [ (1,3)(2,4)(5,6) ] ) 13 SymmetricGroup( [ 1, 3 .. 3 ] ) 14 SymmetricGroup( [ 2, 4 .. 4 ] ) 15 SymmetricGroup( [ ] )