================================================= C9_1_3 1 [ 1 .. 9 ] Group( () ) [ 1, 1 ] 2 [ 1, 2, 2, 4, 5, 6, 6, 5, 4 ] Group( [ (2,3)(4,9)(5,8)(6,7) ] ) [ 2, 1 ] [ 1, 2, 1, 4, 5, 6, 5, 4, 6 ] Group( [ (1,3)(4,8)(5,7)(6,9) ] ) [ 2, 1 ] [ 1, 1, 3, 4, 5, 6, 4, 6, 5 ] Group( [ (1,2)(4,7)(5,9)(6,8) ] ) [ 2, 1 ] 3 [ 0, 2, -2, 4, 5, 6, -6, -5, -4 ] Group( [ ( 2, 3)( 4, 9)( 5, 8)( 6, 7)(10,11) ] ) [ 2, 1 ] [ 1, 0, -1, 4, 5, 6, -5, -4, -6 ] Group( [ ( 1, 3)( 4, 8)( 5, 7)( 6, 9)(10,11) ] ) [ 2, 1 ] [ 1, -1, 0, 4, 5, 6, -4, -6, -5 ] Group( [ ( 1, 2)( 4, 7)( 5, 9)( 6, 8)(10,11) ] ) [ 2, 1 ] 4 [ 1, 1, 1, 4, 4, 4, 7, 7, 7 ] Group( [ (1,2,3)(4,5,6)(7,8,9) ] ) [ 3, 1 ] 5 [ 1, 1, 1, 4, 4, 4, 4, 4, 4 ] Group( [ (1,2,3)(4,5,6)(7,8,9), (2,3)(4,9)(5,8)(6,7) ] ) [ 6, 1 ] 6 [ 0, 0, 0, 4, 4, 4, -4, -4, -4 ] Group( [ (1,2,3)(4,5,6)(7,8,9), ( 2, 3)( 4, 9)( 5, 8)( 6, 7)(10,11) ] ) [ 6, 1 ] 7 [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] Group( [ (1,2,3)(4,5,6)(7,8,9), (2,3)(4,9)(5,8)(6,7), (10,11) ] ) [ 12, 4 ] Subgroup structure [ [ 0, 1, 2, 3, 4, 5, 6, 7 ], [ 1, 1, 1, 1, 1, 1, 1, 1 ], [ 2, 0, 1, 0, 0, 1, 0, 1 ], [ 3, 0, 0, 1, 0, 0, 1, 1 ], [ 4, 0, 0, 0, 1, 1, 1, 1 ], [ 5, 0, 0, 0, 0, 1, 0, 1 ], [ 6, 0, 0, 0, 0, 0, 1, 1 ], [ 7, 0, 0, 0, 0, 0, 0, 1 ] ] 9 7 1 3 3 1 1 1 1 1 2 2 3 6 6 12 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 3 2 9 8 7 6 5 4 1 2 3 4 5 6 7 8 9 3 2 1 8 7 9 5 4 6 1 2 3 4 5 6 7 8 9 2 1 3 7 9 8 4 6 5 1 2 3 4 5 6 7 8 9 -1 -3 -2 -9 -8 -7 -6 -5 -4 1 2 3 4 5 6 7 8 9 -3 -2 -1 -8 -7 -9 -5 -4 -6 1 2 3 4 5 6 7 8 9 -2 -1 -3 -7 -9 -8 -4 -6 -5 1 2 3 4 5 6 7 8 9 2 3 1 5 6 4 8 9 7 3 1 2 6 4 5 9 7 8 1 2 3 4 5 6 7 8 9 1 3 2 9 8 7 6 5 4 2 3 1 5 6 4 8 9 7 3 2 1 8 7 9 5 4 6 2 1 3 7 9 8 4 6 5 3 1 2 6 4 5 9 7 8 1 2 3 4 5 6 7 8 9 -1 -3 -2 -9 -8 -7 -6 -5 -4 2 3 1 5 6 4 8 9 7 -3 -2 -1 -8 -7 -9 -5 -4 -6 -2 -1 -3 -7 -9 -8 -4 -6 -5 3 1 2 6 4 5 9 7 8 1 2 3 4 5 6 7 8 9 -1 -2 -3 -4 -5 -6 -7 -8 -9 3 1 2 6 4 5 9 7 8 -3 -1 -2 -6 -4 -5 -9 -7 -8 3 2 1 8 7 9 5 4 6 -3 -2 -1 -8 -7 -9 -5 -4 -6 1 3 2 9 8 7 6 5 4 -1 -3 -2 -9 -8 -7 -6 -5 -4 2 1 3 7 9 8 4 6 5 -2 -1 -3 -7 -9 -8 -4 -6 -5 2 3 1 5 6 4 8 9 7 -2 -3 -1 -5 -6 -4 -8 -9 -7 Symmetry H=H1, Symmetry type S1, Id(H)=[ 1, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 9, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1 ], maximals , factor group id(H/H')=[ 1, 1 ] Jumps from H1 to H[ ] Typejumps from S1 to S[ ] Symmetry H=H2, Symmetry type S2, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H2, dim Fix_P(U) K = 5, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 2 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 4, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H2, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 2 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H2 to H[ 1 ] Typejumps from S2 to S[ 1 ] Symmetry H=H3, Symmetry type S2, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H3, dim Fix_P(U) K = 5, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 3 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 4, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H3, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 3 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H3 to H[ 1 ] Typejumps from S2 to S[ 1 ] Symmetry H=H4, Symmetry type S2, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H4, dim Fix_P(U) K = 5, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 4 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 4, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H4, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 4 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H4 to H[ 1 ] Typejumps from S2 to S[ 1 ] Symmetry H=H5, Symmetry type S3, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H5, dim Fix_P(U) K = 4, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 5 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 5, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H5, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 5 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H5 to H[ 1 ] Typejumps from S3 to S[ 1 ] Symmetry H=H6, Symmetry type S3, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H6, dim Fix_P(U) K = 4, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 6 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 5, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H6, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 6 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H6 to H[ 1 ] Typejumps from S3 to S[ 1 ] Symmetry H=H7, Symmetry type S3, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H7, dim Fix_P(U) K = 4, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 7 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 5, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H7, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 7 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H7 to H[ 1 ] Typejumps from S3 to S[ 1 ] Symmetry H=H8, Symmetry type S4, Id(H)=[ 3, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H8, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 8 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 NON-REAL projection K=H1, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 3, 1 ] K=H8, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 8 ], maximals 1 , factor group id(H/H')=[ 3, 1 ] Representation 3, dimension of irred subspace U_i = 1 NON-REAL projection K=H1, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 3, 1 ] K=H8, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 8 ], maximals 1 , factor group id(H/H')=[ 3, 1 ] Jumps from H8 to H[ 1, 1 ] Typejumps from S4 to S[ 1, 1 ] Symmetry H=H9, Symmetry type S5, Id(H)=[ 6, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H9, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 9 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H8, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H9, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 8, 9 ], maximals 8 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 2 K=H1, dim Fix_P(U) K = 6, dim Fix_U_i K = 2, id(N_H(K)/K)=[ 6, 1 ] K=H2, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] K=H9, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 2, 9 ], maximals 2 , factor group id(H/H')=[ 6, 1 ] Jumps from H9 to H[ 2, 8 ] Typejumps from S5 to S[ 2, 4 ] Symmetry H=H10, Symmetry type S6, Id(H)=[ 6, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H10, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 10 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H8, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H10, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 8, 10 ], maximals 8 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 2 K=H1, dim Fix_P(U) K = 6, dim Fix_U_i K = 2, id(N_H(K)/K)=[ 6, 1 ] K=H5, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] K=H10, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 5, 10 ], maximals 5 , factor group id(H/H')=[ 6, 1 ] Jumps from H10 to H[ 5, 8 ] Typejumps from S6 to S[ 3, 4 ] Symmetry H=H11, Symmetry type S7, Id(H)=[ 12, 4 ] Representation 3, dimension of irred subspace U_i = 1 K=H9, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H11, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 9, 11 ], maximals 9 , factor group id(H/H')=[ 2, 1 ] Representation 4, dimension of irred subspace U_i = 1 K=H10, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H11, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 10, 11 ], maximals 10 , factor group id(H/H')=[ 2, 1 ] Representation 5, dimension of irred subspace U_i = 2 K=H1, dim Fix_P(U) K = 6, dim Fix_U_i K = 2, id(N_H(K)/K)=[ 12, 4 ] K=H2, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H5, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H11, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 2, 5, 11 ], maximals 2 5 , factor group id(H/H')=[ 12, 4 ] Jumps from H11 to H[ 2, 5, 9, 10 ] Typejumps from S7 to S[ 2, 3, 5, 6 ] fromH : fromS : toK size(N_H(K)/K) ... 1 : 1 : 2 : 2 : 1 2 3 : 2 : 1 2 4 : 2 : 1 2 5 : 3 : 1 2 6 : 3 : 1 2 7 : 3 : 1 2 8 : 4 : 1 3 9 : 5 : 2 1 8 2 10 : 6 : 5 1 8 2 11 : 7 : 2 2 5 2 9 2 10 2 fromH toK size(N_H(K)/K) ... one from each type 1 2 1 2 3 1 2 4 1 2 5 1 2 6 1 2 7 1 2 8 1 3 9 2 1 8 2 10 5 1 8 2 11 2 2 5 2 9 2 10 2 fromS : toS size(N_H(K)/K) id(H/H')... 1 : 2 : 1 2 Z2 3 : 1 2 Z2 4 : 1 3 Z3 5 : 2 1 D3 4 2 Z2 6 : 3 1 D3 4 2 Z2 7 : 2 2 D6 3 2 D6 5 2 Z2 6 2 Z2 Typejumps from S1 to S[ ] Typejumps from S2 to S[ 1 ] Typejumps from S2 to S[ 1 ] Typejumps from S2 to S[ 1 ] Typejumps from S3 to S[ 1 ] Typejumps from S3 to S[ 1 ] Typejumps from S3 to S[ 1 ] Typejumps from S4 to S[ 1, 1 ] Typejumps from S5 to S[ 2, 4 ] Typejumps from S6 to S[ 3, 4 ] Typejumps from S7 to S[ 2, 3, 5, 6 ]