Centralizers of semisimple elements in D4_sc(q) (=Spin8(q)) ----------------------------------------------------------- |G(q)| = q^12 phi1^4 phi2^4 phi3 phi4^2 phi6 Semisimple class types: i = 1: Pi = [ 1, 2, 3, 4 ] j = 1: Omega trivial k = 1: F-action on Pi is () [1,1,1] Dynkin type is D_4(q) Order of center |Z^F|: q^0 times 1, q congruent 0 modulo 2 4, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: q^0 q congruent 1 modulo 2: 4*q^0 Fusion of maximal tori of C^F in those of G^F: [ 1 .. 13 ] elements of other class types in center: [ ] i = 2: Pi = [ 1, 2, 4, 24 ] j = 1: Omega trivial k = 1: F-action on Pi is () [2,1,1] Dynkin type is A_1(q) + A_1(q) + A_1(q) + A_1(q) Order of center |Z^F|: q^0 times 1, q congruent 0 modulo 2 8, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 0 q congruent 1 modulo 2: q^0 Fusion of maximal tori of C^F in those of G^F: [ 1, 4, 4, 2, 4, 7, 8, 6, 4, 8, 7, 6, 2, 6, 6, 3 ] elements of other class types in center: [ [ 1, 1, 1, 1 ] ] k = 2: F-action on Pi is ( 1, 2)( 4,24) [2,1,2] Dynkin type is A_1(q^2) + A_1(q^2) Order of center |Z^F|: q^0 times 1, q congruent 0 modulo 2 4, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 0 q congruent 1 modulo 2: 0 Fusion of maximal tori of C^F in those of G^F: [ 2, 5, 5, 9 ] elements of other class types in center: [ [ 1, 1, 1, 1 ] ] k = 3: F-action on Pi is ( 1, 4)( 2,24) [2,1,3] Dynkin type is A_1(q^2) + A_1(q^2) Order of center |Z^F|: q^0 times 1, q congruent 0 modulo 2 4, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 0 q congruent 1 modulo 2: 0 Fusion of maximal tori of C^F in those of G^F: [ 7, 12, 12, 9 ] elements of other class types in center: [ [ 1, 1, 1, 1 ] ] k = 4: F-action on Pi is ( 1,24)( 2, 4) [2,1,4] Dynkin type is A_1(q^2) + A_1(q^2) Order of center |Z^F|: q^0 times 1, q congruent 0 modulo 2 4, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 0 q congruent 1 modulo 2: 0 Fusion of maximal tori of C^F in those of G^F: [ 8, 13, 13, 9 ] elements of other class types in center: [ [ 1, 1, 1, 1 ] ] i = 3: Pi = [ 1, 2, 3 ] j = 1: Omega trivial k = 1: F-action on Pi is () [3,1,1] Dynkin type is A_3(q) + T(phi1) Order of center |Z^F|: q - 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/2*( q-2 ) q congruent 1 modulo 2: q - 3 Fusion of maximal tori of C^F in those of G^F: [ 1, 4, 2, 10, 5 ] elements of other class types in center: [ [ 1, 1, 1, 1 ] ] k = 2: F-action on Pi is (1,2) [3,1,2] Dynkin type is ^2A_3(q) + T(phi2) Order of center |Z^F|: q + 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/2*q q congruent 1 modulo 2: q - 1 Fusion of maximal tori of C^F in those of G^F: [ 3, 6, 2, 11, 5 ] elements of other class types in center: [ [ 1, 1, 1, 1 ] ] i = 4: Pi = [ 1, 2, 4 ] j = 1: Omega trivial k = 1: F-action on Pi is () [4,1,1] Dynkin type is A_1(q) + A_1(q) + A_1(q) + T(phi1) Order of center |Z^F|: q - 1 times 1, q congruent 0 modulo 2 4, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/2*( q-2 ) q congruent 1 modulo 2: 2*q - 6 Fusion of maximal tori of C^F in those of G^F: [ 1, 4, 4, 8, 4, 7, 2, 6 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ] ] k = 2: F-action on Pi is () [4,1,2] Dynkin type is A_1(q) + A_1(q) + A_1(q) + T(phi2) Order of center |Z^F|: q + 1 times 1, q congruent 0 modulo 2 4, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/2*q q congruent 1 modulo 2: 2*q - 2 Fusion of maximal tori of C^F in those of G^F: [ 4, 2, 7, 6, 8, 6, 6, 3 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ] ] i = 5: Pi = [ 1, 3, 4 ] j = 1: Omega trivial k = 1: F-action on Pi is () [5,1,1] Dynkin type is A_3(q) + T(phi1) Order of center |Z^F|: q - 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/2*( q-2 ) q congruent 1 modulo 2: q - 3 Fusion of maximal tori of C^F in those of G^F: [ 1, 4, 7, 10, 12 ] elements of other class types in center: [ [ 1, 1, 1, 1 ] ] k = 2: F-action on Pi is (1,4) [5,1,2] Dynkin type is ^2A_3(q) + T(phi2) Order of center |Z^F|: q + 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/2*q q congruent 1 modulo 2: q - 1 Fusion of maximal tori of C^F in those of G^F: [ 3, 6, 7, 11, 12 ] elements of other class types in center: [ [ 1, 1, 1, 1 ] ] i = 6: Pi = [ 2, 3, 4 ] j = 1: Omega trivial k = 1: F-action on Pi is () [6,1,1] Dynkin type is A_3(q) + T(phi1) Order of center |Z^F|: q - 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/2*( q-2 ) q congruent 1 modulo 2: q - 3 Fusion of maximal tori of C^F in those of G^F: [ 1, 4, 8, 10, 13 ] elements of other class types in center: [ [ 1, 1, 1, 1 ] ] k = 2: F-action on Pi is (2,4) [6,1,2] Dynkin type is ^2A_3(q) + T(phi2) Order of center |Z^F|: q + 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/2*q q congruent 1 modulo 2: q - 1 Fusion of maximal tori of C^F in those of G^F: [ 3, 6, 8, 11, 13 ] elements of other class types in center: [ [ 1, 1, 1, 1 ] ] i = 7: Pi = [ 1, 2 ] j = 1: Omega trivial k = 1: F-action on Pi is () [7,1,1] Dynkin type is A_1(q) + A_1(q) + T(phi1^2) Order of center |Z^F|: q^2 - 2*q + 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^2-6*q+8 ) q congruent 1 modulo 2: 1/4*( q^2-8*q+15 ) Fusion of maximal tori of C^F in those of G^F: [ 1, 4, 4, 2 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 3, 1, 1, 4 ], [ 4, 1, 1, 4 ] ] k = 2: F-action on Pi is () [7,1,2] Dynkin type is A_1(q) + A_1(q) + T(phi1 phi2) Order of center |Z^F|: q^2 - 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/4*( q^2-2*q ) q congruent 1 modulo 2: 1/2*( q^2-4*q+3 ) Fusion of maximal tori of C^F in those of G^F: [ 4, 8, 7, 6 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 4, 1, 1, 2 ], [ 4, 1, 2, 2 ] ] k = 3: F-action on Pi is () [7,1,3] Dynkin type is A_1(q) + A_1(q) + T(phi2^2) Order of center |Z^F|: q^2 + 2*q + 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^2-2*q ) q congruent 1 modulo 2: 1/4*( q^2-4*q+3 ) Fusion of maximal tori of C^F in those of G^F: [ 2, 6, 6, 3 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 3, 1, 2, 4 ], [ 4, 1, 2, 4 ] ] k = 4: F-action on Pi is (1,2) [7,1,4] Dynkin type is A_1(q^2) + T(phi1 phi2) Order of center |Z^F|: q^2 - 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/4*( q^2-2*q ) q congruent 1 modulo 2: 1/2*( q^2-2*q+1 ) Fusion of maximal tori of C^F in those of G^F: [ 2, 5 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 2, 4 ], [ 3, 1, 1, 2 ], [ 3, 1, 2, 2 ] ] k = 5: F-action on Pi is (1,2) [7,1,5] Dynkin type is A_1(q^2) + T(phi4) Order of center |Z^F|: q^2 + 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/4*q^2 q congruent 1 modulo 2: 1/2*( q^2-1 ) Fusion of maximal tori of C^F in those of G^F: [ 5, 9 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 2, 4 ] ] i = 8: Pi = [ 1, 3 ] j = 1: Omega trivial k = 1: F-action on Pi is () [8,1,1] Dynkin type is A_2(q) + T(phi1^2) Order of center |Z^F|: q^2 - 2*q + 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/2*( q^2-5*q+6 ) q congruent 1 modulo 2: 1/2*( q^2-8*q+15 ) Fusion of maximal tori of C^F in those of G^F: [ 1, 4, 10 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 3, 1, 1, 2 ], [ 5, 1, 1, 2 ], [ 6, 1, 1, 2 ] ] k = 2: F-action on Pi is (1,3) [8,1,2] Dynkin type is ^2A_2(q) + T(phi2^2) Order of center |Z^F|: q^2 + 2*q + 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/2*( q^2-q ) q congruent 1 modulo 2: 1/2*( q^2-4*q+3 ) Fusion of maximal tori of C^F in those of G^F: [ 3, 6, 11 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 3, 1, 2, 2 ], [ 5, 1, 2, 2 ], [ 6, 1, 2, 2 ] ] i = 9: Pi = [ 1, 4 ] j = 1: Omega trivial k = 1: F-action on Pi is () [9,1,1] Dynkin type is A_1(q) + A_1(q) + T(phi1^2) Order of center |Z^F|: q^2 - 2*q + 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^2-6*q+8 ) q congruent 1 modulo 2: 1/4*( q^2-8*q+15 ) Fusion of maximal tori of C^F in those of G^F: [ 1, 4, 4, 7 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 4, 1, 1, 4 ], [ 5, 1, 1, 4 ] ] k = 2: F-action on Pi is (1,4) [9,1,2] Dynkin type is A_1(q^2) + T(phi1 phi2) Order of center |Z^F|: q^2 - 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/4*( q^2-2*q ) q congruent 1 modulo 2: 1/2*( q^2-2*q+1 ) Fusion of maximal tori of C^F in those of G^F: [ 7, 12 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 3, 4 ], [ 5, 1, 1, 2 ], [ 5, 1, 2, 2 ] ] k = 3: F-action on Pi is () [9,1,3] Dynkin type is A_1(q) + A_1(q) + T(phi2^2) Order of center |Z^F|: q^2 + 2*q + 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^2-2*q ) q congruent 1 modulo 2: 1/4*( q^2-4*q+3 ) Fusion of maximal tori of C^F in those of G^F: [ 7, 6, 6, 3 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 4, 1, 2, 4 ], [ 5, 1, 2, 4 ] ] k = 4: F-action on Pi is () [9,1,4] Dynkin type is A_1(q) + A_1(q) + T(phi1 phi2) Order of center |Z^F|: q^2 - 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/4*( q^2-2*q ) q congruent 1 modulo 2: 1/2*( q^2-4*q+3 ) Fusion of maximal tori of C^F in those of G^F: [ 4, 8, 2, 6 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 4, 1, 1, 2 ], [ 4, 1, 2, 2 ] ] k = 5: F-action on Pi is (1,4) [9,1,5] Dynkin type is A_1(q^2) + T(phi4) Order of center |Z^F|: q^2 + 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/4*q^2 q congruent 1 modulo 2: 1/2*( q^2-1 ) Fusion of maximal tori of C^F in those of G^F: [ 12, 9 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 3, 4 ] ] i = 10: Pi = [ 2, 4 ] j = 1: Omega trivial k = 1: F-action on Pi is () [10,1,1] Dynkin type is A_1(q) + A_1(q) + T(phi1^2) Order of center |Z^F|: q^2 - 2*q + 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^2-6*q+8 ) q congruent 1 modulo 2: 1/4*( q^2-8*q+15 ) Fusion of maximal tori of C^F in those of G^F: [ 1, 4, 4, 8 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 4, 1, 1, 4 ], [ 6, 1, 1, 4 ] ] k = 2: F-action on Pi is (2,4) [10,1,2] Dynkin type is A_1(q^2) + T(phi1 phi2) Order of center |Z^F|: q^2 - 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/4*( q^2-2*q ) q congruent 1 modulo 2: 1/2*( q^2-2*q+1 ) Fusion of maximal tori of C^F in those of G^F: [ 8, 13 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 4, 4 ], [ 6, 1, 1, 2 ], [ 6, 1, 2, 2 ] ] k = 3: F-action on Pi is () [10,1,3] Dynkin type is A_1(q) + A_1(q) + T(phi2^2) Order of center |Z^F|: q^2 + 2*q + 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^2-2*q ) q congruent 1 modulo 2: 1/4*( q^2-4*q+3 ) Fusion of maximal tori of C^F in those of G^F: [ 8, 6, 6, 3 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 4, 1, 2, 4 ], [ 6, 1, 2, 4 ] ] k = 4: F-action on Pi is () [10,1,4] Dynkin type is A_1(q) + A_1(q) + T(phi1 phi2) Order of center |Z^F|: q^2 - 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/4*( q^2-2*q ) q congruent 1 modulo 2: 1/2*( q^2-4*q+3 ) Fusion of maximal tori of C^F in those of G^F: [ 4, 7, 2, 6 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 4, 1, 1, 2 ], [ 4, 1, 2, 2 ] ] k = 5: F-action on Pi is (2,4) [10,1,5] Dynkin type is A_1(q^2) + T(phi4) Order of center |Z^F|: q^2 + 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 0 modulo 2: 1/4*q^2 q congruent 1 modulo 2: 1/2*( q^2-1 ) Fusion of maximal tori of C^F in those of G^F: [ 13, 9 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 4, 4 ] ] i = 11: Pi = [ 1 ] j = 1: Omega trivial k = 1: F-action on Pi is () [11,1,1] Dynkin type is A_1(q) + T(phi1^3) Order of center |Z^F|: q^3 - 3*q^2 + 3*q - 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^3-10*q^2+32*q-32 ) q congruent 1 modulo 2: 1/8*( q^3-13*q^2+59*q-87 ) Fusion of maximal tori of C^F in those of G^F: [ 1, 4 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 3, 1, 1, 4 ], [ 4, 1, 1, 6 ], [ 5, 1, 1, 4 ], [ 6, 1, 1, 4 ], [ 7, 1, 1, 8 ], [ 8, 1, 1, 8 ], [ 9, 1, 1, 8 ], [ 10, 1, 1, 8 ] ] k = 2: F-action on Pi is () [11,1,2] Dynkin type is A_1(q) + T(phi1^2 phi2) Order of center |Z^F|: q^3 - q^2 - q + 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^3-4*q^2+4*q ) q congruent 1 modulo 2: 1/8*( q^3-7*q^2+15*q-9 ) Fusion of maximal tori of C^F in those of G^F: [ 4, 7 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 4, 1, 1, 4 ], [ 4, 1, 2, 2 ], [ 5, 1, 1, 4 ], [ 7, 1, 2, 4 ], [ 9, 1, 1, 8 ], [ 10, 1, 4, 4 ] ] k = 3: F-action on Pi is () [11,1,3] Dynkin type is A_1(q) + T(phi1 phi2^2) Order of center |Z^F|: q^3 + q^2 - q - 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^3-2*q^2 ) q congruent 1 modulo 2: 1/8*( q^3-5*q^2+7*q-3 ) Fusion of maximal tori of C^F in those of G^F: [ 2, 6 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 3, 1, 2, 4 ], [ 4, 1, 1, 2 ], [ 4, 1, 2, 4 ], [ 7, 1, 3, 8 ], [ 9, 1, 4, 4 ], [ 10, 1, 4, 4 ] ] k = 4: F-action on Pi is () [11,1,4] Dynkin type is A_1(q) + T(phi1^2 phi2) Order of center |Z^F|: q^3 - q^2 - q + 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^3-4*q^2+4*q ) q congruent 1 modulo 2: 1/8*( q^3-7*q^2+15*q-9 ) Fusion of maximal tori of C^F in those of G^F: [ 4, 8 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 4, 1, 1, 4 ], [ 4, 1, 2, 2 ], [ 6, 1, 1, 4 ], [ 7, 1, 2, 4 ], [ 9, 1, 4, 4 ], [ 10, 1, 1, 8 ] ] k = 5: F-action on Pi is () [11,1,5] Dynkin type is A_1(q) + T(phi1^2 phi2) Order of center |Z^F|: q^3 - q^2 - q + 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^3-4*q^2+4*q ) q congruent 1 modulo 2: 1/8*( q^3-7*q^2+15*q-9 ) Fusion of maximal tori of C^F in those of G^F: [ 4, 2 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 3, 1, 1, 4 ], [ 4, 1, 1, 4 ], [ 4, 1, 2, 2 ], [ 7, 1, 1, 8 ], [ 9, 1, 4, 4 ], [ 10, 1, 4, 4 ] ] k = 6: F-action on Pi is () [11,1,6] Dynkin type is A_1(q) + T(phi1 phi2^2) Order of center |Z^F|: q^3 + q^2 - q - 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^3-2*q^2 ) q congruent 1 modulo 2: 1/8*( q^3-5*q^2+7*q-3 ) Fusion of maximal tori of C^F in those of G^F: [ 8, 6 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 4, 1, 1, 2 ], [ 4, 1, 2, 4 ], [ 6, 1, 2, 4 ], [ 7, 1, 2, 4 ], [ 9, 1, 4, 4 ], [ 10, 1, 3, 8 ] ] k = 7: F-action on Pi is () [11,1,7] Dynkin type is A_1(q) + T(phi2^3) Order of center |Z^F|: q^3 + 3*q^2 + 3*q + 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^3-4*q^2+4*q ) q congruent 1 modulo 2: 1/8*( q^3-7*q^2+19*q-13 ) Fusion of maximal tori of C^F in those of G^F: [ 6, 3 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 3, 1, 2, 4 ], [ 4, 1, 2, 6 ], [ 5, 1, 2, 4 ], [ 6, 1, 2, 4 ], [ 7, 1, 3, 8 ], [ 8, 1, 2, 8 ], [ 9, 1, 3, 8 ], [ 10, 1, 3, 8 ] ] k = 8: F-action on Pi is () [11,1,8] Dynkin type is A_1(q) + T(phi1 phi2^2) Order of center |Z^F|: q^3 + q^2 - q - 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^3-2*q^2 ) q congruent 1 modulo 2: 1/8*( q^3-5*q^2+7*q-3 ) Fusion of maximal tori of C^F in those of G^F: [ 7, 6 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 4, 1, 1, 2 ], [ 4, 1, 2, 4 ], [ 5, 1, 2, 4 ], [ 7, 1, 2, 4 ], [ 9, 1, 3, 8 ], [ 10, 1, 4, 4 ] ] i = 12: Pi = [ ] j = 1: Omega trivial k = 1: F-action on Pi is () [12,1,1] Dynkin type is A_0(q) + T(phi1^4) Order of center |Z^F|: q^4 - 4*q^3 + 6*q^2 - 4*q + 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/192*( q^4-16*q^3+92*q^2-224*q+192 ) q congruent 1 modulo 2: 1/192*( q^4-16*q^3+110*q^2-368*q+465 ) Fusion of maximal tori of C^F in those of G^F: [ 1 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 12 ], [ 3, 1, 1, 8 ], [ 4, 1, 1, 24 ], [ 5, 1, 1, 8 ], [ 6, 1, 1, 8 ], [ 7, 1, 1, 48 ], [ 8, 1, 1, 32 ], [ 9, 1, 1, 48 ], [ 10, 1, 1, 48 ], [ 11, 1, 1, 96 ] ] k = 2: F-action on Pi is () [12,1,2] Dynkin type is A_0(q) + T(phi1^2 phi2^2) Order of center |Z^F|: q^4 - 2*q^2 + 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/32*( q^4-4*q^3+8*q ) q congruent 1 modulo 2: 1/32*( q^4-4*q^3+2*q^2+4*q-3 ) Fusion of maximal tori of C^F in those of G^F: [ 2 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 2, 1, 2, 8 ], [ 3, 1, 1, 4 ], [ 3, 1, 2, 4 ], [ 4, 1, 1, 4 ], [ 4, 1, 2, 4 ], [ 7, 1, 1, 8 ], [ 7, 1, 3, 8 ], [ 7, 1, 4, 16 ], [ 9, 1, 4, 8 ], [ 10, 1, 4, 8 ], [ 11, 1, 3, 16 ], [ 11, 1, 5, 16 ] ] k = 3: F-action on Pi is () [12,1,3] Dynkin type is A_0(q) + T(phi2^4) Order of center |Z^F|: q^4 + 4*q^3 + 6*q^2 + 4*q + 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/192*( q^4-8*q^3+20*q^2-16*q ) q congruent 1 modulo 2: 1/192*( q^4-8*q^3+38*q^2-88*q+57 ) Fusion of maximal tori of C^F in those of G^F: [ 3 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 12 ], [ 3, 1, 2, 8 ], [ 4, 1, 2, 24 ], [ 5, 1, 2, 8 ], [ 6, 1, 2, 8 ], [ 7, 1, 3, 48 ], [ 8, 1, 2, 32 ], [ 9, 1, 3, 48 ], [ 10, 1, 3, 48 ], [ 11, 1, 7, 96 ] ] k = 4: F-action on Pi is () [12,1,4] Dynkin type is A_0(q) + T(phi1^3 phi2) Order of center |Z^F|: q^4 - 2*q^3 + 2*q - 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/16*( q^4-6*q^3+12*q^2-8*q ) q congruent 1 modulo 2: 1/16*( q^4-6*q^3+18*q^2-26*q+13 ) Fusion of maximal tori of C^F in those of G^F: [ 4 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 3, 1, 1, 4 ], [ 4, 1, 1, 6 ], [ 4, 1, 2, 2 ], [ 5, 1, 1, 4 ], [ 6, 1, 1, 4 ], [ 7, 1, 1, 8 ], [ 7, 1, 2, 4 ], [ 8, 1, 1, 8 ], [ 9, 1, 1, 8 ], [ 9, 1, 4, 4 ], [ 10, 1, 1, 8 ], [ 10, 1, 4, 4 ], [ 11, 1, 1, 8 ], [ 11, 1, 2, 8 ], [ 11, 1, 4, 8 ], [ 11, 1, 5, 8 ] ] k = 5: F-action on Pi is () [12,1,5] Dynkin type is A_0(q) + T(phi1 phi2 phi4) Order of center |Z^F|: q^4 - 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^4-2*q^2 ) q congruent 1 modulo 2: 1/8*( q^4-4*q^2+3 ) Fusion of maximal tori of C^F in those of G^F: [ 5 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 2, 4 ], [ 3, 1, 1, 2 ], [ 3, 1, 2, 2 ], [ 7, 1, 4, 4 ], [ 7, 1, 5, 4 ] ] k = 6: F-action on Pi is () [12,1,6] Dynkin type is A_0(q) + T(phi1 phi2^3) Order of center |Z^F|: q^4 + 2*q^3 - 2*q - 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/16*( q^4-2*q^3 ) q congruent 1 modulo 2: 1/16*( q^4-2*q^3+6*q^2-6*q+1 ) Fusion of maximal tori of C^F in those of G^F: [ 6 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 3, 1, 2, 4 ], [ 4, 1, 1, 2 ], [ 4, 1, 2, 6 ], [ 5, 1, 2, 4 ], [ 6, 1, 2, 4 ], [ 7, 1, 2, 4 ], [ 7, 1, 3, 8 ], [ 8, 1, 2, 8 ], [ 9, 1, 3, 8 ], [ 9, 1, 4, 4 ], [ 10, 1, 3, 8 ], [ 10, 1, 4, 4 ], [ 11, 1, 3, 8 ], [ 11, 1, 6, 8 ], [ 11, 1, 7, 8 ], [ 11, 1, 8, 8 ] ] k = 7: F-action on Pi is () [12,1,7] Dynkin type is A_0(q) + T(phi1^2 phi2^2) Order of center |Z^F|: q^4 - 2*q^2 + 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/32*( q^4-4*q^3+8*q ) q congruent 1 modulo 2: 1/32*( q^4-4*q^3+2*q^2+4*q-3 ) Fusion of maximal tori of C^F in those of G^F: [ 7 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 2, 1, 3, 8 ], [ 4, 1, 1, 4 ], [ 4, 1, 2, 4 ], [ 5, 1, 1, 4 ], [ 5, 1, 2, 4 ], [ 7, 1, 2, 8 ], [ 9, 1, 1, 8 ], [ 9, 1, 2, 16 ], [ 9, 1, 3, 8 ], [ 10, 1, 4, 8 ], [ 11, 1, 2, 16 ], [ 11, 1, 8, 16 ] ] k = 8: F-action on Pi is () [12,1,8] Dynkin type is A_0(q) + T(phi1^2 phi2^2) Order of center |Z^F|: q^4 - 2*q^2 + 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/32*( q^4-4*q^3+8*q ) q congruent 1 modulo 2: 1/32*( q^4-4*q^3+2*q^2+4*q-3 ) Fusion of maximal tori of C^F in those of G^F: [ 8 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 4 ], [ 2, 1, 4, 8 ], [ 4, 1, 1, 4 ], [ 4, 1, 2, 4 ], [ 6, 1, 1, 4 ], [ 6, 1, 2, 4 ], [ 7, 1, 2, 8 ], [ 9, 1, 4, 8 ], [ 10, 1, 1, 8 ], [ 10, 1, 2, 16 ], [ 10, 1, 3, 8 ], [ 11, 1, 4, 16 ], [ 11, 1, 6, 16 ] ] k = 9: F-action on Pi is () [12,1,9] Dynkin type is A_0(q) + T(phi4^2) Order of center |Z^F|: q^4 + 2*q^2 + 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/16*( q^4-4*q^2 ) q congruent 1 modulo 2: 1/16*( q^4-10*q^2+9 ) Fusion of maximal tori of C^F in those of G^F: [ 9 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 2, 4 ], [ 2, 1, 3, 4 ], [ 2, 1, 4, 4 ], [ 7, 1, 5, 8 ], [ 9, 1, 5, 8 ], [ 10, 1, 5, 8 ] ] k = 10: F-action on Pi is () [12,1,10] Dynkin type is A_0(q) + T(phi1^2 phi3) Order of center |Z^F|: q^4 - q^3 - q + 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/6*( q^4-q^3-q^2+q ) q congruent 1 modulo 2: 1/6*( q^4-q^3-q^2+q ) Fusion of maximal tori of C^F in those of G^F: [ 10 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 3, 1, 1, 2 ], [ 5, 1, 1, 2 ], [ 6, 1, 1, 2 ], [ 8, 1, 1, 2 ] ] k = 11: F-action on Pi is () [12,1,11] Dynkin type is A_0(q) + T(phi2^2 phi6) Order of center |Z^F|: q^4 + q^3 + q + 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/6*( q^4+q^3-q^2-q ) q congruent 1 modulo 2: 1/6*( q^4+q^3-q^2-q ) Fusion of maximal tori of C^F in those of G^F: [ 11 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 3, 1, 2, 2 ], [ 5, 1, 2, 2 ], [ 6, 1, 2, 2 ], [ 8, 1, 2, 2 ] ] k = 12: F-action on Pi is () [12,1,12] Dynkin type is A_0(q) + T(phi1 phi2 phi4) Order of center |Z^F|: q^4 - 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^4-2*q^2 ) q congruent 1 modulo 2: 1/8*( q^4-4*q^2+3 ) Fusion of maximal tori of C^F in those of G^F: [ 12 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 3, 4 ], [ 5, 1, 1, 2 ], [ 5, 1, 2, 2 ], [ 9, 1, 2, 4 ], [ 9, 1, 5, 4 ] ] k = 13: F-action on Pi is () [12,1,13] Dynkin type is A_0(q) + T(phi1 phi2 phi4) Order of center |Z^F|: q^4 - 1 Numbers of classes in class type: q congruent 0 modulo 2: 1/8*( q^4-2*q^2 ) q congruent 1 modulo 2: 1/8*( q^4-4*q^2+3 ) Fusion of maximal tori of C^F in those of G^F: [ 13 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 4, 4 ], [ 6, 1, 1, 2 ], [ 6, 1, 2, 2 ], [ 10, 1, 2, 4 ], [ 10, 1, 5, 4 ] ]