Centralizers of semisimple elements in G2(q) -------------------------------------------- |G(q)| = q^6 phi1^2 phi2^2 phi3 phi6 Semisimple class types: i = 1: Pi = [ 1, 2 ] j = 1: Omega trivial k = 1: F-action on Pi is () [1,1,1] Dynkin type is G_2(q) Order of center |Z^F|: 1 Numbers of classes in class type: q congruent 1 modulo 6: 1 q congruent 2 modulo 6: 1 q congruent 3 modulo 6: 1 q congruent 4 modulo 6: 1 q congruent 5 modulo 6: 1 Fusion of maximal tori of C^F in those of G^F: [ 1 .. 6 ] elements of other class types in center: [ ] i = 2: Pi = [ 1, 12 ] j = 1: Omega trivial k = 1: F-action on Pi is () [2,1,1] Dynkin type is A_2(q) Order of center |Z^F|: 1 times 1, q congruent 0 modulo 3 3, q congruent 1 modulo 3 1, q congruent 2 modulo 3 Numbers of classes in class type: q congruent 1 modulo 6: 1 q congruent 2 modulo 6: 0 q congruent 3 modulo 6: 0 q congruent 4 modulo 6: 1 q congruent 5 modulo 6: 0 Fusion of maximal tori of C^F in those of G^F: [ 1, 2, 5 ] elements of other class types in center: [ [ 1, 1, 1, 1 ] ] k = 2: F-action on Pi is ( 1,12) [2,1,2] Dynkin type is ^2A_2(q) Order of center |Z^F|: 1 times 1, q congruent 0 modulo 3 1, q congruent 1 modulo 3 3, q congruent 2 modulo 3 Numbers of classes in class type: q congruent 1 modulo 6: 0 q congruent 2 modulo 6: 1 q congruent 3 modulo 6: 0 q congruent 4 modulo 6: 0 q congruent 5 modulo 6: 1 Fusion of maximal tori of C^F in those of G^F: [ 6, 3, 4 ] elements of other class types in center: [ [ 1, 1, 1, 1 ] ] i = 3: Pi = [ 2, 12 ] j = 1: Omega trivial k = 1: F-action on Pi is () [3,1,1] Dynkin type is A_1(q) + ~A_1(q) Order of center |Z^F|: 1 times 1, q congruent 0 modulo 2 2, q congruent 1 modulo 2 Numbers of classes in class type: q congruent 1 modulo 6: 1 q congruent 2 modulo 6: 0 q congruent 3 modulo 6: 1 q congruent 4 modulo 6: 0 q congruent 5 modulo 6: 1 Fusion of maximal tori of C^F in those of G^F: [ 1, 3, 2, 6 ] elements of other class types in center: [ [ 1, 1, 1, 1 ] ] i = 4: Pi = [ 1 ] j = 1: Omega trivial k = 1: F-action on Pi is () [4,1,1] Dynkin type is A_1(q) + T(phi1) Order of center |Z^F|: phi1 Numbers of classes in class type: q congruent 1 modulo 6: 1/2 ( q-5 ) q congruent 2 modulo 6: 1/2 ( q-2 ) q congruent 3 modulo 6: 1/2 ( q-3 ) q congruent 4 modulo 6: 1/2 ( q-4 ) q congruent 5 modulo 6: 1/2 ( q-3 ) Fusion of maximal tori of C^F in those of G^F: [ 1, 2 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 1, 2 ], [ 3, 1, 1, 1 ] ] k = 2: F-action on Pi is () [4,1,2] Dynkin type is A_1(q) + T(phi2) Order of center |Z^F|: phi2 Numbers of classes in class type: q congruent 1 modulo 6: 1/2 phi1 q congruent 2 modulo 6: 1/2 ( q-2 ) q congruent 3 modulo 6: 1/2 phi1 q congruent 4 modulo 6: 1/2 q q congruent 5 modulo 6: 1/2 ( q-3 ) Fusion of maximal tori of C^F in those of G^F: [ 3, 6 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 2, 1, 2, 2 ], [ 3, 1, 1, 1 ] ] i = 5: Pi = [ 2 ] j = 1: Omega trivial k = 1: F-action on Pi is () [5,1,1] Dynkin type is ~A_1(q) + T(phi1) Order of center |Z^F|: phi1 Numbers of classes in class type: q congruent 1 modulo 6: 1/2 ( q-3 ) q congruent 2 modulo 6: 1/2 ( q-2 ) q congruent 3 modulo 6: 1/2 ( q-3 ) q congruent 4 modulo 6: 1/2 ( q-2 ) q congruent 5 modulo 6: 1/2 ( q-3 ) Fusion of maximal tori of C^F in those of G^F: [ 1, 3 ] elements of other class types in center: [ [ 1, 1, 1, 1 ], [ 3, 1, 1, 1 ] ] k = 2: F-action on Pi is () [5,1,2] Dynkin type is ~A_1(q) + T(phi2) Order of center |Z^F|: phi2 Numbers of classes in class type: q congruent 1 modulo 6: 1/2 phi1 q congruent 2 modulo 6: 1/2 q q congruent 3 modulo 6: 1/2 phi1 q congruent 4 modulo 6: 1/2 q q congruent 5 modulo 6: 1/2 phi1 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 ], [ 3, 1, 1, 1 ] ] i = 6: Pi = [ ] j = 1: Omega trivial k = 1: F-action on Pi is () [6,1,1] Dynkin type is A_0(q) + T(phi1^2) Order of center |Z^F|: phi1^2 Numbers of classes in class type: q congruent 1 modulo 6: 1/12 ( q^2-8*q+19 ) q congruent 2 modulo 6: 1/12 ( q^2-8*q+12 ) q congruent 3 modulo 6: 1/12 ( q^2-8*q+15 ) q congruent 4 modulo 6: 1/12 ( q^2-8*q+16 ) q congruent 5 modulo 6: 1/12 ( q^2-8*q+15 ) 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, 2 ], [ 3, 1, 1, 3 ], [ 4, 1, 1, 6 ], [ 5, 1, 1, 6 ] ] k = 2: F-action on Pi is () [6,1,2] Dynkin type is A_0(q) + T(phi1 phi2) Order of center |Z^F|: phi1 phi2 Numbers of classes in class type: q congruent 1 modulo 6: 1/4 phi1^2 q congruent 2 modulo 6: 1/4 q ( q-2 ) q congruent 3 modulo 6: 1/4 phi1^2 q congruent 4 modulo 6: 1/4 q ( q-2 ) q congruent 5 modulo 6: 1/4 phi1^2 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, 2 ], [ 3, 1, 1, 1 ], [ 4, 1, 1, 2 ], [ 5, 1, 2, 2 ] ] k = 3: F-action on Pi is () [6,1,3] Dynkin type is A_0(q) + T(phi1 phi2) Order of center |Z^F|: phi1 phi2 Numbers of classes in class type: q congruent 1 modulo 6: 1/4 phi1^2 q congruent 2 modulo 6: 1/4 q ( q-2 ) q congruent 3 modulo 6: 1/4 phi1^2 q congruent 4 modulo 6: 1/4 q ( q-2 ) q congruent 5 modulo 6: 1/4 phi1^2 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, 2, 2 ], [ 3, 1, 1, 1 ], [ 4, 1, 2, 2 ], [ 5, 1, 1, 2 ] ] k = 4: F-action on Pi is () [6,1,4] Dynkin type is A_0(q) + T(phi6) Order of center |Z^F|: phi6 Numbers of classes in class type: q congruent 1 modulo 6: 1/6 q phi1 q congruent 2 modulo 6: 1/6 phi2 ( q-2 ) q congruent 3 modulo 6: 1/6 q phi1 q congruent 4 modulo 6: 1/6 q phi1 q congruent 5 modulo 6: 1/6 phi2 ( q-2 ) 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, 2, 2 ] ] k = 5: F-action on Pi is () [6,1,5] Dynkin type is A_0(q) + T(phi3) Order of center |Z^F|: phi3 Numbers of classes in class type: q congruent 1 modulo 6: 1/6 phi1 ( q+2 ) q congruent 2 modulo 6: 1/6 q phi2 q congruent 3 modulo 6: 1/6 q phi2 q congruent 4 modulo 6: 1/6 phi1 ( q+2 ) q congruent 5 modulo 6: 1/6 q phi2 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, 1, 2 ] ] k = 6: F-action on Pi is () [6,1,6] Dynkin type is A_0(q) + T(phi2^2) Order of center |Z^F|: phi2^2 Numbers of classes in class type: q congruent 1 modulo 6: 1/12 phi1 ( q-3 ) q congruent 2 modulo 6: 1/12 ( q^2-4*q+4 ) q congruent 3 modulo 6: 1/12 phi1 ( q-3 ) q congruent 4 modulo 6: 1/12 q ( q-4 ) q congruent 5 modulo 6: 1/12 ( q^2-4*q+7 ) 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, 2, 2 ], [ 3, 1, 1, 3 ], [ 4, 1, 2, 6 ], [ 5, 1, 2, 6 ] ]