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 ] ]