The generic automorphism phi of C x g_{5,5} as computed by calculautom6_55xC.r\ ed : phi:= [b(1,1) 0 0 0 0 0 ] [ ] [b(2,1) b(2,2) 0 0 0 0 ] [ ] [b(3,1) b(3,2) b(2,2)*b(1,1) 0 0 0 ] [ ] [ 2 ] [b(4,1) b(4,2) b(3,2)*b(1,1) b(2,2)*b(1,1) 0 0 ] [ ] [ 2 3 ] [b(5,1) b(5,2) b(4,2)*b(1,1) b(3,2)*b(1,1) b(2,2)*b(1,1) b(5,6)] [ ] [b(6,1) b(6,2) 0 0 0 b(6,6)] 4 7 det(phi):=b(6,6)*b(2,2) *b(1,1) generic derivation : delta:= mat((xi(1,1),0,0,0,0,0), (xi(2,1),xi(2,2),0,0,0,0), (xi(3,1),xi(3,2),xi(1,1) + xi(2,2),0,0,0), (xi(4,1),xi(4,2),xi(3,2),2*xi(1,1) + xi(2,2),0,0), (xi(5,1),xi(5,2),xi(4,2),xi(3,2),3*xi(1,1) + xi(2,2),xi(5,6)), (xi(6,1),xi(6,2),0,0,0,xi(6,6))) [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 1 0 0 0 0] adx1 := [ ] [0 0 1 0 0 0] [ ] [0 0 0 1 0 0] [ ] [0 0 0 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] adx2 := [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx3 := [ ] [-1 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx4 := [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] [ ] [0 0 0 0 0 0] The generic nilpotent derivation : the eigenvalues are 0 xi(1,1):=0 xi(2,2):=0 xi(6,6):=0 by subtracting adjoints one then may suppose: xi(3,1):=0,xi(3,2):=0,xi(4,1):=0,xi(5,1):=0 delta:= [ 0 0 0 0 0 0 ] [ ] [xi(2,1) 0 0 0 0 0 ] [ ] [ 0 0 0 0 0 0 ] [ ] [ 0 xi(4,2) 0 0 0 0 ] [ ] [ 0 xi(5,2) xi(4,2) 0 0 xi(5,6)] [ ] [xi(6,1) xi(6,2) 0 0 0 0 ] We denote this delta by the shortform shortformdelta:={xi(2,1), ss, xi(4,2), ss, xi(5,2), xi(5,6), ss, xi(6,1), xi(6,2)} paramindexeslist:={{2,1},{4,2},{5,2},{5,6},{6,1},{6,2}} With the general automorphism one gets$ shortformdeltaprimemodadg:={(b(2,2)*xi(2,1))/b(1,1), ss, b(1,1)**2*xi(4,2), ss, ( - b(6,2)*b(2,2)*b(1,1)**3*xi(5,6) + b(6,6)*b(2,2)*b(1,1)**3*xi(5,2) + b(6,6)*b (5,6)*xi(6,2))/(b(6,6)*b(2,2)), (b(2,2)*b(1,1)**3*xi(5,6))/b(6,6), ss, (b(6,2)*b(2,2)*xi(2,1) - b(6,6)*b(2,1)*xi(6,2) + b(6,6)*b(2,2)*xi(6,1))/(b(2,2)* b(1,1)), (b(6,6)*xi(6,2))/b(2,2)}$ deltaprimemodg(2,1):=(b(2,2)*xi(2,1))/b(1,1)$ deltaprimemodg(4,2):=b(1,1)**2*xi(4,2)$ deltaprimemodg(5,2):=( - b(6,2)*b(2,2)*b(1,1)**3*xi(5,6) + b(6,6)*b(2,2)*b(1,1) **3*xi(5,2) + b(6,6)*b(5,6)*xi(6,2))/(b(6,6)*b(2,2))$ deltaprimemodg(5,6):=(b(2,2)*b(1,1)**3*xi(5,6))/b(6,6)$ deltaprimemodg(6,1):=(b(6,2)*b(2,2)*xi(2,1) - b(6,6)*b(2,1)*xi(6,2) + b(6,6)*b(2 ,2)*xi(6,1))/(b(2,2)*b(1,1))$ deltaprimemodg(6,2):=(b(6,6)*xi(6,2))/b(2,2)$ det(AUTOM):=b(6,6)*b(2,2)**4*b(1,1)**7$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), b(2,2)*xi(2,1) (----------------,0,0,0,0,0), b(1,1) (0,0,0,0,0,0), 2 (0,b(1,1) *xi(4,2),0,0,0,0), 3 3 (0,( - b(6,2)*b(2,2)*b(1,1) *xi(5,6) + b(6,6)*b(2,2)*b(1,1) *xi(5,2) 2 + b(6,6)*b(5,6)*xi(6,2))/(b(6,6)*b(2,2)),b(1,1) *xi(4,2),0,0, 3 b(2,2)*b(1,1) *xi(5,6) ------------------------), b(6,6) b(6,2)*b(2,2)*xi(2,1) - b(6,6)*b(2,1)*xi(6,2) + b(6,6)*b(2,2)*xi(6,1) (-----------------------------------------------------------------------, b(2,2)*b(1,1) b(6,6)*xi(6,2) ----------------,0,0,0,0)) b(2,2) ****************** CASE 2 : xi(6,2) = 0 *************************$ xi(6,2):=0$ With the general automorphism one gets$ shortformdeltaprimemodadg:={(b(2,2)*xi(2,1))/b(1,1), ss, b(1,1)**2*xi(4,2), ss, (b(1,1)**3*( - b(6,2)*xi(5,6) + b(6,6)*xi(5,2)))/b(6,6), (b(2,2)*b(1,1)**3*xi(5,6))/b(6,6), ss, (b(6,2)*xi(2,1) + b(6,6)*xi(6,1))/b(1,1), 0}$ deltaprimemodg(2,1):=(b(2,2)*xi(2,1))/b(1,1)$ deltaprimemodg(4,2):=b(1,1)**2*xi(4,2)$ deltaprimemodg(5,2):=(b(1,1)**3*( - b(6,2)*xi(5,6) + b(6,6)*xi(5,2)))/b(6,6)$ deltaprimemodg(5,6):=(b(2,2)*b(1,1)**3*xi(5,6))/b(6,6)$ deltaprimemodg(6,1):=(b(6,2)*xi(2,1) + b(6,6)*xi(6,1))/b(1,1)$ deltaprimemodg(6,2):=0$ det(AUTOM):=b(6,6)*b(2,2)**4*b(1,1)**7$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), b(2,2)*xi(2,1) (----------------,0,0,0,0,0), b(1,1) (0,0,0,0,0,0), 2 (0,b(1,1) *xi(4,2),0,0,0,0), 3 b(1,1) *( - b(6,2)*xi(5,6) + b(6,6)*xi(5,2)) 2 (0,----------------------------------------------,b(1,1) *xi(4,2),0,0, b(6,6) 3 b(2,2)*b(1,1) *xi(5,6) ------------------------), b(6,6) b(6,2)*xi(2,1) + b(6,6)*xi(6,1) (---------------------------------,0,0,0,0,0)) b(1,1) ****************** SUBCASE 2.1 : xi(2,1) NEQ 0 *************************$ Then one may suppose xi(2,1)=1$ xi(2,1):=1$ and one keeps deltaprime(2,1)=k by taking :$ b(2,2):=b(1,1)*k$ With the general automorphism one gets$ shortformdeltaprimemodadg:={k, ss, b(1,1)**2*xi(4,2), ss, (( - b(6,2)*xi(5,6) + b(6,6)*xi(5,2))*b(1,1)**3)/b(6,6), (b(1,1)**4*xi(5,6)*k)/b(6,6), ss, (b(6,2) + b(6,6)*xi(6,1))/b(1,1), 0}$ deltaprimemodg(2,1):=k$ deltaprimemodg(4,2):=b(1,1)**2*xi(4,2)$ deltaprimemodg(5,2):=(( - b(6,2)*xi(5,6) + b(6,6)*xi(5,2))*b(1,1)**3)/b(6,6)$ deltaprimemodg(5,6):=(b(1,1)**4*xi(5,6)*k)/b(6,6)$ deltaprimemodg(6,1):=(b(6,2) + b(6,6)*xi(6,1))/b(1,1)$ deltaprimemodg(6,2):=0$ det(AUTOM):=b(6,6)*b(1,1)**11*k**4$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (k,0,0,0,0,0), (0,0,0,0,0,0), 2 (0,b(1,1) *xi(4,2),0,0,0,0), 3 ( - b(6,2)*xi(5,6) + b(6,6)*xi(5,2))*b(1,1) 2 (0,----------------------------------------------,b(1,1) *xi(4,2),0,0, b(6,6) 4 b(1,1) *xi(5,6)*k -------------------), b(6,6) b(6,2) + b(6,6)*xi(6,1) (-------------------------,0,0,0,0,0)) b(1,1) and one gets deltaprime(6,1)=0 by taking :$ b(6,2):= - b(6,6)*xi(6,1)$ Hence one can suppose xi(6,1):=0$ xi(6,1):=0$ With the general automorphism one gets$ shortformdeltaprimemodadg:={k, ss, b(1,1)**2*xi(4,2), ss, b(1,1)**3*xi(5,2), (b(1,1)**4*xi(5,6)*k)/b(6,6), ss, 0, 0}$ deltaprimemodg(2,1):=k$ deltaprimemodg(4,2):=b(1,1)**2*xi(4,2)$ deltaprimemodg(5,2):=b(1,1)**3*xi(5,2)$ deltaprimemodg(5,6):=(b(1,1)**4*xi(5,6)*k)/b(6,6)$ deltaprimemodg(6,1):=0$ deltaprimemodg(6,2):=0$ det(AUTOM):=b(6,6)*b(1,1)**11*k**4$ DELTAPRIMEMODADG:= [0 0 0 0 0 0 ] [ ] [k 0 0 0 0 0 ] [ ] [0 0 0 0 0 0 ] [ ] [ 2 ] [0 b(1,1) *xi(4,2) 0 0 0 0 ] [ ] [ 4 ] [ 3 2 b(1,1) *xi(5,6)*k ] [0 b(1,1) *xi(5,2) b(1,1) *xi(4,2) 0 0 -------------------] [ b(6,6) ] [ ] [0 0 0 0 0 0 ] Then necessarily, xi(5,6) neq 0, since we do not consider direct products.$ In fact if xi(6,1)=xi(6,2)=xi(5,6)=0 one has a direct product by Cx_7.$ Then one gets deltaprime(5,6)=k by taking :$ b(6,6):=b(1,1)**4*xi(5,6)$ Then if xi(4,2) neq 0, one gets deltaprime(4,2)=k by taking :$ b(1,1):=sqrt(k/xi(4,2))$ With the general automorphism one gets$ shortformdeltaprimemodadg:={k, ss, k, ss, (xi(5,2)*k**2)/(sqrt(k/xi(4,2))*xi(4,2)**2), k, ss, 0, 0}$ deltaprimemodg(2,1):=k$ deltaprimemodg(4,2):=k$ deltaprimemodg(5,2):=(xi(5,2)*k**2)/(sqrt(k/xi(4,2))*xi(4,2)**2)$ deltaprimemodg(5,6):=k$ deltaprimemodg(6,1):=0$ deltaprimemodg(6,2):=0$ det(AUTOM):=(sqrt(k/xi(4,2))*xi(5,6)*k**11)/xi(4,2)**7$ DELTAPRIMEMODADG:= [0 0 0 0 0 0] [ ] [k 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 k 0 0 0 0] [ ] [ 2 ] [ xi(5,2)*k ] [0 -------------------------- k 0 0 k] [ k 2 ] [ sqrt(---------)*xi(4,2) ] [ xi(4,2) ] [ ] [0 0 0 0 0 0] Finally, if xi(5,2) neq 0, one may suppose xi(5,2):=k by taking :$ k:=xi(4,2)**3/xi(5,2)**2$ With the general automorphism one gets$ shortformdeltaprimemodadg:={xi(4,2)**3/xi(5,2)**2, ss, xi(4,2)**3/xi(5,2)**2, ss, xi(4,2)**3/xi(5,2)**2, xi(4,2)**3/xi(5,2)**2, ss, 0, 0}$ deltaprimemodg(2,1):=xi(4,2)**3/xi(5,2)**2$ deltaprimemodg(4,2):=xi(4,2)**3/xi(5,2)**2$ deltaprimemodg(5,2):=xi(4,2)**3/xi(5,2)**2$ deltaprimemodg(5,6):=xi(4,2)**3/xi(5,2)**2$ deltaprimemodg(6,1):=0$ deltaprimemodg(6,2):=0$ det(AUTOM):=(xi(5,6)*xi(4,2)**27)/xi(5,2)**23$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0 ] [ ] [ 3 ] [ xi(4,2) ] [---------- 0 0 0 0 0 ] [ 2 ] [ xi(5,2) ] [ ] [ 0 0 0 0 0 0 ] [ ] [ 3 ] [ xi(4,2) ] [ 0 ---------- 0 0 0 0 ] [ 2 ] [ xi(5,2) ] [ ] [ 3 3 3 ] [ xi(4,2) xi(4,2) xi(4,2) ] [ 0 ---------- ---------- 0 0 ----------] [ 2 2 2 ] [ xi(5,2) xi(5,2) xi(5,2) ] [ ] [ 0 0 0 0 0 0 ] Hence, we are reduced in the subcase 2.1. under consideration to:$ shortformdeltaprime ={1,SS,epsilon,SS,eta,1,SS,0,0}$ where epsilon=xi(4,2)=0,1 and eta = xi(5,2)=0,1.$ ****************** SUBCASE 2.2 : xi(2,1) = 0 *************************$ xi(2,1):=0$ clear b(2,2),b(6,2),xi(6,1),b(6,6),b(1,1),k$ With the general automorphism one gets$ shortformdeltaprimemodadg:={0, ss, b(1,1)**2*xi(4,2), ss, (b(1,1)**3*( - b(6,2)*xi(5,6) + b(6,6)*xi(5,2)))/b(6,6), (b(2,2)*b(1,1)**3*xi(5,6))/b(6,6), ss, (b(6,6)*xi(6,1))/b(1,1), 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(4,2):=b(1,1)**2*xi(4,2)$ deltaprimemodg(5,2):=(b(1,1)**3*( - b(6,2)*xi(5,6) + b(6,6)*xi(5,2)))/b(6,6)$ deltaprimemodg(5,6):=(b(2,2)*b(1,1)**3*xi(5,6))/b(6,6)$ deltaprimemodg(6,1):=(b(6,6)*xi(6,1))/b(1,1)$ deltaprimemodg(6,2):=0$ det(AUTOM):=b(6,6)*b(2,2)**4*b(1,1)**7$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (0,0,0,0,0,0), (0,0,0,0,0,0), 2 (0,b(1,1) *xi(4,2),0,0,0,0), 3 b(1,1) *( - b(6,2)*xi(5,6) + b(6,6)*xi(5,2)) 2 (0,----------------------------------------------,b(1,1) *xi(4,2),0,0, b(6,6) 3 b(2,2)*b(1,1) *xi(5,6) ------------------------), b(6,6) b(6,6)*xi(6,1) (----------------,0,0,0,0,0)) b(1,1) Suppose first xi(5,6) neq 0.$ Then one may suppose xi(5,6)=1$ xi(5,6):=1$ and one keeps deltaprime(5,6)=k by taking :$ and one gets deltaprime(5,2)=0 by taking :$ b(6,2):=(b(2,2)*b(1,1)**3*xi(5,2))/k$ With the general automorphism one gets$ shortformdeltaprimemodadg:={0, ss, b(1,1)**2*xi(4,2), ss, 0, k, ss, (b(2,2)*b(1,1)**2*xi(6,1))/k, 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(4,2):=b(1,1)**2*xi(4,2)$ deltaprimemodg(5,2):=0$ deltaprimemodg(5,6):=k$ deltaprimemodg(6,1):=(b(2,2)*b(1,1)**2*xi(6,1))/k$ deltaprimemodg(6,2):=0$ det(AUTOM):=(b(2,2)**5*b(1,1)**10)/k$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 2 ] [ 0 b(1,1) *xi(4,2) 0 0 0 0] [ ] [ 2 ] [ 0 0 b(1,1) *xi(4,2) 0 0 k] [ ] [ 2 ] [ b(2,2)*b(1,1) *xi(6,1) ] [------------------------ 0 0 0 0 0] [ k ] Hence, we are reduced in the subcase 2.2. if xi(5,6) neq 0 to:$ shortformdeltaprime ={0,SS,epsilon,SS,0,1,SS,eta,0}$ where epsilon=xi(4,2)=0,1 and eta = xi(6,1)=0,1.$ Suppose now xi(5,6) = 0.$ xi(5,6):=0$ clear b(6,2),b(6,6)$ With the general automorphism one gets$ shortformdeltaprimemodadg:={0, ss, b(1,1)**2*xi(4,2), ss, b(1,1)**3*xi(5,2), 0, ss, (b(6,6)*xi(6,1))/b(1,1), 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(4,2):=b(1,1)**2*xi(4,2)$ deltaprimemodg(5,2):=b(1,1)**3*xi(5,2)$ deltaprimemodg(5,6):=0$ deltaprimemodg(6,1):=(b(6,6)*xi(6,1))/b(1,1)$ deltaprimemodg(6,2):=0$ det(AUTOM):=b(6,6)*b(2,2)**4*b(1,1)**7$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 2 ] [ 0 b(1,1) *xi(4,2) 0 0 0 0] [ ] [ 3 2 ] [ 0 b(1,1) *xi(5,2) b(1,1) *xi(4,2) 0 0 0] [ ] [ b(6,6)*xi(6,1) ] [---------------- 0 0 0 0 0] [ b(1,1) ] Then necessarily, xi(6,1) neq 0, since we do not consider direct products.$ In fact if xi(6,1)=xi(6,2)=xi(5,6)=0 one has a direct product by Cx_7.$ Then one may suppose xi(6,1)=1$ xi(6,1):=1$ Then one keeps deltaprime(6,1)=k by taking :$ b(6,6):=b(1,1)*k$ With the general automorphism one gets$ shortformdeltaprimemodadg:={0, ss, b(1,1)**2*xi(4,2), ss, b(1,1)**3*xi(5,2), 0, ss, k, 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(4,2):=b(1,1)**2*xi(4,2)$ deltaprimemodg(5,2):=b(1,1)**3*xi(5,2)$ deltaprimemodg(5,6):=0$ deltaprimemodg(6,1):=k$ deltaprimemodg(6,2):=0$ det(AUTOM):=b(2,2)**4*b(1,1)**8*k$ DELTAPRIMEMODADG:= [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [ 2 ] [0 b(1,1) *xi(4,2) 0 0 0 0] [ ] [ 3 2 ] [0 b(1,1) *xi(5,2) b(1,1) *xi(4,2) 0 0 0] [ ] [k 0 0 0 0 0] Finally, if xi(4,2) neq 0, one may suppose xi(4,2):=k by taking :$ b(1,1):=sqrt(k/xi(4,2))$ and if xi(5,2) neq 0, one may suppose xi(5,2):=k by taking :$ k:=xi(4,2)**3/xi(5,2)**2$ With the general automorphism one gets$ shortformdeltaprimemodadg:={0, ss, xi(4,2)**3/xi(5,2)**2, ss, xi(4,2)**3/xi(5,2)**2, 0, ss, xi(4,2)**3/xi(5,2)**2, 0}$ deltaprimemodg(2,1):=0$ deltaprimemodg(4,2):=xi(4,2)**3/xi(5,2)**2$ deltaprimemodg(5,2):=xi(4,2)**3/xi(5,2)**2$ deltaprimemodg(5,6):=0$ deltaprimemodg(6,1):=xi(4,2)**3/xi(5,2)**2$ deltaprimemodg(6,2):=0$ det(AUTOM):=(b(2,2)**4*xi(4,2)**11)/xi(5,2)**10$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 3 ] [ xi(4,2) ] [ 0 ---------- 0 0 0 0] [ 2 ] [ xi(5,2) ] [ ] [ 3 3 ] [ xi(4,2) xi(4,2) ] [ 0 ---------- ---------- 0 0 0] [ 2 2 ] [ xi(5,2) xi(5,2) ] [ ] [ 3 ] [ xi(4,2) ] [---------- 0 0 0 0 0] [ 2 ] [ xi(5,2) ] Hence, we are reduced in the subcase 2.2. if xi(5,6) = 0 to:$ shortformdeltaprime ={0,SS,epsilon,SS,eta,0,SS,1,0}$ where epsilon=xi(4,2)=0,1 and eta = xi(5,2)=0,1.$