Action of automorphisms

The group of outer automorphisms of G has order 64

Number of generators: 6

Automorphism number 1

Order of the class of the automorphism in the outer automorphism group: 2

Group generator Image under automorphism

g1 g1

g2 g2 g5

g3 g3

g4 g4

g5 g5

g6 g6

Cohomology generator Image induced by automorphism

z z

y y

x x

w w

v z2 + zy + yx + v

u u

Automorphism number 2

Order of the class of the automorphism in the outer automorphism group: 2

Group generator Image under automorphism

g1 g1 g6

g2 g2

g3 g3

g4 g4 g6

g5 g5

g6 g6

Cohomology generator Image induced by automorphism

z z

y y

x x

w w

v v

u x2 w2 + u

Automorphism number 3

Order of the class of the automorphism in the outer automorphism group: 2

Group generator Image under automorphism

g1 g1

g2 g2 g6

g3 g3 g6

g4 g4 g6

g5 g5

g6 g6

Cohomology generator Image induced by automorphism

z z

y y

x x

w w

v v

u x2 w2 + u

Automorphism number 4

Order of the class of the automorphism in the outer automorphism group: 2

Group generator Image under automorphism

g1 g1

g2 g2 g5

g3 g3 g5

g4 g4 g5

g5 g5

g6 g6

Cohomology generator Image induced by automorphism

z z

y y

x x

w w

v z2 + zy + zx + yx + x2 + v

u u

Automorphism number 5

Order of the class of the automorphism in the outer automorphism group: 4

Group generator Image under automorphism

g1 g1 g4 g6

g2 g2 g6

g3 g2 g3 g6

g4 g4 g5 g6

g5 g5

g6 g6

Cohomology generator Image induced by automorphism

z z

y y + x

x x

w z + w

v v

u x4 + x2 w2 + u

Automorphism number 6

Order of the class of the automorphism in the outer automorphism group: 2

Group generator Image under automorphism

g1 g1 g4

g2 g2 g6

g3 g3 g6

g4 g4 g5

g5 g5

g6 g6

Cohomology generator Image induced by automorphism

z z

y y

x x

w z + w

v v

u u