| 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
|