Class-preserving automorphisms

Definition

A subgroup of a group is termed conjugacy-closed if any two elements of the subgroup that are conjugate in the whole group are also conjugate in the subgroup.

An automorphism ζ ∈ Aut(H) of a group H is termed a class-preserving automorphism if it takes each element to within its conjugacy class, i.e. ζ(h) = h^x for some x ∈ H. Let us call C_p(H) this subgroup of automorphims such that

References

• CLASS PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS: A SURVEY.

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Group: C2

Aut(G): 1

Inn(G): 1

Class Preserving sub(G):1

Group: C3

Aut(G): C2

Inn(G): 1

Class Preserving sub(G):1

Group: C4

Aut(G): C2

Inn(G): 1

Class Preserving sub(G):1

Group: C2 x C2

Aut(G): S3

Inn(G): 1

Class Preserving sub(G):1

Group: C5

Aut(G): C4

Inn(G): 1

Class Preserving sub(G):1

Group: S3

Aut(G): S3

Inn(G): S3

Class Preserving sub(G):S3

Group: C6

Aut(G): C2

Inn(G): 1

Class Preserving sub(G):1

Group: C7

Aut(G): C6

Inn(G): 1

Class Preserving sub(G):1

Group: C8

Aut(G): C2 x C2

Inn(G): 1

Class Preserving sub(G):1

Group: C4 x C2

Aut(G): D8

Inn(G): 1

Class Preserving sub(G):1

Group: D8

Aut(G): D8

Inn(G): C2 x C2

Class Preserving sub(G):C2 x C2

Group: Q8

Aut(G): S4

Inn(G): C2 x C2

Class Preserving sub(G):C2 x C2

Group: C2 x C2 x C2

Aut(G): PSL(3,2)

Inn(G): 1

Class Preserving sub(G):1

Group: C9

Aut(G): C6

Inn(G): 1

Class Preserving sub(G):1

Group: C3 x C3

Aut(G): GL(2,3)

Inn(G): 1

Class Preserving sub(G):1

Group: D10

Aut(G): C5 : C4

Inn(G): D10

Class Preserving sub(G):D10

Group: C10

Aut(G): C4

Inn(G): 1

Class Preserving sub(G):1

Group: C11

Aut(G): C10

Inn(G): 1

Class Preserving sub(G):1

Group: C3 : C4

Aut(G): D12

Inn(G): S3

Class Preserving sub(G):S3

Group: C12

Aut(G): C2 x C2

Inn(G): 1

Class Preserving sub(G):1

Group: A4

Aut(G): S4

Inn(G): A4

Class Preserving sub(G):A4

Group: D12

Aut(G): D12

Inn(G): S3

Class Preserving sub(G):S3

Group: C6 x C2

Aut(G): D12

Inn(G): 1

Class Preserving sub(G):1

Group: C13

Aut(G): C12

Inn(G): 1

Class Preserving sub(G):1

Group: D14

Aut(G): (C7 : C3) : C2

Inn(G): D14

Class Preserving sub(G):D14

Group: C14

Aut(G): C6

Inn(G): 1

Class Preserving sub(G):1

Group: C15

Aut(G): C4 x C2

Inn(G): 1

Class Preserving sub(G):1

Group: C16

Aut(G): C4 x C2

Inn(G): 1

Class Preserving sub(G):1

Group: C4 x C4

Aut(G): (C2 x C2 x A4) : C2

Inn(G): 1

Class Preserving sub(G):1

Group: (C4 x C2) : C2

Aut(G): (C2 x C2 x C2 x C2) : C2

Inn(G): C2 x C2

Class Preserving sub(G):C2 x C2

Group: C4 : C4

Aut(G): (C2 x C2 x C2 x C2) : C2

Inn(G): C2 x C2

Class Preserving sub(G):C2 x C2

Group: C8 x C2

Aut(G): C2 x D8

Inn(G): 1

Class Preserving sub(G):1

Group: C8 : C2

Aut(G): C2 x D8

Inn(G): C2 x C2

Class Preserving sub(G):C2 x C2

Group: D16

Aut(G): (C2 x D8) : C2

Inn(G): D8

Class Preserving sub(G):D8

Group: QD16

Aut(G): C2 x D8

Inn(G): D8

Class Preserving sub(G):D8

Group: Q16

Aut(G): (C2 x D8) : C2

Inn(G): D8

Class Preserving sub(G):D8

Group: C4 x C2 x C2

Aut(G): (((C2 x D8) : C2) : C3) : C2

Inn(G): 1

Class Preserving sub(G):1

Group: C2 x D8

Aut(G): (((C4 x C2) : C2) : C2) : C2

Inn(G): C2 x C2

Class Preserving sub(G):C2 x C2

Group: C2 x Q8

Aut(G): (((C2 x C2 x C2 x C2) : C3) : C2) : C2

Inn(G): C2 x C2

Class Preserving sub(G):C2 x C2

Group: (C4 x C2) : C2

Aut(G): C2 x S4

Inn(G): C2 x C2

Class Preserving sub(G):C2 x C2

Group: C2 x C2 x C2 x C2

Aut(G): A8

Inn(G): 1

Class Preserving sub(G):1

Group: C17

Aut(G): C16

Inn(G): 1

Class Preserving sub(G):1

Group: D18

Aut(G): (C9 : C3) : C2

Inn(G): D18

Class Preserving sub(G):D18

Group: C18

Aut(G): C6

Inn(G): 1

Class Preserving sub(G):1

Group: C3 x S3

Aut(G): D12

Inn(G): S3

Class Preserving sub(G):S3

Group: (C3 x C3) : C2

Aut(G): (((C3 x C3) : Q8) : C3) : C2

Inn(G): (C3 x C3) : C2

Class Preserving sub(G):(C3 x C3) : C2

Group: C6 x C3

Aut(G): GL(2,3)

Inn(G): 1

Class Preserving sub(G):1

Group: C19

Aut(G): C18

Inn(G): 1

Class Preserving sub(G):1

Group: C5 : C4

Aut(G): C2 x (C5 : C4)

Inn(G): D10

Class Preserving sub(G):D10

Group: C20

Aut(G): C4 x C2

Inn(G): 1

Class Preserving sub(G):1

Group: C5 : C4

Aut(G): C5 : C4

Inn(G): C5 : C4

Class Preserving sub(G):C5 : C4

Group: D20

Aut(G): C2 x (C5 : C4)

Inn(G): D10

Class Preserving sub(G):D10

Group: C10 x C2

Aut(G): C4 x S3

Inn(G): 1

Class Preserving sub(G):1

Group: C7 : C3

Aut(G): (C7 : C3) : C2

Inn(G): C7 : C3

Class Preserving sub(G):C7 : C3

Group: C21

Aut(G): C6 x C2

Inn(G): 1

Class Preserving sub(G):1

Group: D22

Aut(G): (C11 : C5) : C2

Inn(G): D22

Class Preserving sub(G):D22

Group: C22

Aut(G): C10

Inn(G): 1

Class Preserving sub(G):1

Group: C23

Aut(G): C22

Inn(G): 1

Class Preserving sub(G):1

Group: C3 : C8

Aut(G): C2 x C2 x S3

Inn(G): S3

Class Preserving sub(G):S3

Group: C24

Aut(G): C2 x C2 x C2

Inn(G): 1

Class Preserving sub(G):1

Group: SL(2,3)

Aut(G): S4

Inn(G): A4

Class Preserving sub(G):A4

Group: C3 : Q8

Aut(G): D8 x S3

Inn(G): D12

Class Preserving sub(G):D12

Group: C4 x S3

Aut(G): C2 x C2 x S3

Inn(G): S3

Class Preserving sub(G):S3

Group: D24

Aut(G): D8 x S3

Inn(G): D12

Class Preserving sub(G):D12

Group: C2 x (C3 : C4)

Aut(G): D8 x S3

Inn(G): S3

Class Preserving sub(G):S3

Group: (C6 x C2) : C2

Aut(G): C2 x C2 x S3

Inn(G): D12

Class Preserving sub(G):D12

Group: C12 x C2

Aut(G): C2 x D8

Inn(G): 1

Class Preserving sub(G):1

Group: C3 x D8

Aut(G): C2 x D8

Inn(G): C2 x C2

Class Preserving sub(G):C2 x C2

Group: C3 x Q8

Aut(G): C2 x S4

Inn(G): C2 x C2

Class Preserving sub(G):C2 x C2

Group: S4

Aut(G): S4

Inn(G): S4

Class Preserving sub(G):S4

Group: C2 x A4

Aut(G): S4

Inn(G): A4

Class Preserving sub(G):A4

Group: C2 x C2 x S3

Aut(G): S3 x S4

Inn(G): S3

Class Preserving sub(G):S3

Group: C6 x C2 x C2

Aut(G): C2 x PSL(3,2)

Inn(G): 1

Class Preserving sub(G):1

Group: C25

Aut(G): C20

Inn(G): 1

Class Preserving sub(G):1

Group: C5 x C5

Aut(G): GL(2,5)

Inn(G): 1

Class Preserving sub(G):1

Group: D26

Aut(G): (C13 : C4) : C3

Inn(G): D26

Class Preserving sub(G):D26

Group: C26

Aut(G): C12

Inn(G): 1

Class Preserving sub(G):1

Group: C27

Aut(G): C18

Inn(G): 1

Class Preserving sub(G):1

Group: C9 x C3

Aut(G): C2 x (((C3 x C3) : C3) : C2)

Inn(G): 1

Class Preserving sub(G):1

Group: (C3 x C3) : C3

Aut(G): (((C3 x C3) : Q8) : C3) : C2

Inn(G): C3 x C3

Class Preserving sub(G):C3 x C3

Group: C9 : C3

Aut(G): ((C3 x C3) : C3) : C2

Inn(G): C3 x C3

Class Preserving sub(G):C3 x C3

Group: C3 x C3 x C3

Aut(G): GL(3,3)

Inn(G): 1

Class Preserving sub(G):1

Group: C7 : C4

Aut(G): C2 x ((C7 : C3) : C2)

Inn(G): D14

Class Preserving sub(G):D14

Group: C28

Aut(G): C6 x C2

Inn(G): 1

Class Preserving sub(G):1

Group: D28

Aut(G): C2 x ((C7 : C3) : C2)

Inn(G): D14

Class Preserving sub(G):D14

Group: C14 x C2

Aut(G): C6 x S3

Inn(G): 1

Class Preserving sub(G):1

Group: C29

Aut(G): C28

Inn(G): 1

Class Preserving sub(G):1

Group: C5 x S3

Aut(G): C4 x S3

Inn(G): S3

Class Preserving sub(G):S3

Group: C3 x D10

Aut(G): C2 x (C5 : C4)

Inn(G): D10

Class Preserving sub(G):D10

Group: D30

Aut(G): (C5 : C4) x S3

Inn(G): D30

Class Preserving sub(G):D30

Group: C30

Aut(G): C4 x C2

Inn(G): 1

Class Preserving sub(G):1

Group: C31

Aut(G): C30

Inn(G): 1

Class Preserving sub(G):1

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Group: C32

Aut(G) es: C8 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: (C4 x C2) : C4

Aut(G) es: (C2 x C2 x ((C2 x C2 x C2 x C2) : C3)) : C2

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: C8 x C4

Aut(G) es: (C2 x (((C4 x C2) : C2) : C2)) : C2

Inn(G): 1

Class Preserving subgroup: 1

Group: C8 : C4

Aut(G) es: (C2 x (((C4 x C2) : C2) : C2)) : C2

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: (C8 x C2) : C2

Aut(G) es: C2 x ((C2 x C2 x C2 x C2) : C2)

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: ((C4 x C2) : C2) : C2

Aut(G) es: (((C4 x C2) : C2) : C2) : C2

Inn(G): (C4 x C2) : C2

Class Preserving subgroup: (C4 x C2) : C2

Group: (C8 : C2) : C2

Aut(G) es: (D8 x D8) : C2

Inn(G): (C4 x C2) : C2

Class Preserving subgroup: (C4 x C2) : C2

Group: C2 . ((C4 x C2) : C2) = (C2 x C2) . (C4 x C2)

Aut(G) es: (D8 x D8) : C2

Inn(G): (C4 x C2) : C2

Class Preserving subgroup: (C4 x C2) : C2

Group: (C8 x C2) : C2

Aut(G) es: C2 x C2 x C2 x D8

Inn(G): D8

Class Preserving subgroup: D8

Group: Q8 : C4

Aut(G) es: C2 x C2 x C2 x D8

Inn(G): D8

Class Preserving subgroup: D8

Group: (C4 x C4) : C2

Aut(G) es: C2 x C2 x D8

Inn(G): D8

Class Preserving subgroup: D8

Group: C4 : C8

Aut(G) es: C2 x ((C2 x C2 x C2 x C2) : C2)

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: C8 : C4

Aut(G) es: (C2 x C2 x C2 x D8) : C2

Inn(G): D8

Class Preserving subgroup: D8

Group: C8 : C4

Aut(G) es: (C2 x C2 x C2 x D8) : C2

Inn(G): D8

Class Preserving subgroup: D8

Group: C4 . D8 = C4 . (C4 x C2)

Aut(G) es: C2 x ((C2 x D8) : C2)

Inn(G): D8

Class Preserving subgroup: D8

Group: C16 x C2

Aut(G) es: C2 x ((C4 x C2) : C2)

Inn(G): 1

Class Preserving subgroup: 1

Group: C16 : C2

Aut(G) es: C2 x ((C4 x C2) : C2)

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: D32

Aut(G) es: (C16 : C4) : C2

Inn(G): D16

Class Preserving subgroup: D16

Group: QD32

Aut(G) es: (C4 x D8) : C2

Inn(G): D16

Class Preserving subgroup: D16

Group: Q32

Aut(G) es: (C16 : C4) : C2

Inn(G): D16

Class Preserving subgroup: D16

Group: C4 x C4 x C2

Aut(G) es: (((C2 x C2 x ((C2 x C2 x C2 x C2) : C2)) : C2) : C3) : C2

Inn(G): 1

Class Preserving subgroup: 1

Group: C2 x ((C4 x C2) : C2)

Aut(G) es: ((C2 x (((C4 x C2) : C2) : C2)) : C4) : C2

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: C2 x (C4 : C4)

Aut(G) es: (((C2 x C2 x ((C4 x C2) : C2)) : C2) : C2) : C2

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: (C4 x C4) : C2

Aut(G) es: (C2 x C2 x ((C2 x C2 x C2 x C2) : C2)) : C2

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: C4 x D8

Aut(G) es: C2 x C2 x ((C2 x C2 x C2 x C2) : C2)

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: C4 x Q8

Aut(G) es: C2 x C2 x (((C2 x C2 x C2 x C2) : C3) : C2)

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: (C2 x C2 x C2 x C2) : C2

Aut(G) es: (C2 x C2 x ((C2 x C2 x C2 x C2) : C3)) : C2

Inn(G): C2 x C2 x C2

Class Preserving subgroup: C2 x C2 x C2

Group: (C4 x C2 x C2) : C2

Aut(G) es: C2 x C2 x ((C2 x C2 x C2 x C2) : C2)

Inn(G): C2 x C2 x C2

Class Preserving subgroup: C2 x C2 x C2

Group: (C2 x Q8) : C2

Aut(G) es: C2 x C2 x ((C2 x C2 x C2 x C2) : C2)

Inn(G): C2 x C2 x C2

Class Preserving subgroup: C2 x C2 x C2

Group: (C4 x C2 x C2) : C2

Aut(G) es: (C2 x C2 x C2 x C2 x C2 x C2) : C2

Inn(G): C2 x C2 x C2

Class Preserving subgroup: C2 x C2 x C2

Group: (C4 x C4) : C2

Aut(G) es: (C2 x C2 x ((C2 x C2 x C2 x C2) : C2)) : C2

Inn(G): C2 x C2 x C2

Class Preserving subgroup: C2 x C2 x C2

Group: (C2 x C2) . (C2 x C2 x C2)

Aut(G) es: (C2 x ((((C4 x C2) : C2) : C2) : C2)) : C2

Inn(G): C2 x C2 x C2

Class Preserving subgroup: C2 x C2 x C2

Group: (C4 x C4) : C2

Aut(G) es: C2 x C2 x ((C2 x C2 x C2 x C2) : C3)

Inn(G): C2 x C2 x C2

Class Preserving subgroup: C2 x C2 x C2

Group: (C4 x C4) : C2

Aut(G) es: ((((((C4 x C4) : C2) : C2) : C3) : C2) : C2) : C2

Inn(G): C2 x C2 x C2

Class Preserving subgroup: C2 x C2 x C2

Group: C4 : Q8

Aut(G) es: ((C2 x (((C8 : C2) : C2) : C2)) : C2) : C2

Inn(G): C2 x C2 x C2

Class Preserving subgroup: C2 x C2 x C2

Group: C8 x C2 x C2

Aut(G) es: C2 x ((((C2 x D8) : C2) : C3) : C2)

Inn(G): 1

Class Preserving subgroup: 1

Group: C2 x (C8 : C2)

Aut(G) es: C2 x ((((C4 x C2) : C2) : C2) : C2)

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: (C8 x C2) : C2

Aut(G) es: C2 x C2 x S4

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: C2 x D16

Aut(G) es: ((C2 x ((C2 x D8) : C2)) : C2) : C2

Inn(G): D8

Class Preserving subgroup: D8

Group: C2 x QD16

Aut(G) es: (C2 x C2 x C2 x D8) : C2

Inn(G): D8

Class Preserving subgroup: D8

Group: C2 x Q16

-----THIS WAS THE END FOR GAP!

Group: C33

Aut(G):C10 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: D34

Aut(G):C17 : C16

Inn(G): D34

Class Preserving subgroup: D34

Group: C34

Aut(G):C16

Inn(G): 1

Class Preserving subgroup: 1

Group: C35

Aut(G):C12 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: C9 : C4

Aut(G):C2 x ((C9 : C3) : C2)

Inn(G): D18

Class Preserving subgroup: D18

Group: C36

Aut(G):C6 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: (C2 x C2) : C9

Aut(G):C3 x S4

Inn(G): A4

Class Preserving subgroup: A4

Group: D36

Aut(G):C2 x ((C9 : C3) : C2)

Inn(G): D18

Class Preserving subgroup: D18

Group: C18 x C2

Aut(G):C6 x S3

Inn(G): 1

Class Preserving subgroup: 1

Group: C3 x (C3 : C4)

Aut(G):C2 x C2 x S3

Inn(G): S3

Class Preserving subgroup: S3

Group: (C3 x C3) : C4

Aut(G):C2 x ((((C3 x C3) : Q8) : C3) : C2)

Inn(G): (C3 x C3) : C2

Class Preserving subgroup: (C3 x C3) : C2

Group: C12 x C3

Aut(G):C2 x GL(2,3)

Inn(G): 1

Class Preserving subgroup: 1

Group: (C3 x C3) : C4

Aut(G):((C3 x C3) : C8) : C2

Inn(G): (C3 x C3) : C4

Class Preserving subgroup: (C3 x C3) : C4

Group: S3 x S3

Aut(G):(S3 x S3) : C2

Inn(G): S3 x S3

Class Preserving subgroup: S3 x S3

Group: C3 x A4

Aut(G):S4 x S3

Inn(G): A4

Class Preserving subgroup: A4

Group: C6 x S3

Aut(G):C2 x C2 x S3

Inn(G): S3

Class Preserving subgroup: S3

Group: C2 x ((C3 x C3) : C2)

Aut(G):C2 x ((((C3 x C3) : Q8) : C3) : C2)

Inn(G): (C3 x C3) : C2

Class Preserving subgroup: (C3 x C3) : C2

Group: C6 x C6

Aut(G):GL(2,3) x S3

Inn(G): 1

Class Preserving subgroup: 1

Group: C37

Aut(G):C36

Inn(G): 1

Class Preserving subgroup: 1

Group: D38

Aut(G):(C19 : C9) : C2

Inn(G): D38

Class Preserving subgroup: D38

Group: C38

Aut(G):C18

Inn(G): 1

Class Preserving subgroup: 1

Group: C13 : C3

Aut(G):(C13 : C4) : C3

Inn(G): C13 : C3

Class Preserving subgroup: C13 : C3

Group: C39

Aut(G):C12 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: C5 : C8

Aut(G):C2 x C2 x (C5 : C4)

Inn(G): D10

Class Preserving subgroup: D10

Group: C40

Aut(G):C4 x C2 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: C5 : C8

Aut(G):C2 x (C5 : C4)

Inn(G): C5 : C4

Class Preserving subgroup: C5 : C4

Group: C5 : Q8

Aut(G):(C5 : C4) x D8

Inn(G): D20

Class Preserving subgroup: D20

Group: C4 x D10

Aut(G):C2 x C2 x (C5 : C4)

Inn(G): D10

Class Preserving subgroup: D10

Group: D40

Aut(G):(C5 : C4) x D8

Inn(G): D20

Class Preserving subgroup: D20

Group: C2 x (C5 : C4)

Aut(G):(C5 : C4) x D8

Inn(G): D10

Class Preserving subgroup: D10

Group: (C10 x C2) : C2

Aut(G):C2 x C2 x (C5 : C4)

Inn(G): D20

Class Preserving subgroup: D20

Group: C20 x C2

Aut(G):C4 x D8

Inn(G): 1

Class Preserving subgroup: 1

Group: C5 x D8

Aut(G):C4 x D8

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: C5 x Q8

Aut(G):C4 x S4

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: C2 x (C5 : C4)

Aut(G):C2 x (C5 : C4)

Inn(G): C5 : C4

Class Preserving subgroup: C5 : C4

Group: C2 x C2 x D10

Aut(G):(C5 : C4) x S4

Inn(G): D10

Class Preserving subgroup: D10

Group: C10 x C2 x C2

Aut(G):C4 x PSL(3,2)

Inn(G): 1

Class Preserving subgroup: 1

Group: C41

Aut(G):C40

Inn(G): 1

Class Preserving subgroup: 1

Group: (C7 : C3) : C2

Aut(G):(C7 : C3) : C2

Inn(G): (C7 : C3) : C2

Class Preserving subgroup: (C7 : C3) : C2

Group: C2 x (C7 : C3)

Aut(G):(C7 : C3) : C2

Inn(G): C7 : C3

Class Preserving subgroup: C7 : C3

Group: C7 x S3

Aut(G):C6 x S3

Inn(G): S3

Class Preserving subgroup: S3

Group: C3 x D14

Aut(G):C2 x ((C7 : C3) : C2)

Inn(G): D14

Class Preserving subgroup: D14

Group: D42

Aut(G):((C7 : C3) : C2) x S3

Inn(G): D42

Class Preserving subgroup: D42

Group: C42

Aut(G):C6 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: C43

Aut(G):C42

Inn(G): 1

Class Preserving subgroup: 1

Group: C11 : C4

Aut(G):C2 x ((C11 : C5) : C2)

Inn(G): D22

Class Preserving subgroup: D22

Group: C44

Aut(G):C10 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: D44

Aut(G):C2 x ((C11 : C5) : C2)

Inn(G): D22

Class Preserving subgroup: D22

Group: C22 x C2

Aut(G):C10 x S3

Inn(G): 1

Class Preserving subgroup: 1

Group: C45

Aut(G):C12 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: C15 x C3

Aut(G):C4 x GL(2,3)

Inn(G): 1

Class Preserving subgroup: 1

Group: D46

Aut(G):(C23 : C11) : C2

Inn(G): D46

Class Preserving subgroup: D46

Group: C46

Aut(G):C22

Inn(G): 1

Class Preserving subgroup: 1

Group: C47

Aut(G):C46

Inn(G): 1

Class Preserving subgroup: 1

Group: C3 : C16

Aut(G):C2 x C4 x S3

Inn(G): S3

Class Preserving subgroup: S3

Group: C48

Aut(G):C4 x C2 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: (C4 x C4) : C3

Aut(G):((((C4 x C4) : C3) : C2) : C2) : C2

Inn(G): (C4 x C4) : C3

Class Preserving subgroup: (C4 x C4) : C3

Group: C8 x S3

Aut(G):C2 x C2 x C2 x S3

Inn(G): S3

Class Preserving subgroup: S3

Group: C24 : C2

Aut(G):C2 x C2 x C2 x S3

Inn(G): D12

Class Preserving subgroup: D12

Group: C24 : C2

Aut(G):C2 x D8 x S3

Inn(G): D24

Class Preserving subgroup: D24

Group: D48

Aut(G):((C2 x D8) : C2) x S3

Inn(G): D24

Class Preserving subgroup: D24

Group: C3 : Q16

Aut(G):((C2 x D8) : C2) x S3

Inn(G): D24

Class Preserving subgroup: D24

Group: C2 x (C3 : C8)

Aut(G):C2 x D8 x S3

Inn(G): S3

Class Preserving subgroup: S3

Group: (C3 : C8) : C2

Aut(G):C2 x D8 x S3

Inn(G): D12

Class Preserving subgroup: D12

Group: C4 x (C3 : C4)

Aut(G):((C2 x C2 x C2 x C2) : C2) x S3

Inn(G): S3

Class Preserving subgroup: S3

Group: (C3 : C4) : C4

Aut(G):C2 x C2 x C2 x C2 x S3

Inn(G): D12

Class Preserving subgroup: D12

Group: C12 : C4

Aut(G):((C2 x C2 x C2 x C2) : C2) x S3

Inn(G): D12

Class Preserving subgroup: D12

Group: (C12 x C2) : C2

Aut(G):C2 x C2 x C2 x C2 x S3

Inn(G): D12

Class Preserving subgroup: D12

Group: (C3 x D8) : C2

Aut(G):C2 x D8 x S3

Inn(G): (C6 x C2) : C2

Class Preserving subgroup: (C6 x C2) : C2

Group: (C3 : C8) : C2

Aut(G):C2 x D8 x S3

Inn(G): (C6 x C2) : C2

Class Preserving subgroup: (C6 x C2) : C2

Group: (C3 x Q8) : C2

Aut(G):C2 x D8 x S3

Inn(G): (C6 x C2) : C2

Class Preserving subgroup: (C6 x C2) : C2

Group: C3 : Q16

Aut(G):C2 x D8 x S3

Inn(G): (C6 x C2) : C2

Class Preserving subgroup: (C6 x C2) : C2

Group: (C2 x (C3 : C4)) : C2

Aut(G):((C2 x C2 x C2 x C2) : C2) x S3

Inn(G): D12

Class Preserving subgroup: D12

Group: C12 x C4

Aut(G):C2 x ((C2 x C2 x A4) : C2)

Inn(G): 1

Class Preserving subgroup: 1

Group: C3 x ((C4 x C2) : C2)

Aut(G):C2 x ((C2 x C2 x C2 x C2) : C2)

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: C3 x (C4 : C4)

Aut(G):C2 x ((C2 x C2 x C2 x C2) : C2)

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: C24 x C2

Aut(G):C2 x C2 x D8

Inn(G): 1

Class Preserving subgroup: 1

Group: C3 x (C8 : C2)

Aut(G):C2 x C2 x D8

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: C3 x D16

Aut(G):C2 x ((C2 x D8) : C2)

Inn(G): D8

Class Preserving subgroup: D8

Group: C3 x QD16

Aut(G):C2 x C2 x D8

Inn(G): D8

Class Preserving subgroup: D8

Group: C3 x Q16

Aut(G):C2 x ((C2 x D8) : C2)

Inn(G): D8

Class Preserving subgroup: D8

Group: C2 . S4 = SL(2,3) . C2

Aut(G):C2 x S4

Inn(G): S4

Class Preserving subgroup: S4

Group: GL(2,3)

Aut(G):C2 x S4

Inn(G): S4

Class Preserving subgroup: S4

Group: A4 : C4

Aut(G):C2 x S4

Inn(G): S4

Class Preserving subgroup: S4

Group: C4 x A4

Aut(G):C2 x S4

Inn(G): A4

Class Preserving subgroup: A4

Group: C2 x SL(2,3)

Aut(G):C2 x S4

Inn(G): A4

Class Preserving subgroup: A4

Group: SL(2,3) : C2

Aut(G):C2 x S4

Inn(G): A4

Class Preserving subgroup: A4

Group: C2 x (C3 : Q8)

Aut(G):S3 x ((((C4 x C2) : C2) : C2) : C2)

Inn(G): D12

Class Preserving subgroup: D12

Group: C2 x C4 x S3

Aut(G):((C2 x C2 x C2 x C2) : C2) x S3

Inn(G): S3

Class Preserving subgroup: S3

Group: C2 x D24

Aut(G):S3 x ((((C4 x C2) : C2) : C2) : C2)

Inn(G): D12

Class Preserving subgroup: D12

Group: (C12 x C2) : C2

Aut(G):C2 x D8 x S3

Inn(G): D12

Class Preserving subgroup: D12

Group: D8 x S3

Aut(G):C2 x D8 x S3

Inn(G): C2 x C2 x S3

Class Preserving subgroup: C2 x C2 x S3

Group: (C2 x (C3 : C4)) : C2

Aut(G):C2 x D8 x S3

Inn(G): C2 x C2 x S3

Class Preserving subgroup: C2 x C2 x S3

Group: Q8 x S3

Aut(G):C2 x S3 x S4

Inn(G): C2 x C2 x S3

Class Preserving subgroup: C2 x C2 x S3

Group: (C4 x S3) : C2

Aut(G):C2 x S3 x S4

Inn(G): C2 x C2 x S3

Class Preserving subgroup: C2 x C2 x S3

Group: C2 x C2 x (C3 : C4)

Aut(G):((((C2 x D8) : C2) : C3) : C2) x S3

Inn(G): S3

Class Preserving subgroup: S3

Group: C2 x ((C6 x C2) : C2)

Aut(G):((C2 x C2 x C2 x C2) : C2) x S3

Inn(G): D12

Class Preserving subgroup: D12

Group: C12 x C2 x C2

Aut(G):C2 x ((((C2 x D8) : C2) : C3) : C2)

Inn(G): 1

Class Preserving subgroup: 1

Group: C6 x D8

Aut(G):C2 x ((((C4 x C2) : C2) : C2) : C2)

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: C6 x Q8

Aut(G):C2 x ((((C2 x C2 x C2 x C2) : C3) : C2) : C2)

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: C3 x ((C4 x C2) : C2)

Aut(G):C2 x C2 x S4

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: C2 x S4

Aut(G):C2 x S4

Inn(G): S4

Class Preserving subgroup: S4

Group: C2 x C2 x A4

Aut(G):S4 x S3

Inn(G): A4

Class Preserving subgroup: A4

Group: (C2 x C2 x C2 x C2) : C3

Aut(G):(((C2 x C2 x C2 x C2) : A5) : C3) : C2

Inn(G): (C2 x C2 x C2 x C2) : C3

Class Preserving subgroup: (C2 x C2 x C2 x C2) : C3

Group: C2 x C2 x C2 x S3

Aut(G):S3 x ((C2 x C2 x C2) : PSL(3,2))

Inn(G): S3

Class Preserving subgroup: S3

Group: C6 x C2 x C2 x C2

Aut(G):C2 x A8

Inn(G): 1

Class Preserving subgroup: 1

Group: C49

Aut(G):C42

Inn(G): 1

Class Preserving subgroup: 1

Group: C7 x C7

Aut(G):GL(2,7)

Inn(G): 1

Class Preserving subgroup: 1

Group: D50

Aut(G):(C25 : C5) : C4

Inn(G): D50

Class Preserving subgroup: D50

Group: C50

Aut(G):C20

Inn(G): 1

Class Preserving subgroup: 1

Group: C5 x D10

Aut(G):C4 x (C5 : C4)

Inn(G): D10

Class Preserving subgroup: D10

Group: (C5 x C5) : C2

Aut(G):((C5 x C5) : SL(2,5)) : C4

Inn(G): (C5 x C5) : C2

Class Preserving subgroup: (C5 x C5) : C2

Group: C10 x C5

Aut(G):GL(2,5)

Inn(G): 1

Class Preserving subgroup: 1

Group: C51

Aut(G):C16 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: C13 : C4

Aut(G):C2 x ((C13 : C4) : C3)

Inn(G): D26

Class Preserving subgroup: D26

Group: C52

Aut(G):C12 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: C13 : C4

Aut(G):(C13 : C4) : C3

Inn(G): C13 : C4

Class Preserving subgroup: C13 : C4

Group: D52

Aut(G):C2 x ((C13 : C4) : C3)

Inn(G): D26

Class Preserving subgroup: D26

Group: C26 x C2

Aut(G):C12 x S3

Inn(G): 1

Class Preserving subgroup: 1

Group: C53

Aut(G):C52

Inn(G): 1

Class Preserving subgroup: 1

Group: D54

Aut(G):(C27 : C9) : C2

Inn(G): D54

Class Preserving subgroup: D54

Group: C54

Aut(G):C18

Inn(G): 1

Class Preserving subgroup: 1

Group: C3 x D18

Aut(G):C2 x ((C9 : C3) : C2)

Inn(G): D18

Class Preserving subgroup: D18

Group: C9 x S3

Aut(G):C6 x S3

Inn(G): S3

Class Preserving subgroup: S3

Group: ((C3 x C3) : C3) : C2

Aut(G):(((C3 x C3) : C3) : C2) : C2

Inn(G): ((C3 x C3) : C3) : C2

Class Preserving subgroup: ((C3 x C3) : C3) : C2

Group: (C9 : C3) : C2

Aut(G):(C9 : C3) : C2

Inn(G): (C9 : C3) : C2

Class Preserving subgroup: (C9 : C3) : C2

Group: (C9 x C3) : C2

Aut(G):(((C3 x ((C3 x C3 x C3) : C3)) : C3) : C2) : C2

Inn(G): (C9 x C3) : C2

Class Preserving subgroup: (C9 x C3) : C2

Group: ((C3 x C3) : C3) : C2

Aut(G):(((C3 x C3) : Q8) : C3) : C2

Inn(G): (C3 x C3) : C2

Class Preserving subgroup: (C3 x C3) : C2

Group: C18 x C3

Aut(G):C2 x (((C3 x C3) : C3) : C2)

Inn(G): 1

Class Preserving subgroup: 1

Group: C2 x ((C3 x C3) : C3)

Aut(G):(((C3 x C3) : Q8) : C3) : C2

Inn(G): C3 x C3

Class Preserving subgroup: C3 x C3

Group: C2 x (C9 : C3)

Aut(G):((C3 x C3) : C3) : C2

Inn(G): C3 x C3

Class Preserving subgroup: C3 x C3

Group: C3 x C3 x S3

Aut(G):GL(2,3) x S3

Inn(G): S3

Class Preserving subgroup: S3

Group: C3 x ((C3 x C3) : C2)

Aut(G):C2 x ((((C3 x C3) : Q8) : C3) : C2)

Inn(G): (C3 x C3) : C2

Class Preserving subgroup: (C3 x C3) : C2

Group: (C3 x C3 x C3) : C2

Aut(G):((C3 x C3 x C3) : PSL(3,3)) : C2

Inn(G): (C3 x C3 x C3) : C2

Class Preserving subgroup: (C3 x C3 x C3) : C2

Group: C6 x C3 x C3

Aut(G):GL(3,3)

Inn(G): 1

Class Preserving subgroup: 1

Group: C11 : C5

Aut(G):(C11 : C5) : C2

Inn(G): C11 : C5

Class Preserving subgroup: C11 : C5

Group: C55

Aut(G):C20 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: C7 : C8

Aut(G):C2 x C2 x ((C7 : C3) : C2)

Inn(G): D14

Class Preserving subgroup: D14

Group: C56

Aut(G):C6 x C2 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: C7 : Q8

Aut(G):((C7 : C3) : C2) x D8

Inn(G): D28

Class Preserving subgroup: D28

Group: C4 x D14

Aut(G):C2 x C2 x ((C7 : C3) : C2)

Inn(G): D14

Class Preserving subgroup: D14

Group: D56

Aut(G):((C7 : C3) : C2) x D8

Inn(G): D28

Class Preserving subgroup: D28

Group: C2 x (C7 : C4)

Aut(G):((C7 : C3) : C2) x D8

Inn(G): D14

Class Preserving subgroup: D14

Group: (C14 x C2) : C2

Aut(G):C2 x C2 x ((C7 : C3) : C2)

Inn(G): D28

Class Preserving subgroup: D28

Group: C28 x C2

Aut(G):C6 x D8

Inn(G): 1

Class Preserving subgroup: 1

Group: C7 x D8

Aut(G):C6 x D8

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: C7 x Q8

Aut(G):C6 x S4

Inn(G): C2 x C2

Class Preserving subgroup: C2 x C2

Group: (C2 x C2 x C2) : C7

Aut(G):((C2 x C2 x C2) : C7) : C3

Inn(G): (C2 x C2 x C2) : C7

Class Preserving subgroup: (C2 x C2 x C2) : C7

Group: C2 x C2 x D14

Aut(G):((C7 : C3) : C2) x S4

Inn(G): D14

Class Preserving subgroup: D14

Group: C14 x C2 x C2

Aut(G):C6 x PSL(3,2)

Inn(G): 1

Class Preserving subgroup: 1

Group: C19 : C3

Aut(G):(C19 : C9) : C2

Inn(G): C19 : C3

Class Preserving subgroup: C19 : C3

Group: C57

Aut(G):C18 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: D58

Aut(G):(C29 : C7) : C4

Inn(G): D58

Class Preserving subgroup: D58

Group: C58

Aut(G):C28

Inn(G): 1

Class Preserving subgroup: 1

Group: C59

Aut(G):C58

Inn(G): 1

Class Preserving subgroup: 1

Group: C5 x (C3 : C4)

Aut(G):C2 x C4 x S3

Inn(G): S3

Class Preserving subgroup: S3

Group: C3 x (C5 : C4)

Aut(G):C2 x C2 x (C5 : C4)

Inn(G): D10

Class Preserving subgroup: D10

Group: C15 : C4

Aut(G):C2 x ((C5 : C4) x S3)

Inn(G): D30

Class Preserving subgroup: D30

Group: C60

Aut(G):C4 x C2 x C2

Inn(G): 1

Class Preserving subgroup: 1

Group: A5

Aut(G):S5

Inn(G): A5

Class Preserving subgroup: A5

Group: C3 x (C5 : C4)

Aut(G):C2 x (C5 : C4)

Inn(G): C5 : C4

Class Preserving subgroup: C5 : C4

Group: C15 : C4

Aut(G):(C5 : C4) x S3

Inn(G): C15 : C4

Class Preserving subgroup: C15 : C4

Group: S3 x D10

Aut(G):S3 x (C5 : C4)

Inn(G): S3 x D10

Class Preserving subgroup: S3 x D10

Group: C5 x A4

Aut(G):C4 x S4

Inn(G): A4

Class Preserving subgroup: A4

Group: C6 x D10

Aut(G):C2 x C2 x (C5 : C4)

Inn(G): D10

Class Preserving subgroup: D10

Group: C10 x S3

Aut(G):C2 x C4 x S3

Inn(G): S3

Class Preserving subgroup: S3

Group: D60

Aut(G):C2 x ((C5 : C4) x S3)

Inn(G): D30

Class Preserving subgroup: D30

Group: C30 x C2

Aut(G):C2 x C4 x S3

Inn(G): 1

Class Preserving subgroup: 1

Group: C61

Aut(G):C60

Inn(G): 1

Class Preserving subgroup: 1

Group: D62

Aut(G):((C31 : C5) : C3) : C2

Inn(G): D62

Class Preserving subgroup: D62

Group: C62

Aut(G):C30

Inn(G): 1

Class Preserving subgroup: 1

Group: C7 : C9

Aut(G):C3 x ((C7 : C3) : C2)

Inn(G): C7 : C3

Class Preserving subgroup: C7 : C3

Group: C63

Aut(G):C6 x C6

Inn(G): 1

Class Preserving subgroup: 1

Group: C3 x (C7 : C3)

Aut(G):((C7 : C3) : C2) x S3

Inn(G): C7 : C3

Class Preserving subgroup: C7 : C3

Group: C21 x C3

Aut(G):C6 x GL(2,3)

Inn(G): 1

Class Preserving subgroup: 1