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Consider the character table of the cyclic permutation group $\mathbb{Z}_4$

Z4 = CyclicPermutationGroup(4)
Z4.character_table()

gives

[     1      1      1      1]
[     1     -1      1     -1]
[     1  zeta4     -1 -zeta4]
[     1 -zeta4     -1  zeta4]

if $e$ is the identity and $r$ is a reflection then the columns correspond to transformations $e$, $r$, $r^2$ and $r^3$, which of course is intuitive.

$\mathbb{Z}_4$ can for example be obtained from the centralizer of G((1,2,3,4)) with respect to the dihedral group $D_4$

G = DihedralGroup(4)
n = G.centralizer(G((1,2,3,4)))
ctable = n.character_table()

yields the same table as above with the second and fourth columns swapped.

[     1      1      1      1]
[     1     -1      1     -1]
[     1  -zeta4     -1 zeta4]
[     1 zeta4     -1  -zeta4]

n.list() yields [(), (1,3)(2,4), (1,4,3,2), (1,2,3,4)] or $[e,r^2,r^3,r]$ while the columns above are ordered as $[e,r^3,r^2,r]$, indicating that the columns are not given standard ordering nor are they given ordering with respect to the normalizer list. Generally, what is the convention for ordering?

Consider the character table of the cyclic permutation group $\mathbb{Z}_4$

Z4 = CyclicPermutationGroup(4)
Z4.character_table()

gives

[     1      1      1      1]
[     1     -1      1     -1]
[     1  zeta4     -1 -zeta4]
[     1 -zeta4     -1  zeta4]

if $e$ is the identity and $r$ is a reflection rotation then the columns correspond to transformations $e$, $r$, $r^2$ and $r^3$, which of course is intuitive.

$\mathbb{Z}_4$ can for example be obtained from the centralizer of G((1,2,3,4)) with respect to the dihedral group $D_4$

G = DihedralGroup(4)
n = G.centralizer(G((1,2,3,4)))
ctable = n.character_table()

yields the same table as above with the second and fourth columns swapped.

[     1      1      1      1]
[     1     -1      1     -1]
[     1  -zeta4     -1 zeta4]
[     1 zeta4     -1  -zeta4]

n.list() yields [(), (1,3)(2,4), (1,4,3,2), (1,2,3,4)] or $[e,r^2,r^3,r]$ while the columns above are ordered as $[e,r^3,r^2,r]$, indicating that the columns are not given standard ordering nor are they given ordering with respect to the normalizer list. Generally, what is the convention for ordering?