# Revision history [back]

### character table of normalizer

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?

### character table of normalizer

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?