# 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 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?

Isn't this the same question as in https://ask.sagemath.org/question/735... The columns are in the order given by the list of conjugacy classes:

`n.conjugacy_classes()`

or`n.conjugacy_classes_representatives()`

. The documentation for the second of those methods says "The ordering is that given by GAP."The documentation for GAP at https://docs.gap-system.org/doc/ref/c... does not provide further information on the ordering.

It just seems inconsistent. Like

`G.conjugacy_classes_representatives()`

for`G = CyclicPermutationGroup(4)`

and`n.conjugacy_classes_representatives()`

for above both yield`[(), (1,2,3,4), (1,3)(2,4), (1,4,3,2)]`

. But the 2nd and 4th columns for`n.character_table()`

are swapped relative to`G.character_table()`

(I made a new post just to point out the inconsistency)

How can you tell whether columns 2 and 4 have been switched, or whether rows 3 and 4 have been switched? Couldn't it be the latter?