# 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?

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1

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."

( 2023-09-28 21:49:54 +0200 )edit

The documentation for GAP at https://docs.gap-system.org/doc/ref/c... does not provide further information on the ordering.

( 2023-09-28 21:55:00 +0200 )edit

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()

( 2023-09-29 06:08:51 +0200 )edit

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

( 2023-09-29 06:09:58 +0200 )edit

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?

( 2023-09-29 19:56:01 +0200 )edit