The character table of a finite group will be called *integral* if all its entries are integers. There are $11$ such groups up to order $16$, namely $C\_1$, $C\_2$, $C\_2^2$, $S\_3$, $D\_8$, $Q\_8$, $C\_2^3$, $D\_{12}$, $C\_2 \times D\_8$, $C\_2 \times Q\_8$ and $C\_2^3$. There are $76$ such groups up to order $120$ (see Appendix), the last one of the list, namely $S\_5$, is of specific interest as it is the first non-solvable one. In fact, every symmetric group has an integral character table. It can be proved using the following result (Lemma 7.15 in [this note](https://eugeniomaths.files.wordpress.com/2020/07/ghsvrevised12july.pdf) by [Sam Raskin](https://web.ma.utexas.edu/users/sraskin/)):



> 
> **Lemma**: Let $G$ be a finite group such that for all $g \in G$ of order $n$
> and for all $m$ coprime to $n$, $g^m$ is conjugate to $g$. Then $G$
> has an integral character table.
> 
> 
> 


**Question**: What is known about the finite groups with integral character table? Are they classified?


I mainly ask for references reviewing what is known about such groups. Otherwise here is a list of questions:


* Is the converse of above lemma true?
* I guess that $C\_2$ is the only such group which is simple. Is it true?
* I guess that $C\_1$ is the only such group of odd order. Is it true?
* Is there such a group with a non-abelian and non-alternating simple normal subgroup?
* Is every finite group a normal subgroup of such a group?




---


**Appendix**: List of all the finite groups with integral character table up to order $120$.



```
gap> for o in [1..120] do
>       if o=1 then Print("\n","|G| ","Nr ","G ","\n","\n");fi;
>       n:=NrSmallGroups(o);;
>       for i in [1..n] do
>               G:=SmallGroup(o,i);;
>               irr:=Irr(G);;
>               s:=Size(irr);;
>               c:=0;;
>               for j in [1..s] do
>                       if Conductor(irr[j])<>1 then
>                               c:=1;;
>                               break;
>                       fi;
>               od;
>               if c=0 then
>                       Print(o,"   ",i,"   ",StructureDescription(G),"\n");
>               fi;
>       od;
> od;

|G| Nr G

1   1   1
2   1   C2
4   2   C2 x C2
6   1   S3
8   3   D8
8   4   Q8
8   5   C2 x C2 x C2
12   4   D12
16   11   C2 x D8
16   12   C2 x Q8
16   14   C2 x C2 x C2 x C2
18   4   (C3 x C3) : C2
24   12   S4
24   14   C2 x C2 x S3
32   27   (C2 x C2 x C2 x C2) : C2
32   34   (C4 x C4) : C2
32   35   C4 : Q8
32   43   C8 : (C2 x C2)
32   44   (C2 x Q8) : C2
32   46   C2 x C2 x D8
32   47   C2 x C2 x Q8
32   49   (C2 x C2 x C2) : (C2 x C2)
32   50   (C2 x Q8) : C2
32   51   C2 x C2 x C2 x C2 x C2
36   10   S3 x S3
36   13   C2 x ((C3 x C3) : C2)
48   38   D8 x S3
48   40   Q8 x S3
48   48   C2 x S4
48   51   C2 x C2 x C2 x S3
54   14   (C3 x C3 x C3) : C2
64   134   ((C4 x C4) : C2) : C2
64   137   (C4 : Q8) : C2
64   138   ((C2 x C2 x C2 x C2) : C2) : C2
64   177   ((C4 x C4) : C2) : C2
64   178   (C4 : Q8) : C2
64   182   C8 : Q8
64   202   C2 x ((C2 x C2 x C2 x C2) : C2)
64   211   C2 x ((C4 x C4) : C2)
64   212   C2 x (C4 : Q8)
64   215   (C2 x C2 x D8) : C2
64   216   (C2 x ((C4 x C2) : C2)) : C2
64   217   ((C4 x C4) : C2) : C2
64   218   (C2 x ((C4 x C2) : C2)) : C2
64   224   ((C2 x Q8) : C2) : C2
64   225   (C4 : Q8) : C2
64   226   D8 x D8
64   230   Q8 x D8
64   239   Q8 x Q8
64   241   ((C4 x C2 x C2) : C2) : C2
64   242   ((C4 x C4) : C2) : C2
64   243   ((C4 x C2 x C2) : C2) : C2
64   244   (C4 : Q8) : C2
64   245   (C2 x C2) . (C2 x C2 x C2 x C2)
64   254   C2 x (C8 : (C2 x C2))
64   255   C2 x ((C2 x Q8) : C2)
64   261   C2 x C2 x C2 x D8
64   262   C2 x C2 x C2 x Q8
64   264   C2 x ((C2 x C2 x C2) : (C2 x C2))
64   265   C2 x ((C2 x Q8) : C2)
64   267   C2 x C2 x C2 x C2 x C2 x C2
72   40   (S3 x S3) : C2
72   41   (C3 x C3) : Q8
72   43   (C3 x A4) : C2
72   46   C2 x S3 x S3
72   49   C2 x C2 x ((C3 x C3) : C2)
96   209   C2 x D8 x S3
96   212   C2 x Q8 x S3
96   226   C2 x C2 x S4
96   227   ((C2 x C2 x C2 x C2) : C3) : C2
96   230   C2 x C2 x C2 x C2 x S3
108   17   ((C3 x C3) : C3) : (C2 x C2)
108   39   ((C3 x C3) : C2) x S3
108   44   C2 x ((C3 x C3 x C3) : C2)
120   34   S5

```


Finite groups with integral character table
There is no complete classification, but some structural results are known. To give you something to search for: such groups are called $\mathbb{Q}$-groups. There is a whole book devoted to their structure: [Structure and Representations of $\mathbb{Q}$-Groups](https://link.springer.com/book/10.1007%2FBFb0103426) by D. Kletzing. You will find there answers to many, if not all, of your questions.


In particular, the converse of the lemma is indeed true. It follows from two facts: firstly, characters "separate" conjugacy classes, i.e. if two elements are not conjugate, then there exists an irreducible character that takes different values on them; and secondly, if $m$ is coprime to the order of $g$, then for every irreducible character $\chi$ the values $\chi(g)$ and $\chi(g^m)$ are Galois conjugates.


