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 by Sam Raskin):
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
