This post is motivated by an exchange with Zhengwei Liu. It is more than the dual version of this post, because we consider any subgroup (instead of just *maximal*), and even more at the end...

Let's first define what we mean by *dual depth of a subgroup* (coming from this book p141).

Let $G$ be a finite group, and $H$ a subgroup. The double cosets $HgH$ form a partition of $G$. Let $R_H \subset G$ be a subset of representatives for these double cosets, i.e.
$$ G=\coprod_{g \in R_H} HgH. $$
Now, let $K_g \subset H$ be the subgroup $H \cap gHg^{-1}$, for $g \in R_H$.

Consider the bipartite graph $\mathcal{G}$ whose odd vertices are (up to isom.) the irreducible complex representations (irreps) $V$ of $H$, and the even vertices are the irreps $W$ of $K_g$ with $g \in R_H$, and with $d$ edges between $V$ and $W$ if $\langle V\vert_{K_g},W \rangle = d$. Let $*$ be the even vertex which is the trivial representation of $K_{e} = H$. Let $\mathcal{G}_0$ be the connected component of $\mathcal{G}$ containing $*$, it will be called the dual principal graph. Note that $\Vert \mathcal{G}_0 \Vert^2 = |G:H|$.

**Definition**: The *dual depth* of $H⊂G$ is the distance between $*$ and a farthest vertex in $\mathcal{G}_0$.

Note that the dual depth of $H⊂G$ is $2$ if and only if $H$ is a normal subgroup. Moreover, the dual principal graph (and so the dual depth) is invariant if we quotient by the core of $H$ in $G$, so that we can assume $H$ core-free.

The dual principal graph for $\{e\} \subset C_3$ (index $3$, dual depth $2$):

In general, the dual principal graph of $\{e\} \subset G$ is just a star with $|G|$ arms of length one.

The dual principal graph of $\langle (1,2)(3,4) \rangle \subset A_4$ (index $6$, dual depth $4$):

The dual principal graph for $A_4 \subset A_5$ (index $5$, dual depth $6$):

We computed (with the code below) the dual depth of every core-free proper subgroup $H \subset G$ in the following cases:

- $|G:H|<32$ and $|G|<10^4$,
- $H$ maximal, $|G:H|<100$ and $|G|<10^6$,
- $G$ simple, $H$ maximal, and $|G|<2\cdot 10^6$.

Surprisingly, we found no one with a dual depth $3$. In fact, we found no one with an odd dual depth.

*Question 1*: Is there a subgroup of dual depth $3$?

If no:

*Question 2*: Is there a proper subgroup of odd dual depth?

If yes (to Question 2):

*Question 3*: Is there a maximal subgroup of odd dual depth?

**Code**

```
PrincipalGraph:=function(G,H)
local mat,edges;
mat:=MatScalarProducts(Irr(H),RestrictedClassFunctions(Irr(G),H)); #Print(mat);
edges := Filtered( Cartesian([1..Size(mat)],-[1..Size(mat[1])]), ij -> not IsZero(mat[ij[1]][-ij[2]]));
return edges;
end;;
DualPrincipalGraph:=function(G,H)
local dc,l,rep,grp,g,L,K,ed,edd,n,edges;
dc:=DoubleCosetRepsAndSizes(G,H,H);
rep:=List(dc,l->l[1]);
grp:=List(rep,g->Intersection(H,ConjugateGroup(H,g)));
L:=List(grp,K->PrincipalGraph(H,K));
edges:=[];
n:=0;
for ed in L do
edd:=List(ed,l->[l[1],l[2]-n]);
Append(edges,edd);
n:=n+Length(Set(List(ed,l->l[2])));
od;
edges:=List(edges,l->[-l[2],-l[1]]);
return edges;
end;;
DepthGraph:=function(Gr)
local P,dd,c,cc,PP,a;
P:=Gr;
dd:=0;
c:=[1];
while P<>[] do
PP:=[];
cc:=[];
for a in P do
if a[1] in c then
Add(cc,a[2]);
elif a[2] in c then
Add(cc,a[1]);
else
Add(PP,a);
fi;
od;
c:=cc;
P:=PP;
dd:=dd+1;
od;
return dd;
end;;
DepthDualSubgroup:=function(G,H)
local Gr;
Gr:=DualPrincipalGraph(G,H);
return DepthGraph(Gr);
end;;
DepthDualTransitive:=function(d,r)
local G,H,dd;
G:=TransitiveGroup(d,r);
H:=Stabilizer(G,1);
dd:=DepthDualSubgroup(G,H);
return dd;
end;;
DepthDualPrimitive:=function(d,r)
local G,H,dd;
G:=PrimitiveGroup(d,r);
H:=Stabilizer(G,1);
dd:=DepthDualSubgroup(G,H);
return dd;
end;;
DualDepthOddMaxSubSimple:=function(n,m)
local it,i,G,T,M,x,H,l,L,d;
it:=SimpleGroupsIterator(n,m);
l:=[]; L:=[];
for i in it do
Add(l,i);
od;
for G in l do
M:=MaximalSubgroupClassReps(G);
for H in M do
d:=DepthDualSubgroup(G,H);
Add(L,d);
if d mod 2 = 1 then
Print([d,G,H,IdGroup(H)]); #Add(L,Order(G)/Order(H));
fi;
od;
od;
L:=List(Set(L));
return L;
end;;
```