Presentation of $C_p\ :\ C_q$ , where $p,q$ are primes and $q|p-1$

Question

There are the following presentations of some groups $C_p\ :\ C_q$ :

gap> StructureDescription(f/[a^23,b^11,b*a*b^(-1)*a^(-2)]);
"C23 : C11"
gap> StructureDescription(f/[a^29,b^7,b*a*b^(-1)*a^(-7)]);
"C29 : C7"
gap> StructureDescription(f/[a^43,b^7,b*a*b^(-1)*a^(-4)]);
"C43 : C7"

So, it seems that $ < a^p , b^q , bab^{-1}a^k >$ is a presentation for $C_p\ :\ C_q$ for some $k$. Since GAP can produces an error like this :

gap> StructureDescription(f/[a^23,b^11,b*a*b^(-1)*a^21]);
Error, exponent too large, Modified Todd-Coxeter aborted in
  ApplyRel2( app2, triple[2], triple[1] ); called from
EnterDeduction(  ); called from
LoopOverAllCosets(  ); called from
AugmentedCosetTableRrs( G, table, 0, "_x" ) called from
RelatorMatrixAbelianizedSubgroupRrs( G, H ) called from
AbelianInvariantsSubgroupFpGroup( G, H ) called from
...  at line 215 of *stdin*

it seems to be better to choose a negative value for $k$, as it is the case for the presentations above.

Question : For which $k<-1$ is $< a^p,b^q,bab^{-1}a^k >$ is a presentation for $C_p\ :\ C_q$ with primes $p,q$ and $q|p-1$ ? In particular, which is the largest such $k$ doing the job ?