Subdirect subgroup of base group

[DominikBernhardt]

Implement a function that checks if a given permutation group that is a subgroup of the base group is a subdirect subgroup of the base group of a generic wreath product. This could be done with 2 versions: A no-check function and a function that checks the input. This could be a first draft:

__IsSubdirectSubgroupOnGivenSetGenericWrNC := function( sizeComponentG, H,ell )
    local componentsH;
    componentsH := List([1 .. ell], i-> Group(List(GeneratorsOfGroup(H), gen -> Base(gen,i))));
    if ForAll( [1..ell], i-> sizeComponentG=Size(componentsH[i])) then 
        return true;
    else 
        return false;
    fi;
end;

Please note that this function only works if the base group is G^ell, where |G|=sizeComponentG but it can be adjusted easily.