Ordering of finite groups by the sum of orders of their elements

groupordering.g

# Code for https://math.stackexchange.com/q/2933753/

TestOneOrder:=function(n)
# find the smallest example among the groups of order n
local s,i,m,d,x;
# Calculate lists of sums of element orders.
# Avoid using AllSmallGroups(n) which potentially may be very large
s := List([1..NrSmallGroups(n)],i->Sum(List(SmallGroup(n,i),Order)));
if Length(Set(s))=NrSmallGroups(n) then
  # Sum of element orders uniquely defines each group
  return fail;
else
  # There are duplicates - find them first
  d := Filtered( Collected(s), x -> x[2] > 1 );
  # Find the minimal possible value of the sum of element orders
  m := Minimum( List( d, x-> x[1] ) );
  # Find positions of m in the list
  # Return the list of group IDs
  return List( Positions(s,m), x -> [n,x] );
fi;
end;

FindSmallestPair:=function(n)
# check all groups of order up to n
local i, res;
for i in [1..n] do
  # \r at the end of the print returns to the beginning of the line
  Print("Checking groups of order ", i, "\r");
  res := TestOneOrder(i);
  if res<>fail then
	# print new line before displaying the output 
	Print("\n");
	return res;
  fi;
od;
return fail;
end;