Identify the space group isomorphism between the the group created by AffineCrystGroup and the one given by cryst package

Question

I use the following code snippet to create the diamond space group in GAP with the help of cryst package:

gap> M1:=[[0, 0, 1, 0],[1, 0, 0, 0],[0, -1, 0, 0],[1/4, 1/4, 1/4, 1]];;
gap> M2:=[[0,0,-1,0],[0,-1,0,0],[1,0,0,0],[0,0,0,1]];;
gap> S:=AffineCrystGroup([M1,M2]);
<matrix group with 2 generators>

The above code snippet comes from page 21 of the book Computer Algebra and Materials Physics, as shown below:

# As for the diamond case, in the GAP computation, the 
# crystallographic group is defined as follows. (The minimal
# generating set is used for simplicity.)
gap> M1:=[[0,0,1,0],[1,0,0,0],[0,-1,0,0],[1/4,1/4,1/4,1]];;
gap> M2:=[[0,0,-1,0],[0,-1,0,0],[1,0,0,0],[0,0,0,1]];;
gap> S:=AffineCrystGroup([M1,M2]);
<matrix group with 2 generators>
gap> P:=PointGroup(S);
Group([ [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, -1, 0 ] ],
[ [ 0, 0, -1 ], [ 0, -1, 0 ], [ 1, 0, 0 ] ] ])

It's well-known that diamond has the space group Fd-3m (No. 227). I wonder how I can verify/confirm/check this fact in GAP after I've created the above AffineCrystGroup.

Regards, HZ

Answer 1

Based on the command ConjugatorSpaceGroups provided by the cryst package, as described here, I figured out the following solution:

gap> M1OnRight:=[[0,0,1,0],[1,0,0,0],[0,-1,0,0],[1/4,1/4,1/4,1]];;
gap> M2OnRight:=[[0,0,-1,0],[0,-1,0,0],[1,0,0,0],[0,0,0,1]];;
gap> SG227OnRight:=AffineCrystGroupOnRight([M1OnRight,M2OnRight]);
<matrix group with 2 generators>
gap> ConjugatorSpaceGroups(SG227OnRight, SpaceGroupOnRightIT(3,227));
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 3/8, 3/8, 7/8, 1 ] ]