Listing symmetries geometrically and analytically, what do I do?

[JohnMcBetty]

I have found this question and not sure where to begin in terms of solving it. PLEASE HELP!

Consider a double square pyramid . Introduce a coordinate P system so that the
vertices of P are:

A=(2,0,0)
B=(0,2,0)
C=(-2,0,0)
D=(0,-2,0)
E=(0,0,1)
F=(0,0,-1)

List the symmetries of P. Do this both geometrically and analytically.

 

[Deveno]

hint number 1: E and F are closer to the origin than the other vertices. so they either remain fixed by a symmetry, or swapped.

hint number 2: by hint number 1, all symmetries act on A,B,C and D entirely in the xy-plane. A,B,C and D form a square.

 

[JohnMcBetty]

Thanks Deveno!

Ok so far this is what I have.

If we rotate the plane containing A,B,C,D

Rotation 1, 90 degrees = A maps to B maps to C maps to D maps back to A
Rotation 2, 180 degrees = A maps to C back to A again, and B maps to D and back to B.
Rotation 3, 270 degrees = A maps to D maps to C maps to B maps back to A
Rotation 4 would just be our initial point (epsilon)

Reflection 1, through plane containing E and F, A maps to D and back to A. B maps to C and back again.
Reflection 2, through the plane containing A,B,C, and D, E maps to F and maps back to E.

I would then compose all of the rotations with all reflections

Would this be all I would have to do? Better question, did I solve this problem in a correct manner?

 

[JohnMcBetty]

Thanks Deveno!

Ok so far this is what I have.

If we rotate the plane containing A,B,C,D

Rotation 1, 90 degrees = A maps to B maps to C maps to D maps back to A
Rotation 2, 180 degrees = A maps to C back to A again, and B maps to D and back to B.
Rotation 3, 270 degrees = A maps to D maps to C maps to B maps back to A
Rotation 4 would just be our initial point (epsilon)

Reflection 1, through plane containing E and F, A maps to D and back to A. B maps to C and back again.
Reflection 2, through the plane containing A,B,C, and D, E maps to F and maps back to E.

I would then compose all of the rotations with all reflections

Would this be all I would have to do? Better question, did I solve this problem in a correct manner?

 

[Deveno]

E and F don't determine a plane, they are only two points.

you can regard these symmetries as a subgroup of S2 x S4, since the reflection through the xy-plane doesn't change A,B,C or D, and the rotations in the xy-plane and the reflections in the xy-plane don't change E or F (that is, swapping E and F commutes with all operations that only affect the xy-plane).

i count 16 distinct symmetries in all (the subgroup corresponding to the subgroup of S4 isn't all that big, we can't allow "twisted" mappings of vertices).

 

[JohnMcBetty]

So should I just say that reflection 1 is a reflection through the origin?