2 added 228 characters in body
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JavaScript (Node.js), 2222 17 bytes

(x,y)=>(y&1?y-x:y+x)&7=>y+x*7**y&7

Try it online!Try it online! Port of my answer to Cayley Table of the Dihedral Group \$D_3\$ but golfed down using the suggestions on my Python answer. Uses the following mapping:

 id | r1 | r2 | r3 | s0 | s1 | s2 | s3 
----+----+----+----+----+----+----+----
 0  | 2  | 4  | 6  | 1  | 3  | 5  | 7  

Older versions of JavaScript can be supported in a number of ways for 22 bytes:

(x,y)=>(y&1?y-x:y+x)&7
(x,y)=>y-x*(y&1||-1)&7
(x,y)=>y+x*(y<<31|1)&7

JavaScript (Node.js), 22 bytes

(x,y)=>(y&1?y-x:y+x)&7

Try it online! Port of my answer to Cayley Table of the Dihedral Group \$D_3\$. Uses the following mapping:

 id | r1 | r2 | r3 | s0 | s1 | s2 | s3 
----+----+----+----+----+----+----+----
 0  | 2  | 4  | 6  | 1  | 3  | 5  | 7  

JavaScript (Node.js), 22 17 bytes

(x,y)=>y+x*7**y&7

Try it online! Port of my answer to Cayley Table of the Dihedral Group \$D_3\$ but golfed down using the suggestions on my Python answer. Uses the following mapping:

 id | r1 | r2 | r3 | s0 | s1 | s2 | s3 
----+----+----+----+----+----+----+----
 0  | 2  | 4  | 6  | 1  | 3  | 5  | 7  

Older versions of JavaScript can be supported in a number of ways for 22 bytes:

(x,y)=>(y&1?y-x:y+x)&7
(x,y)=>y-x*(y&1||-1)&7
(x,y)=>y+x*(y<<31|1)&7
1
source | link

JavaScript (Node.js), 22 bytes

(x,y)=>(y&1?y-x:y+x)&7

Try it online! Port of my answer to Cayley Table of the Dihedral Group \$D_3\$. Uses the following mapping:

 id | r1 | r2 | r3 | s0 | s1 | s2 | s3 
----+----+----+----+----+----+----+----
 0  | 2  | 4  | 6  | 1  | 3  | 5  | 7