The elements are denoted by the symbols derived in the Quaternionic representation of the binary tetrahedral group.
NOTE: each element has a negative, so the table should be four times as large; but -1 commutes with everything, so the missing products are easy to retrieve.
1 i j k a a^2 b b^2 c c^2 d d^2
1 1 i j k a a^2 b b^2 c c^2 d d^2
i i -1 k -j d -c^2 c -d^2 -b a^2 -a b^2
j j -k -1 i b -d^2 -a c^2 d -b^2 -c a^2
k k j -i -1 c -b^2 -d a^2 -a d^2 b -c^2
a a c d b a^2 -1 -c^2 j -d^2 k -b^2 i
a^2 a^2 -d^2 -b^2 -c^2 -1 -a -k d -i b -j c
b b d -c -a -d^2 -j b^2 -1 -a^2 i -c^2 -k
b^2 b^2 -c^2 a^2 d^2 k c -1 -b j d -i a
c c -a b -d -b^2 -k -d^2 -i c^2 -1 -a^2 j
c^2 c^2 b^2 -d^2 a^2 i d -j a -1 -c k b
d d -b -a c -c^2 -i -a^2 k -b^2 -j d^2 -1
d^2 d^2 a^2 c^2 -b^2 j b i c -k a -1 -d
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