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The main purpose of this book is to find Artin's character table Ar(Q2m× D3) when m is an odd number and finding the cyclic decomposition of Artin cokernel AC(Q2m× D3) when m is an odd number. We found that if m = p_1^( _1 ). p_2^( _2 ). ... . p_n^( _n ) where p_1, p_2,..., p_n are distinct primes numbers and _1, _2,..., _n are positive integers and p_i 2 for all 1 i n then:AC (Q2m × D3) = - (2( _1 + 1)( _2 + 1)...( _n + 1)2@ C_4@i = 1) + - (4( _1 + 1)( _2 + 1)...( _n + 1)1@ C_2@i = 1)