# Answer to Question #23880 in Abstract Algebra for Irvin

Question #23880

Let G be the group of order 21 generated by two elements a, b with the relations a^7 = 1, b^3 = 1, and bab^−1 = a^2. How about RG and the real representations of G?

Expert's answer

Tensoring Q

that the 6-dimensional irreducible

rational (resp. real) representationof

*G**∼*Q*×*Q(*ω*)*×*M_{3}(Q(*√−*7))*,*up to R, we get R*G**∼*R*×*C*×*M_{3}(C)*,*which means that each of the irreducible Q-representations above remainsirreducible over R (and these give all irreducible R-representations). We seethat the 6-dimensional irreducible

rational (resp. real) representationof

*G*tensors up into the direct sum of the two 3-dimensional complexrepresentations of*G*. (Of course, this is also easy to see by checkingdirectly that*χ**K*=*χ*4 +*χ*5.)
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