Let $ G $ be a quasisimple finite group. Let $ d\_{min} $ be the minimum dimension of a nontrivial irrep of $ G $. Must it be the case that the image of all (nontrivial) dimension $ d\_{min} $ irreps of $ G $ are conjugate in $ SU(d\_{min}) $?


For example all $ SL(2,5) $ subgroups of $ SU(2) $ are conjugate. This is especially interesting given that there are actually two distinct irreps of $ SL(2,5) $ into $ SU(2) $, they just happen to have conjugate image.


And I'm pretty sure that all $ A\_5,GL(3,2) $ and $ 3.A\_6 $ subgroups of $ SU(3) $ are conjugate.


And I'm pretty sure that all $ SL(2,9) \cong 2.A\_6 , 2.A\_7 $ and $ Sp(4,3) $ subgroups of $ SU(4) $ are also unique up to conjugacy



Image of minimal degree representation of quasisimple group unique up to conjugacy
Looking at the Atlas, I'd say that the smallest counterexample is probably the Mathieu group $M\_{11}$. There are three $10$-dimensional irreducible characters, not conjugate in $SU(10)$.


Edit: Actually, $L\_2(8)$ is smaller, with four 7 dimensional irreducibles. Three of them are conjugate in $SU(7)$ but the fourth isn't. I somehow missed this, first time through. Similarly for $L\_2(16)$.


