Question

it is well known that the subgroups of SL(2,C) can be determined, I am wondering if the same situation is known for SL(4,C).For example, I want to know if group of order 42 can be a subgroup of SL(4,C).Thanks for consideration.

Answer 1

Any group of order 42 is isomorphic to $C_{42}$, $AGL(1,7) = C_{7} \rtimes C_{6}$, $D_{14} \times C_{3}$, $D_{6} \times C_{7}$, $D_{42}$ or $(C_{7} \rtimes C_{3}) \times C_{2}$. (Here $D_{2n}$ denotes the dihedral group of order $2n$.)
Any finite cyclic group can be embedded in $GL(3, \mathbb{C})$ as a group of scalar matrices, so those factors in the Cartesian products given can be taken care of. The largest dimension of an irreducible (complex) representation of a dihedral group is 2, and the largest dimension of an irreducible representation of $C_{7} \rtimes C_{3}$ is 3. These largest-dimensional representations are all faithful. So these groups are all subgroups of $GL(3, \mathbb{C})$, which is isomorphic to a subgroup of $SL(4, \mathbb{C})$.
This leaves only $C_{7} \rtimes C_{6}$. The irreducible representations of this group have dimensions 1, 1, 1, 1, 1, 1 and 6. Any combination of the 1-dimensional representations will be an abelian representation and thus not be faithful. The 6-dimensional representation can't be found in $SL(4, \mathbb{C})$, so any group of order 42 will be isomorphic to a subgroup of $SL(4, \mathbb{C})$ or isomorphic to $AGL(1,7) = C_{7} \rtimes C_{6}$.