Hello everyone. I am asked to find the size of the group

and would like to verify my work.

This group acts on the vector space

with basis

. From this, we have the orbit-stabilizer theorem: if

, then

, that is, the size of

is the size of the orbit of

times the size of the stabilizer of

. Take

. If

,

can possibly be any nonzero vector. There are 48 of these. Therefore,

. Then, if

, we must have

equal to another vector not a multiple of

, since

is invertible. In other words,

cannot be

. There are 42 of these, so

.

We conclude that

.

Comments or suggestions?