I have two independent groups: from one I have drawn a sample of size less than 30 due to its small population, while the second group has sample size 100.
Is it appropriate to use an independent samples t-test in this case?
t-test requires a set of assumptions. It assumes your data is i.i.d and comes from a normal distribution. If you care to compare the means of the two groups (and they follow the assumptions), then yes - you can use that test.
As JohnK, you may wish to note if you want to assume equal variance for the two populations (and it is a reasonable rule of thumb to not make that assumption).
Under the assumptions, this test works for small and large sample sizes (and in the case of large sample sizes will approach the z-test).