Yes, it's 3. With such a small group, and with an explicit homomorphism given, you can explicitly see (and count) what the kernel is.

But there's a more insightful way to see that the kernel has order 3. It's what you'll need when things aren't so simple.

You should understand why this true:

\(\displaystyle \text{Let } \phi : S_3 \rightarrow \mathbb{Z}_2 \text{ be *any* surjective homomorphism. Then } |ker(\phi)| = 3.\)

You should also understand why this true:

\(\displaystyle \text{Let } G_0, G_1 \text{ be *any* groups such that }|G_0| = 12,000,000, \ |G_1| = 1,000.\)

\(\displaystyle \text{Let }\phi : G_0 \rightarrow G_1 \text{ be any surjective homomorphism. Then } |ker(\phi)| = 12,000.\)