Read CAR 256A...penalty 25 penalty units for taking your dog with you unless exempted or authorised (conditions apply).
i have a permit to take my dog with me in my aeroplane which I paid $150 for. My permit has no expiry date but I have heard they are now time-limited like everything else. I have a harness for her which attaches to rear seat belt. I fit a lead to her to restrict her further and she sits on a pad over canvas on the back seat. She treats it like the car and never soils it.
Note many councils have general prohibitions on dogs at large and can get a bit testy about them on an airport.