Small diamond such as 20-22-28mm have a hex end usually, they can have an enlarged end to take an arbor though. Larger diamond such as
38-152mm have an arbor end, with push-fit pilot drill (use the drift to remove it).
Amazon do Blue Spot diamond core bits cheaply, and they resell ok too. They can be "extremely good" or "just good" depending on the luck of the draw, and at the price (=A328 for 152mm?) are very good value.
A diamond core drill has a reduction gearbox to give the huge starting torque - note that the "start" tends to be "instant quick" so make sure you go very gently on uneven brick surfaces because you do not want to shear a tab off. This is one reason to buy a core drill bit rather than hire - breaking the tab off a 152mm Hilti bit is likely to be very expensive and is chargeable.
You can pick used diamond core drill up on Ebay, 110V & 240V, but they seem to look rather the worse for wear. If you have a lot to do (bathroom shower toilet 107mm, dryer 107mm, kitchen extractor 152mm, kitchen fan 152mm, wall mounted HVAC 70mm) it can work out ok. A note re kitchen extractor, go for 152mm because you can sleeve down. BES do a brown gravity flap for 125mm & 150mm which has the same 200mm outside "form-factor-n-holes" so you can just fit a 5" tube through to spigot & wall plate on the inside, if you later go to a bigger cooker hood it is easy to upgrade as needed. Do sleeve the cavity properly, a relative had a slip tube since 1996 and it never sealed properly on a dryer - duct tape unwinds even amalgamating didn't bond fully, going to be a long weekend or two sorting that mess out when we go cool weather again.
Most fun of all, if you have not met diamond dust, you will soon - it makes a mockery of IP66 due to particle size :-) 110V TX vary from pig heavy (1.5kVA) to plain misery (3.3kVA) so you may want a 110V extension if doing a lot of stuff.
A pity no-one has done a "hybrid" SDS drill with 2:1 gearbox or similar. Diamond core drills are rather expensive for what they are - unremarkable motor rating, just a reduction gearbox and heavier duty clutch assembly, =A3300 for a Makita. I believe some are 2 speed and some a re 1 speed (all those I have hired are 1 speed which is rather inappropriate). Big core means very slow & small core means somewhat quicker but not quick. Let the bit do the cutting and keep it straight because they do like to wander if one brick is hard and another very soft. True engineering takes... time to drill thro and cheap core bits will just spark due to "too much glass and too little diamond", screeching is the first indicator.
A final comment is "getting started" with small diameter but very long bits (22x350mm), these lack a guide drill and can wander particularly on textured brickwork. It can be worth having a short small diamond bit to machine a "pocket starter", Ebay do 20mm from Hong Kong (7-8 working days) which are fine for this in a cordless drill. I used a stub 20mm core drill one today to start off a very long 22mm bit (sleeving through a cavity) and again to enlarge the 22mm ends to take
20mm conduit bell-bush (silicone flex to halogen does not like sharp edges). I also used a stub 20mm core drill to create a cutout perimeter for a 152mm bit which was "skipping around" despite pilot drill due to one very hard & uneven brick surface, the 1-speed makita core drill would not soft or slow start enough with such a huge "barrel of a core drill bit".
Wonderful tools with no breakout compared to SDS (you can screw a plank the other side and stitch drill, it needs care to avoid a bell- mouth effect with the middle too small and outer too large). Wear goggles & mask or you will need a JCB to clear your nasal passages out.