Just off the top of my head. You might have several of these in a late game fleet, so you might use several of them at once.

Progenitor Monthership + Iconus Guardian - Greatly increases the durability of an Advent fleet, you should have this in every fleet. Well managed use of it can allow you to tank Missile barrage and other painful abilities. Weak to phase missiles however as they go through shields.

Radiance Battleship + Progenitor Mothership + Rapture Battlecruiser(AKA Holy Trinity) - There are a couple of variations to this, and its not as popular as it used to be, but you need at least two of the three ships. The ideal setup is for the Rapture to cast vengeance on the Radiance, which uses animosity to draw enemy fire while the Mothership casts malice. This not only causes Vengeance to do a lot of damage to enemy ships attacking the Raidance but malice will further enhance it. Add Cleansing Brilliance on the Radiance and it can theoretically do a lot of damage.

In practice it doesn't work so great because animosity is bugged so that new attack orders will override it (though if your friend doesn't know that its still good, alternatively convince him to play with the Project Equilibrium balance mod where it is fixed), and you need a high level mothership to keep both shield restore and malice up. Still worth it to use some of these tricks though if you happen to have these ships out anyways.

Halycon Carrier + Illums/Desdras - More obvious one, this one just seeks to maximize DPS with amplify energy aurora. Not as good now that Illums have been fixed, but you should always have a Halycon anyways and if you aren't facing a lot of strikecraft its a good ability to invest in.

Halycon + Rapture + Carrier Cruisers - If your opponent leaves him self open to strike craft, use this. Halycon gets extra squadrons with a large buff with anima tempest, the carrier cruisers provide most of your drones while the Rapture amplifies the damage they do. Best of all you can park these on the other side of the gravity well while your strikecraft get the benefits and destroy the enemy fleet.

Level 6 Progenitor + Dead Capital Ships - Not really a synergy, but the Progens lvl 6 ability is seriously under utilized if the game lasts long enough to get one. Your level 8 Halycon dies? Build a new one and cast resurrection on it. New level 8 Halycon. Basically makes even high level caps expendable as long as you have one level 6 Progen and can afford to keep replacing Capital ships. Also makes the rest of these synergies much easier as you can maintain the high level capital ships to really exploit them.

haha my friend got pissed at me since i made a basic "battleball" and was repulsing and TK pushing away his bombers and frigates and my carriers live forever. Another question: i usually get a lot of bombers and for some reason he cant ever reallycounter it even with making lots of his own carriers nd fighters... now he wants to ban carriers too . i just tell him tec is boring and bad what does work vs lots of bombers since even his gardas and fighters cant seem to stop the bombers, and i dont even kite my carriers much

Generally speaking Advent is a late-game faction and TEC is an early-game faction.  TEC is going to have a serious fight on its hands just to survive a late-game Advent's power, and if he's not carrying a substantial economic advantage by this point his chances of winning are minimal.  If he wants to play TEC, he needs to really focous on his early-expansion and aggressive pressure tactics.  If you've got repulsion and his fleet isn't a good 30% larger than your's, he may as well throw in the towel because his window of opportunity is over.

With larger swarms of bombers, you'll need a Kol (preferably level 5+; if you don't have this, you'll need two Kols).  Use flak burst to deal massive damage to the strafing bombers, then clean them up with flaks and/or fighters.  One garda per squad of bombers should be sufficient if you have some fighters backing them up.  The problem is that this is expensive and leaves you vulnerable if the Advent diversifies with frigates.  Alternately, you can try to knock out the carriers themselves.  Cobalts and Kodiaks would work well, but they'll just get repulsed so at best they're reserves in this scenario.  The only unit at your disposal that outranges repulsion is the Javelis LRM, which fortunately has some great firepower.  However, the Advent needs only diversify with some destras to beat back this frigate-based counter. 

You can probably see this problem in-game; either way TEC has to overcompensate to beat this combo, and Advent can easily stay one step ahead if their fleet is of equal size.  The bottom line is that TEC isn't going to pull this off without a numerical advantage.  Advent's fully-mature combos are just too strong.