Ttha is the aspirated partner of the retroflex stop that heads the class, and it takes the second slot as the aspirate does in every one of the five groups. Once a learner has the rule of the class, the position tells the sound: same tongue position as the letter before it, voiceless as that letter is, with breath released after the closure.
It is uncommon in the inherited vocabulary of Sanskrit. Where it occurs, the word is often one the language absorbed rather than one it built, which is a general property of the retroflex series and a reason the series has been studied so closely by historical linguists.
Its rarity does not affect its position, which is fixed by the shape of the table. The five classes are each a complete crossing of two features against one point of articulation, and a class missing a member would break the regularity that makes every other member predictable. That regularity is what allows the table to be recited forward from any starting point, which is how it was taught in India and how it was taught in Japan.
On the page it takes the vowel marks and the inherent short a like any consonant sign, and it enters conjuncts, though its low frequency means few of those combinations are common enough to have acquired distinctive shapes.