Group Train Travel- Prague to Vienna

Well, at least for Prague to Vienna, Ceske Drahy (Czech National Railway) has group reservation system.
https://www.cd.cz/en/typy-jizdenek/vnitrostatni-jizdenky/-28230/

Frequent trains from Prague to Vienna. Everyone in the group gets a 50% discount.

Nothing is more romantic than riding the rails. But I suspect a private bus wouldn't cost any more, and you could make a few stops along the way; the Archabbey in Panonhalma for instance.

The 50% discount on the Czech National Railway (www.cd.cz). applies only if there are 20 or more of you traveling in a group, which doesn’t appear to be the case from your post.
However, if you buy your train tickets now, there are great deals available on several mid-morning trains for April 2023 at www.TheTrainline.com
There are tickets for $16.50 on Railjet trains and OBB trains ( Austria’s National Train company) from Prague hl.n. to Wien Hauptbahnhof ( Prague and Vienna’s main train stations) and also for $16.50 for Wien Hauptbahnhof to Budapest Keleti, the main train station in Budapest where international trains arrive and depart from. To buy the train tickets online, be sure to select these particular train stations.
Be sure to be certain of the date your group will be traveling as tickets at these low prices are not refundable. Once you buy them online, you are committed to taking the specific train the ticket is for.
Refundable tickets can be bought at higher prices.

@ Kenko

We will need them for 31 people.

Last time I checked, 31 was more than 20.

In addition to ČD, here is ÖBB's group travel information: https://www.oebb.at/en/tickets-kundenkarten/schueler-gruppen/gruppenreisen

Send them both a message and ask for a price.

Booking seat reservations may be a separate step to buying the tickets. Don't think that buying a ticket means you will get a seat.

We have done the trip from Prague to Vienna and then a week later Vienna to Budapest. On both legs people were sitting on the floor of the carriage as they hadn't paid the extra €5- for a seat reservation.

31– there it is in the second sentence! My apologies for misreading the number as 13.
You all should get an entire train carriage together and make it a party!