日本語 Japanese Ask Questions Thread 3

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Japanese Ask Questions Thread2
http://dis.4chan.org/read/lang/1206158123/1-40

I'm fairly new to the language and I have a grammar question regarding subject vs indirect objects

ぼくは田中ちゃんとゲームをしています。
ぼくと田中ちゃんはゲームをしています。

Am I correct in assuming that Tanaka is the indirect object is the first sentence (I am playing a game with Tanaka-chan)
And in the second sentence, it's just multiple subjects (Tanaka-chan and I are playing a game)?