同どう時じに

一いっ緒しょにN5sense 1
一緒に means 'together' (doing something with someone), while 同時に focuses on the timing of events happening at the same moment, not necessarily together.
一いっ方ぽうN3sense 3
一方 introduces a contrasting or additional point ('on the other hand'), similar to sense 3 of 同時に, but 同時に adds a stronger nuance of simultaneity.
中なかにも
同時に simply means 'at the same time', while 中にも in this sense suggests one feeling or quality is present within another situation, often contrastively.
一いっ斉せいにN2
同時に simply means 'at the same time' without implying coordinated group action, while 一斉に stresses that many people or things act together as one.

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ともにN3
同時に stresses that two things happen at the exact same moment, whereas 共に can express a looser 'along with' relationship over time.
一いち度どにN3
同時に means 'at the same time' for concurrent events, while 一度に emphasizes a single action or batch rather than simultaneity of separate events.
いっぺんにN3
同時に is a formal, precise term for 'simultaneously', whereas 一遍に is casual and often implies bundling tasks into one action rather than strict timing.
一いっ緒しょN5
同時に explicitly and formally means 'at the same time', whereas 一緒に can express simultaneity more casually.

Opposites

交こう互ごにN1
同時に means 'at the same time', the opposite of 交互に, which means the actions happen one after another rather than simultaneously.

Compound of 同時 (どうじ, 'same time') + particle に, functioning adverbially and conjunctionally.