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Character error
Part of the Cyberman worldview is that removal of human emotion is a necessary if not essential part of the upgrade process. However, by his tone of voice if nothing else, Cyber Lumic's emotions are clearly still intact. As Cyber Controller, removal of his emotions should have been essential for him to take proper command of the Cybermen.
While hiding behind the mini-skips, Rose and Pete hold hands. After a cut to close up, they hold hands again.
As the Van drives up to rescue the Doctor and the others, a row of Cybermen are marching towards them in the background. The camera angle changes and the Cybermen are nowhere to be seen.
When Rose gets the message there are two things wrong with what you see on the phones screen: firstly on the top right of the screen you can see the icon for mute even though you heard the beep, and secondly that screen is for when you type in the number you want to call (text), not the screen for receiving a text.
When Mickey and Jake drive off in the van at the end of the episode there is a sign saying "Welcome to Newport" (the series is made by BBC Wales) whereas two minutes before they were on the south bank of the Thames in London.
When Mickey decides to stay behind, The Doctor tells him to take Rose's phone since it has the code. Mickey should already have that code on his phone. He sent it to Rose from his phone.