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What's your favourite group?

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The finite simple group of order 1004913. It doesn't exist, but it's really annoying to show that
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SO(3) is my favorite. It has so many weird properties. It pretends to be a 3-sphere, but it isn’t. It’s a subset of projective space. And yet it’s so useful in engineering. really the first physical non-Euclidean manifold we’re all introduced too, yet it isn’t an n-sphere, it’s non-commutative, non-constant curvature, yet it has so many nice properties like compactness, connectedness, it’s a Lie group, and clearly it has so many physical interpretations. Not to mention it’s completely represented by the space of orthogonal matrices with unit determinant.
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S^1 (circle group)
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Free groups! I've heard free objects in general are pretty cool.
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Absolute Galois group of Q. (related: My first thought was “Z_p doesn’t form a group under multiplication”. Then I remembered I’m a number theorist. Don’t forget to exclude 0, btw.)
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SL2R
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Linkin park
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The Klein four group is the smallest group that’s not trivial or cyclic which makes it interesting imo. Same goes for S_3
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I like GL(n,R) cause i love linear algebra and you can both analyze this group (and it's subgroups) using algebra and using topology
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Z/2Z because orientation sucks (e.g. see the construction of Morse homology).
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