# Presentation of simple group of order 168

I am working on finding the automorphism groups of all small groups. So far I am up to the automorphism group of Z(2)^3. I know that it is the simple group of order 168, but I can't find a presentation of it anywhere. Can anyone help?

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It's generated by an element $x$ of order 2 and and an element $y$ of order $3$ whose product $z$ has order 7, but there are more relations necessary. The elements $x$ and $y$ are as unique as they could be (you can only conjugate them (simultaneously) by elements of $\langle z \rangle$). The group you are looking at is of course the group of invertible $3 \times 3$ matrices over the field of $2$ elements. – Geoff Robinson Aug 4 '12 at 7:04

Geoff Robinson is on the money, and you only need one more relation: $$\langle x,y\ |\ x^2=y^3=(xy)^7=[x,y]^4=1 \rangle.$$