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Names of papers, while these are on the front page
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LSpice
LSpice
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Besides the already accepted answer, another way of looking at this family of groups is as follows:

Let $V_d = (\mathbb Z/2)^d$. General theory says that there is a universal central extension $$ H_2(V_d;\mathbb Z/2) \rightarrow P_d \rightarrow V_d,$$ and the group in the question is $P_3$.

A lot is known about the cohomology rings of these groups (and their analogues at odd primes) by the work of Adem, Karagueuzian, and Minác in a 1999 paper On the cohomology of Galois groups determined by Witt rings in Advances, and I had fun with these also in a 2007 Advances paper Primitives and central detection numbers in group cohomology.

Besides the already accepted answer, another way of looking at this family of groups is as follows:

Let $V_d = (\mathbb Z/2)^d$. General theory says that there is a universal central extension $$ H_2(V_d;\mathbb Z/2) \rightarrow P_d \rightarrow V_d,$$ and the group in the question is $P_3$.

A lot is known about the cohomology rings of these groups (and their analogues at odd primes) by the work of Adem, Karagueuzian, and Minác in a 1999 paper in Advances, and I had fun with these also in a 2007 Advances paper.

Besides the already accepted answer, another way of looking at this family of groups is as follows:

Let $V_d = (\mathbb Z/2)^d$. General theory says that there is a universal central extension $$ H_2(V_d;\mathbb Z/2) \rightarrow P_d \rightarrow V_d,$$ and the group in the question is $P_3$.

A lot is known about the cohomology rings of these groups (and their analogues at odd primes) by the work of Adem, Karagueuzian, and Minác in a 1999 paper On the cohomology of Galois groups determined by Witt rings in Advances, and I had fun with these also in a 2007 Advances paper Primitives and central detection numbers in group cohomology.

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Nicholas Kuhn
Nicholas Kuhn
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Besides the already accepted answer, another way of looking at this family of groups is as follows:

Let $V_d = (\mathbb Z/2)^d$. General theory says that there is a universal central extension $$ H_2(V_d;\mathbb Z/2) \rightarrow P_d \rightarrow V_d,$$ and the group in the question is $P_3$.

A lot is known about the cohomology rings of these groups (and their analogues at odd primes) by the work of Adem, Karagueuzian, and Minác in a 1999 paper in Advances, and I had fun with these also in a 2007 Advances paper.