Group cohomology of elementary abelian group of prime-fourth order

This article gives specific information, namely, group cohomology, about a family of groups, namely: elementary abelian group of prime-fourth order.
View group cohomology of group families | View other specific information about elementary abelian group of prime-fourth order

Suppose $p$ is a prime number. We are interested in the elementary abelian group of prime-fourth order

$E_{p^{4}}=(\mathbb {Z} /p\mathbb {Z} )^{4}=\mathbb {Z} /p\mathbb {Z} \oplus \mathbb {Z} /p\mathbb {Z} \oplus \mathbb {Z} /p\mathbb {Z} \oplus \mathbb {Z} /p\mathbb {Z}$

Particular cases

Value of prime $p$	elementary abelian group of prime-cube order	cohomology information
2	elementary abelian group:E8	group cohomology of elementary abelian group:E8
3	elementary abelian group:E27	group cohomology of elementary abelian group:E27
5	elementary abelian group:E125	group cohomology of elementary abelian group:E125

Homology groups for trivial group action

FACTS TO CHECK AGAINST (homology group for trivial group action):
First homology group: first homology group for trivial group action equals tensor product with abelianization
Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier
General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

Over the integers

The homology groups over the integers are given as follows:

$\!H_{q}(E_{p^{4}};\mathbb {Z} )=\left\lbrace {\begin{array}{rl}(\mathbb {Z} /p\mathbb {Z} )^{\frac {2q^{3}+15q^{2}+34q+45}{24}},&\qquad q=1,3,5,\dots \\(\mathbb {Z} /p\mathbb {Z} )^{\frac {2q^{3}+15q^{2}+34q}{24}},&\qquad q=2,4,6,\dots \\\mathbb {Z} ,&\qquad q=0\\\end{array}}\right.$

The first few homology groups are given as follows:

$q$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$H_{q}$	$\mathbb {Z}$	$(\mathbb {Z} /p\mathbb {Z} )^{4}$	$(\mathbb {Z} /p\mathbb {Z} )^{6}$	$(\mathbb {Z} /p\mathbb {Z} )^{14}$	$(\mathbb {Z} /p\mathbb {Z} )^{21}$	$(\mathbb {Z} /p\mathbb {Z} )^{35}$	$(\mathbb {Z} /p\mathbb {Z} )^{49}$	$(\mathbb {Z} /p\mathbb {Z} )^{71}$	$(\mathbb {Z} /p\mathbb {Z} )^{94}$
rank of $H_{q}$ as an elementary abelian $p$-group	--	4	6	14	21	35	49	71	94