Linear representation theory of groups of prime-fourth order

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of prime-fourth order.
View linear representation theory of group families | View other specific information about groups of prime-fourth order

Particular cases

Value of prime $p$	Value of $p^{4}$	Information on groups of order $p^{4}$	Information on linear representation theory of groups of order $p^{4}$
2	16	groups of order 16	linear representation theory of groups of order 16
3	81	groups of order 81	linear representation theory of groups of order 81
5	625	groups of order 625	linear representation theory of groups of order 625

Degrees of irreducible representations

FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:
Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroup
Size bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
Cumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization

Full listing

It turns out that nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order, so there are only three possibilities for the degrees of irreducible representations for groups of order 81, based on whether the nilpotency class is 1, 2, or 3.

The case $p=2$ is somewhat different from the others, because there are only three maximal class groups of order $2^{4}$ as opposed to four for odd $p$. For the case $p=2$, see linear representation theory of groups of order 16.

Group	GAP ID second part	Nilpotency class	Degrees as list	Number of irreps of degree 1 (= order of abelianization)	Number of irreps of degree $p$	Total number of irreps (= number of conjugacy classes)
cyclic group of prime-fourth order	1	1	1 ($p^{4}$ times)	$p^{4}$	0	$p^{4}$
direct product of cyclic group of prime-square order and cyclic group of prime-square order	2	1	1 ($p^{4}$ times)	$p^{4}$	0	$p^{4}$
SmallGroup(p^4,3)	3	2	1 ($p^{3}$ times), $p$ ($p(p-1)$ times)	$p^{3}$	$p(p-1)$	$p^{3}+p^{2}-p$
SmallGroup(p^4,4)	4	2	1 ($p^{3}$ times), $p$ ($p(p-1)$ times)	$p^{3}$	$p(p-1)$	$p^{3}+p^{2}-p$
direct product of cyclic group of prime-cube order and cyclic group of prime order	5	1	1 ($p^{4}$ times)	$p^{4}$	0	$p^{4}$
semidirect product of cyclic group of prime-cube order and cyclic group of prime order	6	2	1 ($p^{3}$ times), $p$ ($p(p-1)$ times)	$p^{3}$	$p(p-1)$	$p^{3}+p^{2}-p$
SmallGroup(p^4,7)	7	3	1 ($p^{2}$ times), $p$ ($p^{2}-1$ times)	$p^{2}$	$p^{2}-1$	$2p^{2}-1$
SmallGroup(p^4,8)	8	3	1 ($p^{2}$ times), $p$ ($p^{2}-1$ times)	$p^{2}$	$p^{2}-1$	$2p^{2}-1$
SmallGroup(p^4,9)	9	3	1 ($p^{2}$ times), $p$ ($p^{2}-1$ times)	$p^{2}$	$p^{2}-1$	$2p^{2}-1$
SmallGroup(p^4,10)	10	3	1 ($p^{2}$ times), $p$ ($p^{2}-1$ times)	$p^{2}$	$p^{2}-1$	$2p^{2}-1$
direct product of cyclic group of prime-square order and elementary abelian group of prime-square order	11	1	1 ($p^{4}$ times)	$p^{4}$	0	$p^{4}$
SmallGroup(p^4,12)	12	2	1 ($p^{3}$ times), $p$ ($p(p-1)$ times)	$p^{3}$	$p(p-1)$	$p^{3}+p^{2}-p$
SmallGroup(p^4,13)	13	2	1 ($p^{3}$ times), $p$ ($p(p-1)$ times)	$p^{3}$	$p(p-1)$	$p^{3}+p^{2}-p$
SmallGroup(p^4,14)	14	2	1 ($p^{3}$ times), $p$ ($p(p-1)$ times)	$p^{3}$	$p(p-1)$	$p^{3}+p^{2}-p$
elementary abelian group of prime-fourth order	15	1	1 ($p^{4}$ times)	$p^{4}$	0	$p^{4}$

Grouping by degrees of irreducible representations

Number of irreps of degree 1	Number of irreps of degree $p$	Total number of irreps	Total number of groups	Nilpotency class(es) attained by these	Description of groups	List of GAP IDs (ascending order)	Order of inner automorphism group = index of center (bounds square of degree of irreducible representation)	Minimum possible index of abelian normal subgroup (degree of irreducible representation divides index of abelian normal subgroup)
$p^{4}$	0	$p^{4}$	5	1	all the abelian groups	1,2,5,11,15 (for odd $p$) 1,2,5,10,14 (for $p=2$)	1	1
$p^{3}$	$p^{2}-p$	$p^{3}+p^{2}-p$	6	2	all the class two groups	3,4,6,12,13,14 (for odd $p$) 3,4,6,11,12,13 (for $p=2$)	$p^{2}$	$p$
$p^{2}$	$p^{2}-1$	$2p^{2}-1$	4 (for odd $p$) 3 (for $p=2$)	3	all the class three groups	7,8,9,10 (for odd $p$) 7,8,9 (for $p=2$)	$p^{3}$	$p$