Linear representation theory of groups of prime-fifth order

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

Particular cases

Value of prime $p$	Value of $p^5$	Information on groups of order $p^5$	Information on linear representation theory of groups of order $p^5$
2	32	groups of order 32	linear representation theory of groups of order 32
3	243	groups of order 243	linear representation theory of groups of order 243
5	3125	groups of order 3125	linear representation theory of groups of order 3125

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

Grouping by degrees of irreducible representations

Number of irreps of degree 1	Number of irreps of degree $p$	Number of irreps of degree $p^2$	Total number of irreducible representations = number of conjugacy classes	Nilpotency class(es) attained by these	Description of groups	Number of groups case $p=2$	Number of groups case $p=3$	Number of groups case $p\ge 5$
$p^5$	0	0	$p^5$	1	all abelian groups of order $p^5$	7	7	7
$p^4$	$p^3-p^2$	0	$p^4+p^3-p^2$	2		15	15	15
$p^4$	0	$p-1$	$p^4+p-1$	2	the extraspecial groups	2	2	2
$p^3$	$p^3-p$	0	$2p^3-p$	2, 3		19	24	$p+21$
$p^2$	$p^3-1$	0	$p^3+p^2-1$	3		4	10	residue class-dependent
$p^3$	$p^2-p$	$p-1$	$p^3+p^2-1$	3		5	6	6
$p^2$	$p^2-1$	$p-1$	$2p^2+p-2$	4		0	3	6

Grouping by cumulative sums of squares of degrees

Sum of squares of irreps of degree 1	Sum of squares of irreps of degree at most $p$	Sum of squares of degree at most $p^2$	Total number of irreducible representations = number of conjugacy classes	Nilpotency class(es) attained by these	Description of groups	Number of groups case $p=2$	Number of groups case $p=3$	Number of groups case $p\ge 5$
$p^5$	$p^5$	$p^5$	$p^5$	1	all abelian groups of order $p^5$	7	7	7
$p^4$	$p^5$	$p^5$	$p^4+p^3-p^2$	2		15	15	15
$p^4$	$p^4$	$p^5$	$p^4+p-1$	2	the extraspecial groups	2	2	2
$p^3$	$p^5$	$p^5$	$2p^3-p$	2, 3		19	24	$p+21$
$p^2$	$p^5$	$p^5$	$p^3+p^2-1$	3		4	10	residue class-dependent
$p^3$	$p^4$	$p^5$	$p^3+p^2-1$	3		5	6	6
$p^2$	$p^4$	$p^5$	$2p^2+p-2$	4		0	3	6

Note that it is true in this case that the sum of squares of degrees of irreducible representations of degree dividing any number itself divides the order of the group (in particular, all these numbers are powers of $p$). However, this is not true for all groups and in fact an analogous statement fails for groups of prime-sixth order (see linear representation theory of groups of prime-sixth order). For more, see:

• All partial sum values of degrees of irreducible representations divide the order of the group for groups up to prime-fifth order
• There exist groups of prime-sixth order in which the partial sum values of degrees of irreducible representations do not divide the order of the group

Correspondence between degrees of irreducible representations and conjugacy class sizes

See also element structure of groups of prime-fifth order#Conjugacy class sizes.

For groups of order $p^5$, it is true that the list of conjugacy class sizes determines the degrees of irreducible representations. In the case $p=2$, the converse also holds, i.e., the degrees of irreducible representations determine the conjugacy class sizes.

However, for $p\ge 3$, there is one ambiguous case: the case of $p^2$ degree one and $p^3-1$ degree two representations corresponds to two possible lists of conjugacy class sizes: ($p$ of size one, $p^3-1$ of size $p$, $p^2-p$ of size $p^3$), and ($p^2$ of size 1, $p^3-1$ of size $p^2$). For $p=2$, there are no groups fitting the latter case.

Number of conjugacy classes of size 1	Number of conjugacy classes of size $p$	Number of conjugacy classes of size $p^2$	Number of conjugacy classes of size $p^3$	Total number of conjugacy classes = number of irreducible representations	Number of degree 1 irreps	Number of degree $p$ irreps	Number of degree $p^2$ irreps
$p^5$	0	0	0	$p^5$	$p^5$	0	0
$p^3$	$p^4-p^2$	0	0	$p^4+p^3-p^2$	$p^4$	$p^3-p^2$	0
$p$	$p^4-1$	0	0	$p^4+p-1$	$p^4$	0	$p-1$
$p^2$	$p^3-p$	$p^3-p^2$	0	$2p^3-p$	$p^3$	$p^3-p$	0
$p^2$	0	$p^3-1$	0	$p^3+p^2-1$	$p^2$	$p^3-1$	0
$p$	$p^3-1$	0	$p^2-p$	$p^3+p^2-1$	$p^2$	$p^3-1$	0
$p$	$p^2-1$	$p^3-p$	0	$p^3+p^2-1$	$p^3$	$p^2-p$	$p-1$
$p$	$p-1$	$p^2-1$	$p^2-p$	$2p^2+p-2$	$p^2$	$p^2-1$	$p-1$