# Linear representation theory of groups of prime-cube order

From Groupprops

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of prime-cube order.

View linear representation theory of group families | View other specific information about groups of prime-cube order

The discussion here is specifically intended to cater to *odd* primes and excludes the case , which is somewhat different. For more on , see linear representation theory of groups of order 8.

## 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 subgroupSize 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 subgroupCumulative 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

Due to the fact that nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order, for groups of order for a fixed prime , the nilpotency class determines the degrees of irreducible representations. (This also holds for , although we are not discussing that case.

Group | GAP ID second part | Degrees as list | Number of irreps of degree 1 (= order of abelianization) | Number of irreps of degree | Total number of irreps (= number of conjugacy classes) |
---|---|---|---|---|---|

cyclic group of prime-cube order | 1 | 1 ( times) | 0 | ||

direct product of cyclic group of prime-square order and cyclic group of prime order | 2 | 1 ( times) | 0 | ||

unitriangular matrix group:UT(3,p) | 3 | 1 ( times), ( times) | |||

semidirect product of cyclic group of prime-square order and cyclic group of prime order | 4 | 1 ( times), ( times) | |||

elementary abelian group of prime-cube order | 5 | 1 ( times) | 0 |

## Splitting field

Two important notes:

- Odd-order and ambivalent implies trivial, so
*none*of these groups are ambivalent groups, i.e., none of them have all their character real-valued. Therefore, none of them are rational groups or rational-representation groups. This is in sharp contrast to the , where three of the five groups of order are rational groups and two of them are rational-representation groups. - Odd-order p-group implies every irreducible representation has Schur index one: All the irreducible representations have Schur index one, so they can be realized over the field generated by the character values. Note that this fact requires
*both*the odd order*and*the fact that it is a group of prime power order. For , there is an example -- faithful irreducible representation of quaternion group -- that has Schur index two. Also, there is a group of order 63 that has an irreducible representation of Schur index three.

With these facts in mind, we present the list of minimal splitting fields. We denote by a primitive root of unity. Note that the degree of the extension over is the Euler totient function .

Group | GAP ID second part | Nilpotency class | minimal splitting field = field generated by character values (because it's an odd-order p-group) | degree of extension over | minimal sufficiently large field | degree of extension over | Note |
---|---|---|---|---|---|---|---|

cyclic group of prime-cube order | 1 | 1 | both same, because abelian | ||||

direct product of cyclic group of prime-square order and cyclic group of prime order | 2 | 1 | both same, because abelian | ||||

unitriangular matrix group:UT(3,p) | 3 | 2 | |||||

semidirect product of cyclic group of prime-square order and cyclic group of prime order | 4 | 2 | splitting not implies sufficiently large | ||||

elementary abelian group of prime-cube order | 5 | 1 | all same, because abelian |