Linear representation theory of groups of prime-cube order

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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 p and excludes the case p = 2, which is somewhat different. For more on p = 2, see linear representation theory of groups of order 8.

Group GAP ID second part Linear representation theory page
cyclic group of prime-cube order 1 linear representation theory of cyclic group of prime-cube order, linear representation theory of cyclic groups
direct product of cyclic group of prime-square order and cyclic group of prime order 2 linear representation theory of direct product of cyclic group of prime-square order and cyclic group of prime order, linear representation theory of finite abelian groups
unitriangular matrix group:UT(3,p) 3 linear representation theory of unitriangular matrix group:UT(3,p)
semidirect product of cyclic group of prime-square order and cyclic group of prime order 4 linear representation theory of semidirect product of cyclic group of prime-square order and cyclic group of prime order
elementary abelian group of prime-cube order 5 linear representation theory of elementary abelian group of prime-cube order, linear representation theory of elementary abelian groups

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

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 p^3 for a fixed prime p, the nilpotency class determines the degrees of irreducible representations. (This also holds for p = 2, 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 p Total number of irreps (= number of conjugacy classes)
cyclic group of prime-cube order 1 1 (p^3 times) p^3 0 p^3
direct product of cyclic group of prime-square order and cyclic group of prime order 2 1 (p^3 times) p^3 0 p^3
unitriangular matrix group:UT(3,p) 3 1 (p^2 times), p (p - 1 times) p^2 p - 1 p^2 + p - 1
semidirect product of cyclic group of prime-square order and cyclic group of prime order 4 1 (p^2 times), p (p - 1 times) p^2 p - 1 p^2 + p - 1
elementary abelian group of prime-cube order 5 1 (p^3 times) p^3 0 p^3

Splitting field

Two important notes:

With these facts in mind, we present the list of minimal splitting fields. We denote by \zeta_n a primitive n^{th} root of unity. Note that the degree of the extension \mathbb{Q}(\zeta_n) over \mathbb{Q} is the Euler totient function \varphi(n).

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 \mathbb{Q} minimal sufficiently large field degree of extension over \mathbb{Q} Note
cyclic group of prime-cube order 1 1 \mathbb{Q}(\zeta_{p^3}) p^2(p - 1) \mathbb{Q}(\zeta_{p^3}) p^2(p - 1) both same, because abelian
direct product of cyclic group of prime-square order and cyclic group of prime order 2 1 \mathbb{Q}(\zeta_{p^2}) p(p - 1) \mathbb{Q}(\zeta_{p^2}) p(p - 1) both same, because abelian
unitriangular matrix group:UT(3,p) 3 2 \mathbb{Q}(\zeta_p) p - 1 \mathbb{Q}(\zeta_p) p - 1
semidirect product of cyclic group of prime-square order and cyclic group of prime order 4 2 \mathbb{Q}(\zeta_p) p - 1 \mathbb{Q}(\zeta_{p^2}) p(p - 1) splitting not implies sufficiently large
elementary abelian group of prime-cube order 5 1 \mathbb{Q}(\zeta_p) p - 1 \mathbb{Q}(\zeta_p) p - 1 all same, because abelian