Linear representation theory of nontrivial semidirect product of Z4 and Z4

Summary

Item	Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$ )	1,1,1,1,1,1,1,1,2,2 (1 occurs 8 times, 2 occurs 2 times) maximum: 2, lcm: 2, number: 10, sum of squares: 16
Schur index values of irreducible representations	1,1,1,1,1,1,1,1,1,2 (1 occurs 9 times, 2 occurs 1 time)
smallest ring of realization (characteristic zero)	$\mathbb{Z}[i] = \mathbb{Z}[\sqrt{-1}] = \mathbb{Z}[t]/(t^2 + 1)$ -- ring of Gaussian integers same as ring generated by character values
minimal splitting field, i.e., smallest field of realization (characteristic zero)	$\mathbb{Q}(i) = \mathbb{Q}(\sqrt{-1}) = \mathbb{Q}[t]/(t^2 + 1)$ Same as field generated by character values, even though there is a representation of Schur index greater than one, because there are other representations whose character values anyway require us to adjoin the roots of unity that would realize this representation.
condition for a field to be a splitting field	The characteristic should not be equal to 2, and the polynomial $t^2 + 1$ should split. For a finite field of size $q$ , this is equivalent to saying that $q \equiv 1 \pmod 4$
minimal splitting field in characteristic $p \ne 0,2$	Case $p \equiv 1 \pmod 4$ : prime field $\mathbb{F}_p$ Case $p \equiv 3 \pmod 4$ : Field $\mathbb{F}_{p^2}$ , quadratic extension of prime field
smallest size splitting field	field:F5, i.e., the field of five elements
degrees of irreducible representations over the rational numbers	1,1,1,1,2,2,2,4 (1 occurs 4 times, 2 occurs 3 times, 4 occurs 1 time) number: 8
orbits of irreducible representations over a splitting field under action of automorphism group	2 orbits of size 1 of degree 1 representations, 1 orbit of size 2 of degree 1 representations, 1 orbit of size 4 of degree 1 representations, 2 orbits of size 1 of degree 2 representations number: 6

Representations

Summary information

Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2, except in the last column, where we consider what happens in characteristic 2.

Name of representation type	Number of representations of this type	Degree	Schur index	Criterion for field	Kernel	Quotient by kernel (on which it descends to a faithful representation)	Characteristic 2?
trivial	1	1	1	any	whole group	trivial group	works
sign, kernel $\langle x, y^2 \rangle$	1	1	1	any		cyclic group:Z2	works, same as trivial
sign, kernel $\langle x^2, y \rangle$ or $\langle x^2, xy \rangle$	2	1	1	any		cyclic group:Z2	works, same as trivial
kernel $\langle x \rangle$ or $\langle xy^2 \rangle$	4	1	1	must contain a primitive fourth root of unity, i.e., the polynomial $t^2 + 1$ must split		cyclic group:Z4	works, same as trivial
kernel $\langle y^2 \rangle$ , obtained by composing quotient map with faithful irreducible representation of dihedral group:D8	1	2	1	any	subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4	dihedral group:D8	indecomposable but not irreducible
kernel $\langle x^2y^2 \rangle$ , obtained by composing quotient map with faithful irreducible representation of quaternion group	1	2	2	-1 should be expressible as a sum of two squares (sufficient condition). Any finite field works	central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4	quaternion group	indecomposable but not irreducible