Linear representation theory of nontrivial semidirect product of Z4 and Z4
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This article gives specific information, namely, linear representation theory, about a particular group, namely: nontrivial semidirect product of Z4 and Z4.
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This article discusses the linear representation theory of the group nontrivial semidirect product of Z4 and Z4 (GAP ID: (16,4)), given by the presentation:
Summary
Item | Value |
---|---|
degrees of irreducible representations over a splitting field (such as or ) | 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) | -- ring of Gaussian integers same as ring generated by character values |
minimal splitting field, i.e., smallest field of realization (characteristic zero) | 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 should split. For a finite field of size , this is equivalent to saying that |
minimal splitting field in characteristic | Case : prime field Case : Field , 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 | 1 | 1 | 1 | any | cyclic group:Z2 | works, same as trivial | |
sign, kernel or | 2 | 1 | 1 | any | cyclic group:Z2 | works, same as trivial | |
kernel or | 4 | 1 | 1 | must contain a primitive fourth root of unity, i.e., the polynomial must split | cyclic group:Z4 | works, same as trivial | |
kernel , 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 , 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 |