Linear representation theory of nontrivial semidirect product of Z4 and Z4
From Groupprops
This article gives specific information, namely, linear representation theory, about a particular group, namely: nontrivial semidirect product of Z4 and Z4.
View linear representation theory of particular groups | View other specific information about nontrivial semidirect product of Z4 and Z4
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 ![]() ![]() |
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) | ![]() 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 ![]() For a finite field of size ![]() ![]() |
minimal splitting field in characteristic ![]() |
Case ![]() ![]() Case ![]() ![]() |
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 ![]() ![]() |
2 | 1 | 1 | any | cyclic group:Z2 | works, same as trivial | |
kernel ![]() ![]() |
4 | 1 | 1 | must contain a primitive fourth root of unity, i.e., the polynomial ![]() |
cyclic group:Z4 | works, same as trivial | |
kernel ![]() |
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 ![]() |
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 |