Linear representation theory of groups of order 16
This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 16.
View linear representation theory of group families | View linear representation theory of groups of a particular order |View other specific information about groups of order 16
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
The following sets of degrees of irreducible representations works over any splitting field not of characteristic two.
See also nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order. This says that for groups of order , the nilpotency class of the group, and the order, together determine the degrees of irreducible representations. In particular, for groups of order 16, there are three cases: the abelian case, where there are 16 of degree one, the class two case, where there are 8 of degree one and 2 of degree two, and the class three case, where there are 4 of degree one and 3 of degree two.
| Group | GAP ID second part | Hall-Senior number | Nilpotency class | Degrees as list | Number of irreps of degree 1 | Number of irreps of degree 2 |
|---|---|---|---|---|---|---|
| cyclic group:Z16 | 1 | 5 | 1 | 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 | 16 | 0 |
| direct product of Z4 and Z4 | 2 | 3 | 1 | 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 | 16 | 0 |
| SmallGroup(16,3) | 3 | 9 | 2 | 1,1,1,1,1,1,1,1,2,2 | 8 | 2 |
| nontrivial semidirect product of Z4 and Z4 | 4 | 10 | 2 | 1,1,1,1,1,1,1,1,2,2 | 8 | 2 |
| direct product of Z8 and Z2 | 5 | 4 | 1 | 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 | 16 | 0 |
| M16 | 6 | 11 | 2 | 1,1,1,1,1,1,1,1,2,2 | 8 | 2 |
| dihedral group:D16 | 7 | 12 | 3 | 1,1,1,1,2,2,2 | 4 | 3 |
| semidihedral group:SD16 | 8 | 13 | 3 | 1,1,1,1,2,2,2 | 4 | 3 |
| generalized quaternion group:Q16 | 9 | 14 | 3 | 1,1,1,1,2,2,2 | 4 | 3 |
| direct product of Z4 and V4 | 10 | 2 | 1 | 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 | 16 | 0 |
| direct product of D8 and Z2 | 11 | 6 | 2 | 1,1,1,1,1,1,1,1,2,2 | 8 | 2 |
| direct product of Q8 and Z2 | 12 | 7 | 2 | 1,1,1,1,1,1,1,1,2,2 | 8 | 2 |
| central product of D8 and Z4 | 13 | 8 | 2 | 1,1,1,1,1,1,1,1,2,2 | 8 | 2 |
| elementary abelian group:E16 | 14 | 1 | 1 | 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 | 16 | 0 |