Linear representation theory of groups of order 16

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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
Group GAP ID second part Hall-Senior number Linear representation theory page
cyclic group:Z16 1 5 See linear representation theory of cyclic groups
direct product of Z4 and Z4 2 3 See linear representation theory of finite abelian groups
SmallGroup(16,3) 3 9 linear representation theory of SmallGroup(16,3)
nontrivial semidirect product of Z4 and Z4 4 10 linear representation theory of nontrivial semidirect product of Z4 and Z4
direct product of Z8 and Z2 5 4 See linear representation theory of finite abelian groups
M16 6 11 linear representation theory of M16
dihedral group:D16 7 12 linear representation theory of dihedral group:D16
semidihedral group:SD16 8 13 linear representation theory of semidihedral group:SD16
generalized quaternion group:Q16 9 14 linear representation theory of generalized quaternion group:Q16
direct product of Z4 and V4 10 2 See linear representation theory of finite abelian groups
direct product of D8 and Z2 11 6  ?
direct product of Q8 and Z2 12 7  ?
central product of D8 and Z4 13 8  ?
elementary abelian group:E16 14 1  ?

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

Full listing

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 p^k, 0 \le k \le 4, 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.

Note that since number of conjugacy classes in group of prime power order is congruent to order of group modulo prime-square minus one, all the values for the total number of irreducible representations, which equals the number of conjugacy classes, is congruent to 16 mod 3, and hence congruent to 1 mod 3.

Group GAP ID second part Hall-Senior number Hall-Senior symbol Nilpotency class Degrees as list Number of irreps of degree 1 (= order of abelianization) Number of irreps of degree 2 Total number of irreps (= number of conjugacy classes)
cyclic group:Z16 1 5 (4) 1 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 16 0 16
direct product of Z4 and Z4 2 3 (2^2) 1 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 16 0 16
SmallGroup(16,3) 3 9 \Gamma_2c_1 2 1,1,1,1,1,1,1,1,2,2 8 2 10
nontrivial semidirect product of Z4 and Z4 4 10 \Gamma_2c_2 2 1,1,1,1,1,1,1,1,2,2 8 2 10
direct product of Z8 and Z2 5 4 (31) 1 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 16 0 16
M16 6 11 \Gamma_2d 2 1,1,1,1,1,1,1,1,2,2 8 2 10
dihedral group:D16 7 12 \Gamma_3a_1 3 1,1,1,1,2,2,2 4 3 7
semidihedral group:SD16 8 13 \Gamma_3a_2 3 1,1,1,1,2,2,2 4 3 7
generalized quaternion group:Q16 9 14 \Gamma_3a_3 3 1,1,1,1,2,2,2 4 3 7
direct product of Z4 and V4 10 2 (21^2) 1 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 16 0 16
direct product of D8 and Z2 11 6 \Gamma_2a_1 2 1,1,1,1,1,1,1,1,2,2 8 2 10
direct product of Q8 and Z2 12 7 \Gamma_2a_2 2 1,1,1,1,1,1,1,1,2,2 8 2 10
central product of D8 and Z4 13 8 \Gamma_2b 2 1,1,1,1,1,1,1,1,2,2 8 2 10
elementary abelian group:E16 14 1 (1^4) 1 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 16 0 16

Grouping by Hall-Senior families

Note that isoclinic groups have same proportions of degrees of irreducible representations, and in particular isoclinic groups of the same order have precisely the same degrees of irreducible representations. Thus, all groups in the same Hall-Senior family have the same degrees of irreducible representations. However, the same multiset of degrees of irreducible representations could be attained by more than one Hall-Senior family, though this phenomenon does not occur for order 16.

For the first two Hall-Senior families \Gamma_1,\Gamma_2, there are isoclinic groups of smaller order, hence the degrees of irreducible representations can be computed by first computing the degrees of irreducible representations of those isoclinic groups of smaller order and then scaling up the proportions based on the order. For instance, dihedral group:D8 of order 8 and family \Gamma_2 has 4 irreps of degree 1 and 1 of degree 2, so the groups in family \Gamma_2 and of order 16 have 4 \times (16/8) = 8 irreps of degree 1 and 1 \times (16/8) = 2 irreps of degree 2.

For more background on the Hall-Senior families business, see Groups of order 16#Families and classification.

Family name Isomorphism class of inner automorphism group Isomorphism class of derived subgroup Number of members Nilpotency class Members Second part of GAP ID of members (sorted ascending) Hall-Senior numbers of members (sorted ascending) Number of irreps of degree 1 (= order of abelianization) Number of irreps of degree 2 Total number of irreps Smallest order of group isoclinic to it Degrees of irreps of smallest order isoclinic stem group
\Gamma_1 trivial group trivial group 5 1 abelian groups: [SHOW MORE] 1, 2, 5, 10, 14 1--5 16 0 16 1 1
\Gamma_2 Klein four-group cyclic group:Z2 6 2 [SHOW MORE] 3, 4, 6, 11, 12, 13 6--11 8 2 10 8 1 (4 times), 2 (1 time)
\Gamma_3 dihedral group:D8 cyclic group:Z4 3 3 maximal class groups: [SHOW MORE] 7, 8, 9 12--14 4 3 7 16 1 (4 times), 2 (3 times)

Grouping by degrees of irreducible representations

Note that isoclinic groups have same proportions of degrees of irreducible representations, and in particular isoclinic groups of the same order have precisely the same degrees of irreducible representations. Thus, all groups in the same Hall-Senior family have the same degrees of irreducible representations.

For order 16 (and more generally for fixed order p^k with p prime and 0 \le k \le 4), the degrees of irreducible representations uniquely determine the Hall-Senior family, i.e., different Hall-Senior families have different degrees of irreducible representations. This breaks down for p^5 and in particular for 32, where there are some cases of multiple Hall-Senior families having the same degrees of irreducible representations.

Number of irreps of degree 1 Number of irreps of degree 2 Total number of irreps Total number of groups Nilpotency class(es) attained by these List of groups List of GAP IDs (ascending order) List of Hall-Senior numbers (ascending order) List of Hall-Senior families (equivalence classes under isoclinism) Order of inner automorphism group = index of center (bounds square of degree of irreducible representation) Minimum possible index of abelian normal subgroup (degree of irreducible representation divides index of abelian normal subgroup)
16 0 16 5 1 [SHOW MORE] 1, 2, 5, 10, 14 1--5 \Gamma_1 (abelian groups) 1 1
8 2 10 6 2 [SHOW MORE] 3, 4, 6, 11, 12, 13 6--11 \Gamma_2 4 2
4 3 7 3 3 [SHOW MORE] 7, 8, 9 12--14 \Gamma_3 (maximal class groups) 8 2

Splitting field

Characteristic zero case

Note that for the abelian cases, the smallest field of realization of representations is the same as the field generated by the character values, because the irreducible representations are all one-dimensional and can be identified with their character values.

Key shorthands in this table:

  • \mathbb{Q}(i) stands for \mathbb{Q}(\sqrt{-1}) = \mathbb{Q}[t]/(t^2 + 1)
  • \mathbb{Q}(i,\sqrt{2}) stands for \mathbb{Q}(\sqrt{-1},\sqrt{-2}) = \mathbb{Q}(e^{\pi i/4})
  • \mathbb{Q}(\sqrt{2}) is the same as \mathbb{Q}[t]/(t^2 - 2)
  • \mathbb{Q}(\sqrt{-2}) is the same as \mathbb{Q}(\sqrt{2}i) = \mathbb{Q}[t]/(t^2 + 2)
Group GAP ID Hall-Senior number Nilpotency class Field generated by character values Degree of extension over \mathbb{Q} Smallest field of realization (i.e., minimal splitting field) in characteristic zero Degree of extension over \mathbb{Q} Minimal sufficiently large field Degree of extension over \mathbb{Q} Comment
cyclic group:Z16 1 5 1 \mathbb{Q}(e^{\pi i/8}) 8 \mathbb{Q}(e^{\pi i/8}) 8 \mathbb{Q}(e^{\pi i/8}) 8 all same, because abelian
direct product of Z4 and Z4 2 3 1 \mathbb{Q}(i) 2 \mathbb{Q}(i) 2 \mathbb{Q}(i) 2 all same, because abelian
SmallGroup(16,3) 3 9 2 \mathbb{Q}(i) 2 \mathbb{Q}(i) 2 \mathbb{Q}(i) 2
nontrivial semidirect product of Z4 and Z4 4 10 2 \mathbb{Q}(i) 2 \mathbb{Q}(i) 2 \mathbb{Q}(i) 2
direct product of Z8 and Z2 5 4 1 \mathbb{Q}(i,\sqrt{2}) 4 \mathbb{Q}(i,\sqrt{2}) 4 \mathbb{Q}(i,\sqrt{2}) 4 all same, because abelian
M16 6 11 2 \mathbb{Q}(i) 2 \mathbb{Q}(i) 2 \mathbb{Q}(i,\sqrt{2}) 4 splitting not implies sufficiently large
dihedral group:D16 7 12 3 \mathbb{Q}(\sqrt{2}) 2 \mathbb{Q}(\sqrt{2}) 2 \mathbb{Q}(i,\sqrt{2}) 4 splitting not implies sufficiently large, minimal splitting field need not be cyclotomic
semidihedral group:SD16 8 13 3 \mathbb{Q}(\sqrt{-2}) 2 \mathbb{Q}(\sqrt{-2}) 2 \mathbb{Q}(i,\sqrt{2}) 4 splitting not implies sufficiently large, minimal splitting field need not be cyclotomic
generalized quaternion group:Q16 9 14 3 \mathbb{Q}(\sqrt{2}) 2 \mathbb{Q}(i,\sqrt{2}) (and possibly others) 4 \mathbb{Q}(i,\sqrt{2}) 4
direct product of Z4 and V4 10 2 1 \mathbb{Q}(i) 2 \mathbb{Q}(i) 2 \mathbb{Q}(i) 2 all same, because abelian
direct product of D8 and Z2 11 6 2 \mathbb{Q} 1 \mathbb{Q} 1 \mathbb{Q}(i) 2
direct product of Q8 and Z2 12 7 2 \mathbb{Q} 1 \mathbb{Q}(i) or \mathbb{Q}(\sqrt{-2}) or \mathbb{Q}(\sqrt{-m^2 - 1}) where m \in \mathbb{Q}, other possibilities too! 2 \mathbb{Q}(i) 2 minimal splitting field need not be unique, minimal splitting field need not be cyclotomic, splitting not implies sufficiently large
central product of D8 and Z4 13 8 2 \mathbb{Q}(i) 2 \mathbb{Q}(i) 2 \mathbb{Q}(i) 2
elementary abelian group:E16 14 1 1 \mathbb{Q} 1 \mathbb{Q} 1 \mathbb{Q} 1 all same, because abelian

Grouping by minimal splitting field

Note that since minimal splitting field need not be unique, some groups have multiple minimal splitting fields. All the minimal splitting fields for direct product of Q8 and Z2 and generalized quaternion group:Q16 are not mentioned.

Field Cyclotomic extension of rationals? Real? Degree of extension over \mathbb{Q} Groups for which this is a minimal splitting field GAP IDs (second part) Groups for which this is a splitting field (not necessarily minimal) GAP IDs (second part)
\mathbb{Q} Yes Yes 1 direct product of D8 and Z2, elementary abelian group:E16 11, 14 direct product of D8 and Z2, elementary abelian group:E16 11, 14
\mathbb{Q}(i) Yes No 2 direct product of Z4 and Z4, SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, M16, direct product of Z4 and V4, direct product of Q8 and Z2, central product of D8 and Z4 2, 3, 4, 6, 10, 12, 13 direct product of Z4 and Z4, SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, M16, direct product of Z4 and V4, direct product of D8 and Z2, direct product of Q8 and Z2, central product of D8 and Z4, elementary abelian group:E16 2, 3, 4, 6, 10, 11, 12, 13, 14
\mathbb{Q}(\sqrt{2}) No Yes 2 dihedral group:D16 7 dihedral group:D16, direct product of D8 and Z2, elementary abelian group:E16 7, 11, 14
\mathbb{Q}(\sqrt{-2}) No No 2 semidihedral group:SD16, direct product of Q8 and Z2 8, 12 semidihedral group:SD16, direct product of D8 and Z2, direct product of Q8 and Z2, elementary abelian group:E16 8, 11, 12, 14
\mathbb{Q}(i,\sqrt{2}) Yes No 4 direct product of Z8 and Z2, generalized quaternion group:Q16 5, 9 all groups except cyclic group:Z16 2 - 14
\mathbb{Q}(e^{\pi i/8}) Yes No 8 cyclic group:Z16 1 all groups 1 - 14

Grouping by field generated by character values

Field Cyclotomic extension of rationals? Real? Degree of extension over \mathbb{Q} Groups for which this is precisely the field generated by character values GAP IDs (second part) Groups for which this contains the field generated by character values GAP IDs (second part)
\mathbb{Q} Yes Yes 1 direct product of D8 and Z2, direct product of Q8 and Z2, elementary abelian group:E16 11, 12, 14 direct product of D8 and Z2, elementary abelian group:E16 11, 12, 14
\mathbb{Q}(i) Yes No 2 direct product of Z4 and Z4, SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, M16, direct product of Z4 and V4, central product of D8 and Z4 2, 3, 4, 6, 10, 13 direct product of Z4 and Z4, SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, M16, direct product of Z4 and V4, direct product of D8 and Z2, direct product of Q8 and Z2, central product of D8 and Z4, elementary abelian group:E16 2, 3, 4, 6, 10, 11, 12, 13, 14
\mathbb{Q}(\sqrt{2}) No Yes 2 dihedral group:D16, generalized quaternion group:Q16 7, 9 dihedral group:D16, generalized quaternion group:Q16, direct product of D8 and Z2, direct product of Q8 and Z2, elementary abelian group:E16 7, 9, 11, 12, 14
\mathbb{Q}(\sqrt{-2}) No No 2 semidihedral group:SD16 8 semidihedral group:SD16, direct product of D8 and Z2, direct product of Q8 and Z2, elementary abelian group:E16 8, 11, 12, 14
\mathbb{Q}(i,\sqrt{2}) Yes No 4 direct product of Z8 and Z2 5 all groups except cyclic group:Z16 2 - 14
\mathbb{Q}(e^{\pi i/8}) Yes No 8 cyclic group:Z16 1 all groups 1 - 14

Rationals and reals: properties

This table can be completely reconstructed based on the above/previous tables, but is included for additional clarity.

Group GAP ID 2nd part Hall-Senior number nilpotency class rational-representation group (all representations realized over rationals)? rational group (all characters take rational values)? real-representation group (all representations realized over reals)? ambivalent group (all characters take real values)?
cyclic group:Z16 1 5 1 No No No No
direct product of Z4 and Z4 2 3 1 No No No No
SmallGroup(16,3) 3 9 2 No No No No
nontrivial semidirect product of Z4 and Z4 4 10 2 No No No No
direct product of Z8 and Z2 5 4 1 No No No No
M16 6 11 2 No No No No
dihedral group:D16 7 12 3 No No Yes Yes
semidihedral group:SD16 8 13 3 No No No No
generalized quaternion group 9 14 3 No No No Yes
direct product of Z4 and V4 10 2 1 No No No No
direct product of D8 and Z2 11 6 2 Yes Yes Yes Yes
direct product of Q8 and Z2 12 7 2 No Yes No Yes
central product of D8 and Z4 13 8 2 No No No No
elementary abelian group:E16 14 1 1 Yes Yes Yes Yes

General case

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Relation with other orders

Divisors of the order

Divisor Quotient value Information on linear representation theory Relationship (subgroup perspective) Relationship (quotient perspective)
2 8 linear representation theory of cyclic group:Z2
4 4 linear representation theory of groups of order 4 (linear representation theory of cyclic group:Z4, linear representation theory of Klein four-group)
8 2 linear representation theory of groups of order 8

Multiples of the order

Multiplier (other factor) Multiple Information on linear representation theory Relationship (subgroup perspective) Relationship (quotient perspective)
2 32 linear representation theory of groups of order 32
3 48 linear representation theory of groups of order 48
4 64 linear representation theory of groups of order 64
5 80 linear representation theory of groups of order 80
6 96 linear representation theory of groups of order 96