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

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] cyclic group:Z16, direct product of Z4 and Z4, direct product of Z8 and Z2, direct product of Z4 and V4, elementary abelian group:E16	1, 2, 5, 10, 14	1--5	16	0	16	1	1
$\Gamma_2$	Klein four-group	cyclic group:Z2	6	2	[SHOW MORE] SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, M16, direct product of D8 and Z2, direct product of Q8 and Z2, central product of D8 and Z4	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] dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16	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] cyclic group:Z16, direct product of Z4 and Z4, direct product of Z8 and Z2, direct product of Z4 and V4, elementary abelian group:E16	1, 2, 5, 10, 14	1--5	$\Gamma_1$ (abelian groups)	1	1
8	2	10	6	2	[SHOW MORE] SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, M16, direct product of D8 and Z2, direct product of Q8 and Z2, central product of D8 and Z4	3, 4, 6, 11, 12, 13	6--11	$\Gamma_2$	4	2
4	3	7	3	3	[SHOW MORE] dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16	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 $m \in \mathbb{Q}$	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

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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