GAP listing of the finite subgroups of U(3) of order smaller than 2000

We have sorted the SmallGroups library of all the finite groups of order smaller than 2000 to identify the groups that possess a faithful three-dimensional irreducible representation (`irrep') and cannot be written as the direct product of a smaller group times a cyclic group. Using the computer algebra system GAP, we have scanned all the three-dimensional irreps of each of those groups to identify those that are subgroups of SU(3); we have labelled each of those subgroups of SU(3) by using the extant complete classification of the finite subgroups of SU(3). Turning to the subgroups of U(3) that are not subgroups of SU(3), we have found the generators of all of them and classified most of them in series according to their generators and structure.


Introduction
Many high-energy physicists are thrilled by the prospect that the numerical entries of the leptonic mixing matrix (PMNS matrix) might be related to some small (or maybe not so small) finite group. Many specific finite groups have been considered, like for instance A 4 [1], S 4 [2], S 3 [3], T 7 [4], A 5 [5], ∆(27) [6], the group series ∆ (6n 2 ) [7], the groups Σ (nϕ) [8], and so on. Most of the finite groups considered are subgroups of SU (3); those subgroups are especially inviting because a complete classification of them, and their generators, have been known for over a century [9]. On the contrary, there is no complete classification of the finite subgroups of U(3), 1 though a few series of those subgroups have been derived in ref. [10]. At least one finite subgroup of U(3) has already been utilized in particle physics [11].
Although a full theoretical study of each individual group can always be undertaken, for large groups such a study becomes impractical and it is convenient to have recourse to the computer algebra system GAP, which is tailored to deal with finite groups and can readily furnish the structure, irreducible representations ('irreps'), character table, and so on, of each of them. GAP is supplemented by the SmallGroups library, which contains, in particular, all the finite groups of order smaller than 2 000. In that library each finite group has an identifier [o, j], where o ≥ 1 is the order, i.e. the number of elements, of the group and j ≥ 1 is an integer which distinguishes among the non-isomorphic groups of identical order. For instance, the group with SmallGroups identifier [4,1] is the cyclic group 2 4 ∼ = {1, i, −1, −i} while the group with SmallGroups identifier [4,2] is the direct product of cyclic groups 2 × 2 ∼ = { (1,1) , (1, −1) , (−1, 1) , (−1, −1)}; SmallGroups informs us that there are, in fact, only these two non-isomorphic groups with four elements. A SmallGroups listing of all the finite groups of order up to 100, together with their structure, 3 was published in ref. [13]. A SmallGroups listing of the finite groups of order up to 512 that have a faithful three- 1 In this paper, whenever we use the expression "finite subgroups of U (3)" we usually mean only the subgroups of U (3) that are not subgroups of SU (3).
2 SmallGroups uses C n to denote the cyclic group of order n, instead of the more usual notation n . SmallGroups uses the notation E(n) for the n'th root of unity. 3 SmallGroups informs us about the structure of each group. This is given in terms of direct products (denoted '×'), semi-direct products (denoted '⋊'), or group extensions (denoted '.'). A pedagogical explanation of these concepts may be found, for instance, in ref. [12]. dimensional irrep and are not the direct product of a cyclic group and some other group was published in ref. [10].
However, SmallGroups lists the groups of the same order in a way that does not allow one to extract much information on them. For instance, the group [12,3] ∼ = A 4 is a subgroup of SU (3) and has a three-dimensional faithful irrep; the groups [12,1] and [12,4] ∼ = D 6 are subgroups of SU(3) but do not possess three-dimensional irreps; the group [12,2] ∼ = 12 is a subgroup of U(1) ⊂ U (3); the group [12,5] ∼ = 6 × 2 is a subgroup of U(1) × U(1) but not of U(3).
The first step in this work was to survey the whole SmallGroups list of groups of order smaller than 2 000 in order to identify the ones that have at least one faithful three-dimensional irreducible representation; cannot be written as the direct product of a smaller group and a cyclic group.
The second step in this work was to pick each of the groups above and ask GAP to compute the determinant of each of the matrices in each of its threedimensional representations. If there is a three-dimensional representation in which all the matrices have unit determinant, then the group is a subgroup of SU (3); otherwise the group is not a subgroup of SU(3) but it is a subgroup of U(3)-because every representation of a finite group is equivalent to a representation through unitary matrices. In this way, we have separated the subgroups of SU(3) from the subgroups of U (3).
A complete classification of all the finite subgroups of SU(3) has long existed [9]. There are groups (so-called type A) of diagonal matrices, i.e. Abelian groups; they may be written as direct products of cyclic factors and do not concern us here. Then there are the subgroups of U (2), which are called type B; their three-dimensional representations are (just as the ones of type A subgroups) reducible and therefore they also do not concern us. Of interest to us are the type C and type D groups, which were best characterized in ref. [14], and also the 'exceptional' groups. In this work we give the SmallGroups identifiers of all the SU(3) subgroups of types C and D, together with their classification according to ref. [14], and also the SmallGroups identifiers of the exceptional subgroups. This is done in section 3.
There is no theoretical classification of all the finite subgroups of U(3). We feel that having a complete listing of all those subgroups of order less than 2 000, together with their generators, may be a useful step towards achieving such a classification; at the very least, it allows one to get a feeling for what it might look like. Therefore, in this work we give the SmallGroups identifiers of all the finite U(3) subgroups, together with their generators. We also partially unite those subgroups in series, viz. in sets of groups that have related generators depending on one, two, or sometimes three integers. This is done in section 4.
We also give, for every finite subgroup of U (3), the dimensions of all its inequivalent irreps, as determined by GAP.
In section 2 we explain our procedure. In an appendix we provide tables of all the finite subgroups of U(3) that have a faithful three-dimensional irrep and are not isomorphic to the direct product of a smaller group and a cyclic group. We give separate tables for the groups that are subgroups of SU (3) and for the groups that are not subgroups of SU (3). In those tables, we order the groups according to their SmallGroups classification, viz. in increasing order first of o and then of j in their [o, j] identifiers.

GAP procedures
GAP [15] is a computer algebra system that provides a programming language, including many functions that implement algebraic algorithms. It is supplemented by many libraries containing a large amount of data on algebraic objects. Using GAP it is possible to study groups and their representations, display the character tables, find the subgroups of larger groups, identify groups given through their generating matrices, and so on.
GAP allows access to the SmallGroups library through the SmallGroups package [16]. That library contains all the finite groups of 'small' orders, 4 viz. less than a certain upper bound and also orders whose prime factorization is small in some sense. The groups are ordered by their orders; for each of the available orders, a complete list of non-isomorphic groups is given. SmallGroups contains all the groups of order less than 2 000 except order 4 The order of a finite group is the number of its elements. 4 1024, because there are many thousands of millions of groups of order 1024. SmallGroups also contains other groups with some specific orders larger than 2 000.
The SmallGroups library has an identification function which returns the SmallGroups identifier of any given group. For each generic group in the library there are effective recognition algorithms available. To identify encoded and insoluble groups, two approaches are used: one is a general algorithm to solve the isomorphism problem for p-groups, 5 the second one uses the invariants 6 of stored groups [17]. Using these methods, it is possible to identify all the groups in the library, except for orders 512, 1536, and some orders above 2 000. For the identification of groups we use GAP command In our work, firstly we have scanned the SmallGroups library and extracted therefrom all the groups with three-dimensional irreps. Using the GAP command one lets G denote the group with identifer [o, j] in the SmallGroups library. The command allows one to find out how many groups there are for a chosen order o and thus automates the scanning of library. For a given group G, GAP offers the possibility to calculate the irreps by using the command repG := IrreducibleRepresentations(G).
It is possible to display all the irreps by using the GAP command where p is a prime number, is a group in which each element has a power of p as its order. That is, for each element g of a p-group, there is a non-negative integer n such that the product of p n copies of g, and not less, is equal to the identity element e. (But, the integer n is in general different for different elements g of the group.) 6 In the SmallGroups library there is a list of distinguishing invariants for all encoded groups except those of orders 512 and 1536. This list of invariants is compressed. It provides an efficient approach to identify any encoded group in the library. too; however, the labeling of the irreps may differ from the labeling received through the command It is convenient to select all the three-dimensional irreps by using the command One may select all the elements of a given group G through the command elG := Elements(G).
Then, the command where the integer i parameterizes the loop, allows one to list all the elements of the chosen irrep. We have selected the groups from the SmallGroups library that have at least one faithful 7 three-dimensional irrep. Then, by using the GAP command that gives the structure of a group, viz.
we have discarded the groups that are direct products with a cyclic group. There are 10 494 213 groups of order 512 and 408 641 062 groups of order 1536. However, the groups of order 512 do not possess three-dimensional irreps because 512 cannot be divided by three, therefore we did not need to consider them. On the other hand, the number of groups of order 1536 is too large for all of them to be scanned in the way described above. Therefore, we have used the conjecture in ref. [18] that both nilpotent groups and groups with a normal Sylow 3-subgroup 8 do not have three-dimensional faithful irreps. Utilizing the command 7 In order to identify the faithful irreps, we have compared all the matrices in each irrep. If different elements of the group are represented by different matrices in the irrep, then the irrep is faithful. 8 These two concepts of group theory have been explained in ref. [19].

6
one gets the information about the arrangement of the groups of a given order. Using this information, we have determined the scanning range of groups of order 1536. To check whether the group is nilpotent, the command may be used, while gives the nilpotency class of the group G. The Sylow 3-subgroups of a group G may be found by typing the command We have found that only four groups of order 1536 have faithful threedimensional irreps and cannot be written as the direct product of a smaller group and a cyclic group. For groups that have faithful three-dimensional irreps, we have asked GAP to compute the determinant of each of the matrices in each of its threedimensional representations. This was done through the command If there is a three-dimensional representation in which all the matrices have unit determinant, then the group is a subgroup of SU (3); if there is no such representation, then the group is not a subgroup of SU (3), but it is a subgroup of U(3) because it has a three-dimensional representation and because all the representations of finite groups are equivalent to representations through unitary matrices.
We have used different methods in order to classify the groups in the lists of the subgroups of U(3) and SU(3). One of the methods is the analysis of the generators of the three-dimensional irreps. The command returns a list of generators of the group G. The generators of the threedimensional irreps may be listed through the command By looking at these lists we have tried to find regularities in the generators. Another strategy was looking at the structures of the groups and sorting groups with analogous structures. When one has some generators, say three matrices M1, M2, and M3, a group G may be generated through the command Afterwards this group may be identified by finding its order, using the command Order(G) (19) or by counting the elements of the group through Size(elG).
Afterwards one may discover the SmallGroups identifier of G by using the command IdGroup(G).
The identification of some groups with large order may require a long computational time, therefore some hints about the group classification may be acquired by analyzing the group structure-using the command (10)-or by comparing the traces of the group matrices, determined through the command List(elG, x-> Trace(x)).
3 Finite subgroups of SU (3) In this section we give the generators and the SmallGroups identifiers of all the finite subgroups of SU(3) that • have a faithful three-dimensional irrep, • cannot be written as the direct product of a smaller group and a cyclic group, • have less than 2 000 elements. 8

Generators
We firstly define a few 3 × 3 matrices that act as generators of the various SU(3) subgroups. All those matrices have, of course, unit determinant.
The matrices are especially useful. Let n ≥ 1 be an integer. Then, Let n ≥ 1 and k ≥ 1 be integers. We define Let n ≥ 1 and r ≥ 1 be integers. We define i.e. G n,r = (L n ) −r .
3.2 The groups ∆ 3n 2 and ∆ 6n 2 For n ≥ 1, the groups ∆ (3n 2 ) have structure ( n × n ) ⋊ 3 and order 3n 2 ; 9 the groups ∆ (6n 2 ) have structure [( n × n ) ⋊ 3 ] ⋊ 2 and order 6n 2 . The group ∆ (3n 2 ) is generated by the matrices E and L n ; the group ∆ (6n 2 ) is generated by the matrices E, I, and L n . The SmallGroups identifiers of the groups ∆ (3n 2 ) of order smaller than 2 000 are given in table 1; 10 the SmallGroups identifiers of the groups ∆ (6n 2 ) of order smaller than 2 000 are given in table 2. 11 9 We adopt the convention that 1 is the trivial group, i.e. the group that has only one element, viz. the identity element e. 10 The group ∆ 3 × 1 2 ∼ = 3 ∼ = [3,1] is not included in table 1 because it is a cyclic group. 11 The group ∆ 6 × 1 2 ∼ = S 3 ∼ = [6,1] is not included in  The group ∆ (3 × 2 2 ) is isomorphic to A 4 , the group of the even permutations of four objects, and also to the symmetry group of the regular tetrahedron. The group ∆ (6 × 2 2 ) is isomorphic to S 4 , the group of all the permutations of four objects, and also to the symmetry group of the cube and of the regular octahedron.

The groups C
(k) n,l We use the notation of ref. [14]. The groups C (k) n,l have structure ( n × l )⋊ 3 and order 3nl. The integer l is positive. The integer n may be written n = rl, where r is another positive integer. The integer r may be either 1. a product of prime numbers p 1 , p 2 , . . . which are of the form p j = 6i j +1, where the numbers i j are integers, or 2. three times a product of prime numbers as in 1.
In case 1, l may be any positive integer; in case 2, l must be a multiple of three. The integer k is a function of r defined by 1 + k + k 2 = 0 mod r and k ≤ (r − 1)/ 2. For most values of r there is only one possible k, but for some r more than one (usually two) k are possible. The values of r, k, and l that produce groups C (k) n,l with order smaller than 2 000 are given in tables 3 and 4. There is a very large number of groups C (k) n,l of order smaller than 2 000, therefore we opt for giving their SmallGroups identifiers only in the appendix.
The groups C (k) n,l only have singlet and triplet irreps. The number of inequivalent singlet irreps is three when l cannot be divided by three and nine when l is a multiple of three.

The groups D
(1) 3l,l We continue to use the notation of ref. [14]. For an integer l that is a multiple of three, the groups D (1) 3l,l have structure [( 3l × l ) ⋊ 3 ]⋊ 2 and order 18l 2 . They are generated by the matrices E, I, and B 3l,1 = diag (ν, ν, ν −2 ) for ν = exp [2iπ/(3l) ]. There are only three groups D (1) 3l,l of order smaller than 2 000: The groups D 3l,l have six inequivalent singlets and three inequivalent doublets for any value of l. Besides, they have 6(l − 1) inequivalent triplet irreps and l(l − 3)/2 + 1 inequivalent six-plets.    (3) The groups ∆ (3n 2 ) and C (k) n,l form the class C of finite subgroups of SU (3). The groups ∆ (6n 2 ) and D (1) 3l,l form the class D of finite subgroups of SU (3). Both classes C and D contain infinite numbers of subgroups. Besides these infinite classes of subgroups, SU(3) has six 'exceptional' subgroups; 13,14 their SmallGroups identifiers are given in table 5. The generators of the exceptional subgroups are given, for instance, in ref. [10], together with the references to the original papers.

The exceptional subgroups of SU
The group Σ (60) is isomorphic to A 5 , the group of the even permutations of five objects, and to the symmetry group of the regular icosahedron and regular dodecahedron. The group Σ (168) is isomorphic to the projective special linear group P SL (2,7) and also to the general linear group GL (3,2).
The number of inequivalent p-dimensional irreps of the exceptional finite subgroups of SU (3) is given in table 6 [21].

Finite subgroups of U (3)
In this section we give the generators and the SmallGroups identifiers of all the finite subgroups of U(3) that • are not subgroups of SU (3), • have a faithful three-dimensional irrep, • cannot be written as the direct product of a smaller group and a cyclic group,  (3).
For most groups, we also give the numbers of inequivalent irreps of each dimension.
There is at present no mathematical classification of the finite subgroups of U(3). Therefore, we will just classify the various subgroups that we have found using the SmallGroups library and GAP, by constructing 'series' of subgroups that have generators, structures, and numbers of irreps related among themselves. Unfortunately, there is some degree of ambiguity in this task, since any group may always be generated by different sets of generators. It is moreover often found that groups with related generators end up having quite different structures. Still, we hope to be able to shed some light on the possible types of subgroups of U(3).

The generators
We firstly define some 3 × 3 matrices that often appear as generators of the U(3) subgroups. Let • r be a product of prime numbers p 1 , p 2 , . . . which are of the form p j = 6i j + 1, where the numbers i j are integers; • k be an integer which is a function of r defined by 1 + k + k 2 = 0 mod r and k ≤ (r − 1)/ 2. For most values of r there is only one possible k, but for some r more than one k are possible.
The lowest r and the corresponding k are given in table 7. In this section, whenever we let r and k denote a pair of integers, we will be referring to one of the pairs in table 7. The matrix appears as generator of many U (3) subgroups. Notice that B r,k ∈ SU (3).
We use the definition of L n in equation (24). Notice that L n ∈ SU (3).
is especially useful. We will also encounter  Let m be an integer. We define The matrix E ≡ E 0 in equation (23a) is especially useful. Both E 0 and E 1 have unit determinant, but E m / ∈ SU (3) for m > 1. Let m and j be integers. We define Notice that F m,j / ∈ SU(3) for m ≥ 2 or j ≥ 1. The matrix I ≡ F 0,0 in equation (23b) has already been useful; also useful is Let ω = exp (2iπ/3) and µ = exp [2iπ / (3 m )]. We define Let ω = exp (2iπ/3). We define Notice that K ∈ SU (3).
Notice that det Q m,j = ξ 3 = 1 in general.

The series of groups that Ludl has discovered
Ludl [10] has proved the existence of the following series of finite subgroups of U (3).
where m is an integer larger than 1, 15 has structure r ⋊ 3 m and order 3 m r. The groups T (k) r (m) of order smaller than 2 000 are given in table 8. Each of these groups has two generators, which may be chosen to be B r,k in equation (28) and E m in equation (31a).
The groups T (k) r (m) have 3 m inequivalent singlet irreps; all the remaining irreps of those groups are triplets.
Groups ∆ (3n 2 , m): The group ∆ (3n 2 , m), where the integer n cannot be divided by 3 and m > 1, 16 has structure ( n × n ) ⋊ 3 m and order 3 m n 2 . The groups ∆ (3n 2 , m) of order less than 2 000 are listed in table 9. The group ∆ (3n 2 , m) is generated by the matrices L n in equation (24) and E m in equation (31a).
Groups S 4 (j): The group S 4 (j), where j > 1, 17 has structure A 4 ⋊ 2 j and order 3 × 2 j+2 . 18 There are six groups S 4 (j) of order smaller than 2 000; they are given in table 10. The group S 4 (j) is generated by the matrices E in equation (23a), L 2 in equation (29), and −F 0,j , where F m,j is given in equation (32).

New series of groups that we have discovered
Ludl [10] has derived the existence of the series of groups in the previous subsection by applying mathematical theorems that he demonstrated. We have discovered some further series of groups through a careful inspection of the list of all the finite subgroups of U(3) of order smaller than 2 000 that we have produced, together with some guesswork. Clearly, since there are no theorems supporting our method, we cannot be sure that our series of groups extend to groups of order larger than 2 000. Still, the series of groups in this subsection seem to us to be on firm standing, since they are quite large and display no exceptions up to group order 2 000.
Groups L (k) r (n, m): For an integer n that cannot be divided by 3 and for m > 1, these are groups with structure ( rn × n ) ⋊ 3 m and order 3 m rn 2 . While the groups T (k) r (m) are generated by the matrices B r,k and E m , and the groups ∆ (3n 2 , m) are generated by the matrices L n and E m , the groups L (k) r (n, m) are generated by all three matrices B r,k , L n , and E m . Thus, the groups L              Groups X(n): There are several groups that have a three-dimensional irrep where all the matrices are of one of the following types [10]: where ν = exp (2iπ/n). We call them 'groups RVW'. The groups X(n) are groups RVW where • n is a multiple of 3, • the matrices R (n, a, b, c) have a + b + c = (n/3) mod n, • the matrices V (n, a, b, c) have a + b + c = (2n/3) mod n, • the matrices W (n, a, b, c) have a + b + c = 0 mod n.
The groups X(n) have order 3n 2 ; the identifiers of the groups of order less than 2 000 are in table 15. The structure of X(n) is n/3 × n/3 ⋊ 9 ⋊ 3 provided n is not a multiple of 9; otherwise it is more complicated. The groups X(n) are generated by the matrices L n in equation (24) and Z 1 in equation (31b).
The groups X(n) have nine inequivalent singlets; their remaining irreps are all triplets.

Tentative series of groups
We have found a few more series of groups through inspection of the list of the finite subgroups of U(3) of order less than 2 000. However, these series have few groups each and we can hardly ascertain whether and how they extend to groups of order larger than 2 000.

Groups S
; they all have order 3 m+2 r. The generators are the matrices B r,k in equation (28), together with • E in equation (23a), L 3 in equation (30), and X 3 (m) in equation (34e) for S 19 (2)         Groups W (n, m): The groups W (n, m), where n cannot be divided by 3 and m > 1, are generated by the matrices E in equation (23a), L n in equation (24), and Y 1 (m) in equation (34c). They have structure ( 3 m n × n )⋊ 3 and order 3 m+1 n 2 . The groups W (n, m) with order smaller than 2 000 are listed in table 18.
Each of the groups W (n, m) has 3 m inequivalent singlets; the remaining irreps of those groups are triplets.
Groups Z (n, m), Z ′ (n, m), and Z ′′ (n, m): These groups, where n is a multiple of 3 and m > 1, have structure 23 ( 3 m−1 n × n ) ⋊ 3 and order 3 m n 2 . The groups with order smaller than 2 000 are listed in table 19. The generators of Z (n, m) are just the same as those of W (n, m), viz. E, L n , and Y 1 (m)-the only difference being that for Z (n, m) the integer n is a multiple of 3 while for W (n, m) the integer n cannot be divided by 3. The groups Z ′ (n, m) are generated by the matrices E, L n , and X 1 (m). The groups Z ′′ (n, m) are generated by the matrices E, L n , and X 2 (m). Notice that, for m = 2, Z ′′ (n, m) is generated by matrices with unit determinant and therefore it is a subgroup of SU (3).
Each of the groups Z (n, m) and Z ′′ (n, m) has 3 m+1 inequivalent singlets. The groups Z ′ (n, m) have 3 m inequivalent singlets. All the remaining irreps of all those groups are triplets.
• n is a multiple of 3, • m > 1, • j is an integer, have order 3 m 2 j n 2 . The groups Z (n, m, j) and Z ′ (n, m, j) with order smaller than 2 000 are in table 20. 24 The groups Z (n, m, j) and Z ′ (n, m, j) are generated by the same matrices as the groups Z ′ (n, m) and Z ′′ (n, m), respectively, with the addition of the further generator −F 1,j , where F m,j is given in equation (32). Notice that there are no groups Z ′ (n, 2, 1) in table 20, because all the matrices generating Z ′ (n, 2, 1), viz. E, L n , X 2 (2), and −F 1,1 have unit determinant and therefore Z ′ (n, 2, 1) is a subgroup of SU (3).
Groups H (n, m, j): When we use generators E, L n , X 1 (2), and −F m,j with m > 1, we obtain groups that we call H (n, m, j) and list in    The SmallGroups identifiers of the groups G (m, j) with order smaller than 2 000. order 3 m+1 × 2n 2 . The groups H (n, m, j) with j > 1 are described in the paragraph of groups G (m, j) below.
The groups H (n, m, j) have exactly the same number of inequivalent irreps of each dimension as the groups Z (n, m + 1, j) and Z ′ (n, m + 1, j).
Groups Y (m, j): The groups Y (m, j), where m ≥ 2 and j ≥ 1, have structure [( 2 j × 2 j ) ⋊ 3 m+1 ] ⋊ 3 and order 3 m+2 4 j . There are only three groups Y (m, j) of order smaller than 2 000: The groups Y (m, j) are generated by L 3 in equation (30) [1944,707] appear in table 21 too.) The groups G (m, j) are generated by the matrices E, −F m,j , where F m,j is given in equation (32), and diag (1, 1, ω). For m = 1 and j = 2 one may add a fourth generator L 2 , given in equation (29), to obtain the group [1296,699], which has structure The groups G (m, j) have exactly the same number of inequivalent irreps of each dimension as the groups Z (3, m + 1, j) and Z ′ (3, m + 1, j).
Groups U (n, m, j): The groups U (n, m, j), where n is a multiple of 3, m > 1, and 1 < j ≤ m, have structure ( 3 m−1 n × n × 3 ) ⋊ 3 and order 3 m+1 n 2 . We have found the following groups U (n, m, j) with order smaller than 2 000: The generators of U (n, m, j) are the matrix E together with diag ν, ν, ν 2 , where ν = exp (2iπ / n ), and Notice that, when j = m-this happens in three out of the four groups U (n, m, j) in (42)-the matrix (44) reduces to the matrix Y 1 (m) in equation (34c). The groups U (n, m, j) possess 3 j+1 inequivalent singlet irreps; all their remaining irreps are triplets.
Groups V (j): The groups V (j) have order 81 × 4 j and structure There are three groups V (j) with order smaller than 2 000: The generators of V (j) are the matrices Z 1 , X 2 (2), and L 2 j . The groups V (j) have nine singlet irreps. All their other irreps are triplets.
Groups D(j): The groups D(j) have structure ( 9×2 j × 9×2 j ) ⋊ 3 and order 243×4 j . They are generated by the matrices E 2 , L 2 j , and T 1 (2). There are two groups of order smaller than 2 000: Both these groups have nine inequivalent singlets; their other irreps are triplets.
Groups J(m): The groups J(m) have structure 3 m . [( 9 × 3 ) ⋊ 3 ] and order 81 × 3 m . They are generated by the matrices Z m and L 9 . There are two groups of order smaller than 2 000: Notice that J(1) coincides with X(9) in table 15. The groups J(m) have 3 m+1 singlets; their other irreps are triplets.

The generators of a few more groups
In this subsection we collect a few more groups together with their generators.
The groups Θ(m) have as many inequivalent irreps of each dimension as groups Π (m, 1).
Notice that all three generators of Υ ′ (2) have unit determinant and therefore Υ ′ (2) is a subgroup of SU(3).

Conclusion
In this paper we have used the SmallGroups library to search for all the finite subgroups of U(3) of order less than 2 000 that have a faithful threedimensional irreducible representation and that cannot be written as the direct product of some smaller group and a cyclic group. We have found that there are three types of finite subgroups of U(3): • Groups that have a three-dimensional representation consisting solely of matrices of the forms (37) for some value of n. Those groups only have singlet and triplet irreducible representations.
• Groups that have a three-dimensional representation consisting solely of matrices of the forms (37) and (52) for some value of n. Those groups only have singlet, doublet, triplet, and six-plet irreducible representations.
• Groups that do not have a three-dimensional representation consisting solely of matrices of the forms (37) and (52) We were able to group most finite subgroups of U(3) in many series depending on one, two, or sometimes three integers; the groups in each series have related generators and related numbers of irreps of each dimension. Unfortunately, many of these series have very few groups and we do not know whether and how they extend to groups of order higher than 2 000. It is possible (and it would be desirable) that some of these series may be further unified among themselves.