Scientific paper
Group Theory for Tetramethylethylene, II
Mohammad Reza Darafsheh1*, Ali Moghani2 and Soroor Naghdi Sedeh2
1 School of Mathematics, College of Science, University of Tehran, 14174 Tehran, Iran
2 Department of Color Physics, Institute for Colorant, Paint and Coating (ICPC),
16765 Tehran, Iran
* Corresponding author: E-mail: darafsheh@ut.ac.ir
Received: 07-05-2008
Abstract
The maturated and unmaturated groups have been introduced by S. Fujita who used them in the markaracter table and the Q-conjugacy character table of a finite group. Fujita introduced more concise forms called the Q-conjugacy characters with integer-valued of the irreducible characters of finite groups and applied his results in this area of research to enumerate isomers of molecules. In this paper using GAP program all integer-valued characters of the full non-rigid group (f-NRG) of tetramethylethylene (2,3-dimethylbut-2-ene) is calculated by the Q-conjugacy relationships. It is shown that this group has 29 dominant classes (similarly, Q-conjugacy characters) such that 16 of them are unmaturated (similarly, Q-conjugacy characters such that they are the sum of two irreducible characters). Then the markaracter table and Q-conjugacy character table of the f-NRG of tetramethylethylene are derived for the first time.
Keywords: Full non-rigid group, markaracter table, Q-conjugacy character table, tetramethylethylene (2,3-dimethyl-but-2-ene).
1. Introduction
In order to develop new methods of combinatorial enumeration of isomers, some relationship between character tables containing characters for irreducible representations and mark tables containing marks for coset representations have been clarified by S. Fujita who proposed not only markaracter tables, which enable us to discuss characters and marks on a common basis, but also Q-conjugacy character tables, which are obtained for finite groups. The enumeration of chemical compounds has been accomplished by various methods, but the Pólya-Redfield theorem has been a standard method for combinatorial enumerations of graphs and chemical compounds. A dominant class is defined as a disjoint union of conjugacy classes that corresponds to the same cyclic subgroup, which is selected as a representative of conjugate cyclic subgroups. Let G be a finite group and hj, h2 e G. We say hj, h2 are Q-conjugate if there exists t e G such that t-1 <h1> t = <h2>. The Q-conjugacy is an equivalence relation on G and generates equivalence classes which are called dominant classes, i.e. the group G is partitioned into dominant classes as follows: G = KJ+ K2+ _ + Ks in
which Kj corresponds to the cyclic (dominant) subgroup Gi selected from a non-redundant set of cyclic subgroups of G denoted by SCSG.1-14
A molecule is said to be non-rigid if there are several local minima on the potential energy surface easily surmountable by the molecular system via a tunneling rearrangement. A non-rigid molecule typically possesses several potential valleys separated by relatively low energy barriers, and thus exhibits large amplitude tunneling dynamics among various potential minima. Because of this deformability, the non-rigid molecules exhibit some interesting properties of intramolecular dynamics, spec-troscopy, dynamical NMR etc., all of which can be interpreted resorting to group theory. Group theory is one of the most powerful mathematical tools in quantum chemistry and spectroscopy. It can predict, interpret, and simplify complex theories and data. Group theory is the best formal method to describe the symmetry concept of molecular structures. Group theory for non-rigid molecules is becoming increasingly relevant and its numerous applications to large amplitude vibrational spectroscopy of small organic molecules are described in the literature.15-19 The molecular symmetry group of a non-rigid molecule was
first defined by Longuet-Higgins20 although there have been earlier works that suggested the need for such a framework by Hougen.21 Bunker and Papou{ek22 extended the definition of the molecular symmetry group to linear molecules using an extended molecular symmetry. The operations of the molecular symmetry group and the three-dimensional rotation group are used together to treat the symmetry properties of molecules in electric and magnetic fields by Watson.23 The complete set of the molecular conversion operations that commute with the nuclear motion operator will contain overall rotation operations that describe a molecule rotating as a whole, and intramolecular motion operations that describe molecular moieties moving with respect to the rest of the molecule. These operations form a group which is called the full non-rigid molecule group (f-NRG) by Smeyers.24 Calculating the f-NRGs using wreath product formalism was first introduced by Balasubramanian. He also computed the character table of non-rigid groups under consideration.25-27
The present study investigates the Q-conjugacy character tables of tetramethylethylene (Figure 1), the f-NRG of which has been previously introduced.29 In order to derive all of its integer-valued characters, it is shown that its unmaturated group has 16 row- and column-reductions in its character table. The reader is encouraged to consult re-ferences26-32 for background materials. The notation we use is standard and mainly taken from references.33-34
For any g e G, the set of all permutations
G(/H) = (
Hg„
)
Figure 1. Geometry of Tetramethylethylene (2,3-Dimethylbut-2-
ene)
2. Results and Discussion
In this section we first describe some notation. Suppose X be a set, a permutation representation P of a finite group G is obtained when the group G acts on a finite set X = {x1, x2, xt} from the right, which means that we are given a mapping P: X x G ^ X via (x, g) ^ xg such that the following holds: (xg)g' = x(gg') and x1 = x, for each g, g' e G and x e X. Now let it is assumed that one is given an action P of G on X and a subgroup H of G. One considers the set of its right cosets Hg and the corresponding partition of G into these cosets: G ^ H + H +_+ H
g1	g2	gm
Hg,
constructs a permutation representation of G, which is called a coset representation of G by H denoted by G(/H). The degree of G(/H) is |G| / |H|, where |G| is the number of elements in G. Obviously, the coset representation G(/H) is transitive, i.e. has just one orbit.1 To denote the consecutive classes of elements of order n, for example if an element g has order n, then its class is denoted by nx, where x runs over the letters a, b, etc. If M is a normal subgroup of G and K is another subgroup of G such that MOK = {e} and G = MN = <M, N>, then G is called a semi direct product of N by M which is denoted by N : M = M x N.
Let K and H be groups and suppose H acts on the set r. Then the wreath product of K by H, denoted by K ~ H is defined to be the semi direct product Kr : H such that Kr = {f | f: r ^ K}, see references33-34 for more details.
Let C be a u x u matrix of character table of G. Then, C is transformed into a more concise form called the Q-conjugacy character table that we denote its s x s matrix by CQ (s < u) as follows: If u = s, then C = CQ i.e. G is a maturated group. Otherwise s < u, for each Gi e SC-SG (the corresponding dominant class Ki) set ti = m(Gi) / ^(|Gj|) where m(Gj) = |Ng(Gj)|/|Cg(Gj)| (called the maturity discriminant), ^ is the Euler function and finally Ng(Gj) and Cg(Gj) denote the normalizer and centralizer of Gj in G, respectively for i = 1, s. If tj = 1 then Kj is exactly a conjugacy class so there is no reduction in row and column of C, but if ti > 1 then Ki is a union of ti-conju-gacy classes of G (i.e. reduction in column) therefore the sum of tJ rows of irreducible characters via the same degree in C (reduction in rows) gives us a reducible character which is called the Q-conjugacy character with integer-valued.
It has been shown that the f-NRG of tetramethy-lethylene is a wreath product of the cyclic group of order three with direct product of two copies of cyclic group of order two,29 i.e. C3 ~ (C2 x C2) as follow:
Referring to Figure 1, the group of each CH3 at the four corners of the framework is given in terms of permutations as follows: A1 = <(1, 2, 3)>, A2 = <(4, 5, 6)>, A3 = <(7,8,9)>, A4 = <(10,11,12)>, where A21, A2, A3 and A4 are the symmetry groups of the CH3 whose carbon atom is marked as 13, 14, 15 and 16, respectively. Let T be the f-NRG of tetramethylethylene, therefore T has the following structure: T = (A1x A2 x A3 x A4) : V, where V = {id, (13, 14)(15, 16)(a, b), (13, 16)(14, 15)(a, b), (13, 15)(14, 16)(a, b)} is the Klein's four group, so it is obvious that every element of T is as a vector (a1, a2, a3, a4, v) such that aJ e Gj and v e V, i.e. T can be written in terms of wreath product T = C3 ~ (C2 x C2). Now, the computations of the symmetry properties of molecules were carried out with the aid of GAP SYSTEM,35 a group theory software package which is free and extendable. We run the following
Table 1: The Markaracter Table for Tetramethylethylene
MC	Gi	G2	G3	G4	G5	G6	G7	G8	G9	Gi0	Gii	Gi2	Gi3	Gi4	Gi5
T(/Gi)	324	0	0	0	0	0	0	0	0	0	0	0	0	0	0
T(/G2)	162	18	0	0	0	0	0	0	0	0	0	0	0	0	0
T(/G3)	162	0	18	0	0	0	0	0	0	0	0	0	0	0	0
T(/G4)	162	0	0	18	0	0	0	0	0	0	0	0	0	0	0
T(/G5)	108	0	0	0	54	0	0	0	0	0	0	0	0	0	0
T(/G6)	108	0	0	0	0	54	0	0	0	0	0	0	0	0	0
T(/G7)	108	0	0	0	0	0	54	0	0	0	0	0	0	0	0
T(/G8)	108	0	0	0	0	0	0	27	0	0	0	0	0	0	0
T(/G9)	108	0	0	0	0	0	0	0	27	0	0	0	0	0	0
T(/Gio)	108	0	0	0	0	0	0	0	0	54	0	0	0	0	0
T(/Gii)	108	0	0	0	0	0	0	0	0	0	54	0	0	0	0
T(/Gi2)	108	0	0	0	0	0	0	0	0	0	0	27	0	0	0
T(/Gi3)	108	0	0	0	0	0	0	0	0	0	0	0	108	0	0
T(/Gi4)	108	0	0	0	0	0	0	0	0	0	0	0	0	54	0
T(/Gi5)	108	0	0	0	0	0	0	0	0	0	0	0	0	0	27
T(/Gi6)	108	0	0	0	0	0	0	0	0	0	0	0	0	0	0
T(/Gi7)	108	0	0	0	0	0	0	0	0	0	0	0	0	0	0
T(/Gi8)	108	0	0	0	0	0	0	0	0	0	0	0	0	0	0
T(/Gi9)	108	0	0	0	0	0	0	0	0	0	0	0	0	0	0
t(/g2O)	108	0	0	0	0	0	0	0	0	0	0	0	0	0	0
T(/G2i)	54	6	0	0	0	0	0	0	0	0	27	0	0	0	0
T(/G22)	54	18	0	0	0	0	0	0	0	0	0	0	0	27	0
T(/G23)	54	0	18	0	0	0	0	0	0	0	0	0	0	0	0
T(/G24)	54	0	6	0	0	0	27	0	0	0	0	0	0	0	0
T(/G25)	54	0	0	6	0	27	0	0	0	0	0	0	0	0	0
T(/G26)	54	0	0	18	0	0	0	0	0	27	0	0	0	0	0
T(/G27)	54	6	0	0	0	0	0	0	0	0	0	0	54	0	0
T(/G28)	54	18	0	0	0	0	0	0	0	0	0	0	0	0	0
T(/G29)	54	0	0	6	0	0	0	0	0	0	0	0	54	0	0
MC	Gi6	Gi7	Gi8	Gi9	G20	G2i	G22	G23	G24	G25	G26	G27	G28	G29	
T(/Gi)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/G2)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/Gj)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/G4)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/G5)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/G6)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/G7)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/G8)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/G9)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/Gio)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/Gii)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/Gi2)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/Gij)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/Gi4)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/Gi5)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/Gi6)	108	0	0	0	0	0	0	0	0	0	0	0	0	0	
T(/Gi7)	0	108	0	0	0	0	0	0	0	0	0	0	0	0	
T(/Gi8)	0	0	108	0	0	0	0	0	0	0	0	0	0	0	
T(/Gi9)	0	0	0	27	0	0	0	0	0	0	0	0	0	0	
t(/g2O)	0	0	0	0	27	0	0	0	0	0	0	0	0	0	
T(/G2i)	0	0	0	0	0	3	0	0	0	0	0	0	0	0	
T(/G22)	0	0	0	0	0	0	9	0	0	0	0	0	0	0	
T(/G23)	54	0	0	0	0	0	0	18	0	0	0	0	0	0	
T(/G24)	0	0	0	0	0	0	0	0	3	0	0	0	0	0	
T(/G25)	0	0	0	0	0	0	0	0	0	3	0	0	0	0	
T(/G26)	0	0	0	0	0	0	0	0	0	0	9	0	0	0	
T(/G27)	0	0	0	0	0	0	0	0	0	0	0	6	0	0	
T(/G28)	0	0	54	0	0	0	0	0	0	0	0	0	18	0	
T(/G29)	0	0	0	0	0	0	0	0	0	0	0	0	0	6	
Table 2: The Q-Conjugacy Character Table for Tetramethylethylene
d Ti T2 T3 T4
T5 T6
T7 T8 T9
T10 Tii T12 T13 T14 T15 T16 T17 T18 T19 T20 T21 T22 T23 T24 T25 T26 T27 T28 T
D1	d2	D3	D4	D5	D6	d7	D8	D9	D10	D11	D12	D1j	D14	D15
1		1	1	1	1	1	1	1	1	1	1	1	1	1
1		-1	-1	1	1	1	1	1	1	1	1	1	1	1
1	-1	1	-1	1	1	1	1	1	1	1	1	1	1	1
1	-1	-1	1	1	1	1	1	1	1	1	1	1	1	1
2	2	2	2	-1	-1	2	-1	2		2	-1		-1	-1
2	2	-2	-2	-1	-1	2	-1	2		2	-1	2	-1	-1
2	-2	2	-2	-1	-1	2	-1	2		2	-1	2	-1	-1
2	-2	-2	2	-1	-1	2	-1	2		2	-1		-1	-1
2	0	-2	0	-1	2	-1	-1	2	2				2	-1
2	0	2	0	-1	2	-1	-1	2	2	-1	-1	-1	2	-1
2	-2	0	0	-1	-1	2	2	-1					-1	2
2	2	0	0	-1	-1	2	2	-1		-1	-1	-1	-1	2
2	0	0	-2	-1	2	-1	2	-1	-1	-1	2	2	-1	-1
2	0	0	2	-1	2	-1	2	-1			2	2	-1	-1
4	0	-4	0	1	-2	-2	1	4	-2	-2	1	-2	-2	1
4	0	4	0	1	-2	-2	1	4	-2	-2	1	-2	-2	1
4	-4	0	0	1	1	4	-2	-2	-2	-2	1	-2	1	-2
4	4	0	-4	1	1	4	-2	-2	-2	-2	1	-2	1	-2
4	0	0	4	1	-2	-2	-2	-2	1	-2	-2	4	1	1
4	0	0	0	1	-2	-2	-2	-2	1	-2	-2	4	1	1
4	0	0	0	1	-2	-2	-2	-2	4	1	1	1	-2	-2
4	0	0	0	1	-2	-2	4	1	-2	1	-2	-2	1	-2
4	0	0	0	1	4	1	-2	-2	-2	1	-2	-2	-2	1
8	0	0	0	-1	2	-4	2	-4	-4	2	-1	2	2	2
8	0	0	0	-1	2	-4	-4	2	2	2		-4	-1	2
8	0	0	0	-1	-4	2	2	-4	2	2		-4	2	-1
8	0	0	0	5	2	2	2	2	2	-1		2	-1	-1
8	0	0	0	-1	-4	2	-4	2	-4	-1		2	2	2
8	0	0	0	-4	2	2	2	2	2	-1	-1	2	-1	-1
D16	D17	D18	D19	D20	D21	D22	D23	D24	D25	D26	D27	D28	D29	
1		1	1	1	1	1	1	1	1	1	1	1	1	
1		1	1	1	1	1	1	-1		-1			-1	
1		1	1	1	-1	-1	-1	1	1	1			-1	
1		1	1	1	-1	-1	-1	-1	-1	-1	1	1	1	
-1	-1	2	2	2	-1	-1	2	-1		2	-1	-1	2	
-1	-1	2	2	2	-1	-1	2	1	1	-2	1	1	-2	
-1	-1	2	2	2	1	1	-2	-1	-1	2	1	1	-2	
-1	-1	2	2	2	1	1	-2	1	1	-2	-1	-1	2	
2	-1	2	-1	2	0	0	0	1	-2	1	0	0	0	
2	-1	2	-1	2	0	0	0	-1	2	-1	0	0	0	
2	-1	-1	2	2	1	-2	1	0	0	0	0	0	0	
2	-1	-1	2	2	-1	2	-1	0	0	0	0	0	0	
2	-1	2	2	-1	0	0	0	0	0	0	1	-2	1	
2	-1	2	2	-1	0	0	0	0	0	0	-1	2	-1	
-2		4	-2	4	0	0	0	-1	2	2	0	0	0	
-2		4	-2	4	0	0	0	1	-2	-2	0	0	0	
-2		-2	4	4	-1	2	2	0	0	0	0	0	0	
-2		-2	4	4	1	-2	-2	0	0	0	0	0	0	
-2		4	4	-2	0	0	0	0	0	0	-1	2	2	
-2		4	4	-2	0	0	0	0	0	0	1	-2	-2	
4		-2	-2	4	0	0	0	0	0	0	0	0	0	
4		-2	4	-2	0	0	0	0	0	0	0	0	0	
4		4	-2	-2	0	0	0	0	0	0	0	0	0	
-4	-1	-4	-4	8	0	0	0	0	0	0	0	0	0	
-4	-1	-4	8	-4	0	0	0	0	0	0	0	0	0	
-4	-1	8	-4	-4	0	0	0	0	0	0	0	0	0	
-4	-4	-4	-4	-4	0	0	0	0	0	0	0	0	0	
8	-1	-4	-4	-4	0	0	0	0	0	0	0	0	0	
-4		-4	-4	-4	0	0	0	0	0	0	0	0	0	
CQ
T1 T2 T3
T4 T5 T6
T7 T8 T9
T10 T11 T12 T13 T14 T15 T16 T17 T18 T19 T20 T21 T22 T23 T24 T25 T26 T27 T28 T
program at the GAP prompt to compute the mark table, the character table and the set SCSG of the f-NRG of te-tramethylethylene.
U:=ConjugacyClassesSubgroups{T); V:=List(ConjugacvClassesSubgroups(T),x->Elements(x)); if lsCydic(V[il[1])then Add(y,i);
After running the program, the following elements belong to the non-redundant set of cyclic subgroups of T: G1 = id, G2 = <(1, 7)(2, 8)(3, 9)(4, 10)(5, 11)(6, 12)>, G3 = <(1, 10)(2, 11)(3, 12)(4, 7)(5, 8)(6, 9)>, G4 = <(1, 4)(2,
5)(3,	6)(7, 10)(8, 11)(9, 12)>, G5 = <(1, 2, 3)(4, 6, 5)(7, 9,
8)(10,	11, 12)>, G6 = <(1, 2, 3)(4, 5, 6)(7, 9, 8)(10, 12,
11)>,	G7 = <(1, 2, 3)(4, 6, 5)(7, 8, 9)(10, 12, 11)>, G8 = <(1, 2, 3)(4, 5, 6)(7, 8, 9)(10, 11, 12)>, G9 = <(7, 8, 9)(10, 11, 12)>, G10 = <(4, 5, 6)(7, 9, 8)>, G11 = <(4, 5,
6)(10,12,11)>,	G12 = <(4, 5, 6)(7, 8, 9)>, G13 = <(7, 8,
9)(10,	12, 11)>, (^14 = <(4, 5, 6)(10, 11, 12)>, G15 = <(4, 5, 6)(7, 9, 8)(10, 11, 12)>, G1g = <(4, 5, 6)(7, 8, 9)(10, 11,
12)>,	G17 = <(4, 5, 6)(7, 8, 9)(10, 12, 11)>, G18 = <(1, 2, 3)(4, 5, 6)(7, 8, 9)(10, 12, 11)>, G19 = <(10, 11, 12)>, G20 = <(4, 5, 6)(7, 9, 8)(10, 12, 11)>, G21 = <(1, 2, 3)(4, 5,
6)(7,	9, 8)(10, 12, 11), (1, 4)(2, 5)(3, 6)(7,10)(8,11)(9, 12)>, G22 = <(1, 2, 3)(4, 6, 5)(7, 8, 9)(10, 12, 11), (1,
7)(2,	8)(3, 9)(4, 10)(5, 11)(6, 12)>, G23 = <(1, 2, 3)(4, 6, 5)(7, 9, 8)(10, 11, 12), (1, 10)(2, 11)(^, 12)(4, 7)(5, 8)(6, 9)>, G24 = <(1, 2, 3)(4, 5, 6)(7, 8, 9)(10, 11, 12), (1, 4)(2, 5)(3, 6)(7, 10)(8, 11)(9, 12)>, G.j = <(1, 2, 3)(4, 5, 6)(7, 8, 9)(10, 11, 12), (1, 10)(2, 11)(3, 12)(4, 7)(5, 8)(6, 9)>, G2g = <(1, 2, 3)(4, 5, 6)(7, 8, 9)(10, 11, 12), (1, 7)(2, 8)(3,
9)(4,	10)(5, 11)(6, 12)>, G27 = <(4, 5, 6)(7, 8, 9), (1,
10)(2,	11)(3, 12)(4, 7)(5, 8)((5, 9)>, G28 = <(4, 5, 6)(10, 11, 12), (1, 7)(2, 8)(3, 9)(4, 10)(5, 11)((5, 12)> and G,^ = <(7, 8, 9)(10, 11, 12), (1, 4)(2, 5)(3, 6)(7, 10)(8, 11)(9, 12)>.
See MC the markaracter table of tetramethylethylene which is derived from M174 ^ 174, the mark table of T in Table 1. Besides, we can see that T has exactly 29 dominant classes as follow:
D1 = 1a, D2 = 2a, D3 = 2b, D4 = 2c, D5 = 3aU3b, Dg = 3cU3e, D7 = 3d, D8 = 3fUU 3n, D9 = 3g, D10 = 3hU3q, D16 = 3iU3s, D12 = 3jU3r, D13 = 3k, D14 = 3lU3p, D15 = 3mU3o, D1g = 3tUJ3z, D17 = 3uUJ3y, D18 == 3v, D19 = 3w, D20 = 3x, D21 = 6aU6b, D22 = 6cU6e, D23 = 6d, D24 = 6fU6g, D.^ = 61iU6j, D2g = 6i, I327 = 6kU6l, = 6mU(5o, D29 = 6n such that the dominant classes Dj for i e {5, 6, 8, 10, 11, 12, 14, 15, 16, 17, 21, 22, 24, 25, 17, 28} are unmaturated which shows 16 column-reductions (similarly, row-reductions) in C45^45, the character table of T = C3 ~ (C: x C:) in the reference.29 There are sixteen unmaturated integer-valued characters in CQ the Q-conjugacy character table of T with the sum of two irreducible characters via same degrees. All integer-valued characters of tetramethylethylene are presented in Table 2.
3. Conclusions
In this paper using GAP program all integer-valued characters of the f-NRG of tetramethylethylene are calculated by the Q-conjugacy relationships. It is shown that this group has 29 dominant classes (similarly, Q-conjugacy characters) such that 16 of them are unmaturated (similarly, Q-conjugacy characters such that they are the sum of two irreducible characters) and the complete Q-conjugacy character table and the markaracter table of this group is computed successfully. The derived markaracter table and Q-conjugacy character table would also be valuable in other applications such as in the context of chemical applications of graph theory and aromatic compounds.14-::
Furthermore, we introduce the following conjecture.
Conjecture. Let Gj be a finite group for i = 1, n and W = G1 ~ G: ~ ~ Gn. If there exists k e {1, n} such that Gk is an unmaturated group, then W is an unma-turated group too.
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Povzetek
Zrele in nezrele grupe je S. Fujita vpeljal v tabele markarakterjev in Q-konjugiranosti končnih grup. Fujita je vpeljal tudi bolj zgoščene oblike, imenovane Q-konjugacijski karakterji s celoštevilčnimi vrednostmi nereducibilnih karakterjev končnih grup ter uporabil svoje rezultate za oštevilčenje izomerov molekul.
V tem prispevku z uporabo programskega paketa GAP in z upoštevanjem odnosov med Q-konjugiranostjo izračunamo vse celoštevilčne karakterje popolne netoge grupe (f-NRG) tetrametiletilena (2,3-dimetilbut-2-en). Pokažemo tudi, da ima ta grupa 29 dominantnih razredov (oz. Q-konjugacijskih karakterjev) in sicer tako, da jih je 16 nezrelih (Q-konju-gacijski karakterji so taki, da so vsota dveh nereducibilnih karakterjev). Nato kot prvi izpeljemo tabelo markarakterjev in Q-konjugacijskih karakterjev za f-NRG tetrametiletilena.