International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2006). Vol. D, ch. 1.10, pp. 255-264

Section 1.10.5. Tables

T. Janssena*

aInstitute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail: ted@sci.kun.nl

1.10.5. Tables

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In this section are presented the irreducible representations of point groups of quasiperiodic structures up to rank six that do not occur as three-dimensional crystallographic point groups.

Table 1.10.5.1[link] gives the characters of the point groups [C_n] with n = 5, 8, 10, 12, [D_n] with n = 5, 8, 10, 12, and the icosahedral group I. The direct products with [{\bb Z}_2] then follow easily. Although these direct products of a group K with [{\bb Z}_2] do not belong to the isomorphism class of K, their irreducible representations are nevertheless given in the table for K because these irreducible representations have the same labels as those for K apart from an additional subindex u. The reresentations of the subgroup K of [K\times {\bb Z}_2] are the same as for K itself, those for the cosets get an additional minus sign. In the tables, the characters for the groups [K\times {\bb Z}_2] are separated from those for K by a horizontal rule. In addition to the characters are given the realizations of crystallographic point groups, and the irreducible components of the vector representations in direct space [V_E] and internal space [V_I] for these realizations. The vector representation in [V_I] is called the perpendicular representation.

Table 1.10.5.1| top | pdf |
Character tables of some point groups for quasicrystals

(a) [C_5] [[\omega =\exp (2\pi i/5)]].

[C_{5}][\varepsilon][\alpha][\alpha^{2}][\alpha^{3}][\alpha^{4}]
n11111
Order 15555
[\Gamma_{1}] 1 1 1 1 1
[\Gamma_{2}] 1 [\omega] [\omega^{2}] [\omega^{3}] [\omega^{4}]
[\Gamma_{3}] 1 [\omega^{2}] [\omega^{4}] [\omega] [\omega^{3}]
[\Gamma_{4}] 1 [\omega^{3}] [\omega] [\omega^{4}] [\omega^{2}]
[\Gamma_{5}] 1 [\omega^{4}] [\omega^{3}] [\omega^{2}] [\omega]

 Generators Vector representationPerpendicular representation
5 [\alpha = C_{5z}] [\Gamma_{1}\oplus\Gamma_{2}\oplus\Gamma_{5}] [\Gamma_{3}\oplus\Gamma_{4}]

(b) [D_5] [[\tau = (\sqrt{5}-1)/2]].

[D_{5}][\varepsilon][\alpha][\alpha^{2}] [\beta]
n1225
Order 1 5 5 2
[\Gamma_{1}] 1 1 1 1
[\Gamma_{2}] 1 1 1 [-1]
[\Gamma_{3}] 2 [\tau] [-1-\tau] 0
[\Gamma_{4}] 2 [-1-\tau] [\tau] 0

 Generators Vector representationPerpendicular representation
52 [\alpha = C_{5z}] [\Gamma_{2}\oplus \Gamma_{3}] [\Gamma_{4}]
  [\beta = C_{2x}]    
5m [\alpha = C_{5z}] [\Gamma_{1}\oplus \Gamma_{3}] [\Gamma_{4}]
  [\beta = m_{x}]    
[\bar{5}m] [\sim 52 \times {\bb Z}_{2}] [\Gamma_{1u}\oplus\Gamma_{3u}] [\Gamma_{4u}]

(c) [C_8] [[\omega =\exp (\pi i/4) = (1+i)/\sqrt{2}]].

[C_{8}][\varepsilon][\alpha][\alpha^{2}][\alpha^{3}][\alpha^{4}][\alpha^{5}][\alpha^{6}][\alpha^{7}]
[ n]11111111
Order 18482868
[\Gamma_{1}] 1 1 1 1 1 1 1 1
[\Gamma_{2}] 1 [\omega] i [\omega^{3}] [-1] [\omega^{5}] [-i] [\omega^{7}]
[\Gamma_{3}] 1 i [-1] [-i] [1] i [-1] [-i]
[\Gamma_{4}] 1 [\omega^{3}] [-i] [\omega] [-1] [\omega^{7}] i [\omega^{5}]
[\Gamma_{5}] 1 [-1] [1] [-1] [1] [-1] [1] [-1]
[\Gamma_{6}] 1 [\omega^{5}] i [\omega^{7}] [-1] [\omega] [-i] [\omega^{3}]
[\Gamma_{7}] 1 [-i] [-1] i [1] [-i] [-1] i
[\Gamma_{8}] 1 [\omega^{7}] [-i] [\omega^{5}] [-1] [\omega^{3}] i [\omega]

 Generators Vector representationPerpendicular representation
8 [\alpha = C_{8z}] [\Gamma_{1}\oplus\Gamma_{2}\oplus\Gamma_{8}] [\Gamma_{4}\oplus\Gamma_{6}]
[\bar{8}] [\alpha =S_{8z}] [\Gamma_4\oplus\Gamma_5\oplus\Gamma_6] [\Gamma_2\oplus\Gamma_8]
[8/m] [\sim 8\times {\bb Z}_2] [\Gamma_{1u}\oplus\Gamma_{2u}\oplus\Gamma_{8u}] [\Gamma_{4u}\oplus\Gamma_{6u}]

(d) [D_8]

[D_{8}][\varepsilon][\alpha] [\alpha^{2}][\alpha^{3}][\alpha^{4}][\beta][\alpha \beta]
[ n]1222144
Order 1 8 4 8 2 2 2
[\Gamma_{1}] 1 1 1 1 1 1 1
[\Gamma_{2}] 1 1 1 1 1 [-1] [-1]
[\Gamma_{3}] 1 [-1] 1 [-1 ] 1 1 [-1]
[\Gamma_{4}] 1 [-1 ] 1 [-1] 1 [-1] 1
[\Gamma_{5}] 2 [\sqrt{2}] 0 [-\sqrt{2}] [-2] 0 0
[\Gamma_{6}] 2 0 [-2 ] 0 2 0 0
[\Gamma_{7}] 2 [-\sqrt{2}] 0 [\sqrt{2}] [-2] 0 0

 GeneratorsVector representationPerpendicular representation
822 [\alpha = C_{8z}] [\Gamma_{2}\oplus\Gamma_{5}] [\Gamma_{7}]
  [\beta = C_{2x}]    
[8mm] [\alpha = C_{8z}] [\Gamma_{1}\oplus\Gamma_{5}] [\Gamma_{7}]
  [\beta = m_{x}]    
[\bar{8}2m] [\alpha = S_{8z}] [\Gamma_{3}\oplus\Gamma_{7}] [\Gamma_{5}]
  [\beta = C_{2x}]    
[8/mmm] [\sim 822\times{\bb Z}_{2}] [\Gamma_{2u}\oplus\Gamma_{5u}] [\Gamma_{7u}]

(e) [C_{10}] [[\omega=exp(2\pi i/5) ]].

[C_{10}][ \varepsilon][\alpha^2][\alpha^4][\alpha^6][\alpha^8]
n11111
Order 15555
[\Gamma_1] 1 1 1 1 1
[\Gamma_2] 1 [\omega] [\omega^2] [\omega^3] [\omega^4]
[\Gamma_3] 1 [\omega^2] [\omega^4] [\omega] [\omega^3]
[\Gamma_4] 1 [\omega^3] [\omega] [\omega^4] [\omega^2]
[\Gamma_5] 1 [\omega^4] [\omega^3] [\omega^2] [\omega]
[\Gamma_6] 1 1 1 1 1
[\Gamma_7] 1 [\omega] [\omega^2] [\omega^3] [\omega^4]
[\Gamma_8] 1 [\omega^2] [\omega^4] [\omega] [\omega^3]
[\Gamma_9] 1 [\omega^3] [\omega] [\omega^4] [\omega^2]
[\Gamma_{10}] 1 [\omega^4] [\omega^3] [\omega^2] [\omega]

[C_{10}][\alpha^5][\alpha^7][\alpha^9][\alpha][\alpha^3]
n11111
Order 210101010
[\Gamma_1] 1 1 1 1 1
[\Gamma_2] 1 [\omega] [\omega^2] [\omega^3] [\omega^4]
[\Gamma_3] 1 [\omega^2] [\omega^4] [\omega] [\omega^3]
[\Gamma_4] 1 [\omega^3] [\omega] [\omega^4] [\omega^2]
[\Gamma_5] 1 [\omega^4] [\omega^3] [\omega^2] [\omega]
[\Gamma_6] [-1] [-1] [-1] [-1] [-1]
[\Gamma_7] [-1] [-\omega] [-\omega^2] [-\omega^3] [-\omega^4]
[\Gamma_8] [-1] [-\omega^2] [-\omega^4] [-\omega] [-\omega^3]
[\Gamma_9] [-1] [-\omega^3] [-\omega] [-\omega^4] [-\omega^2]
[\Gamma_{10}] [-1] [-\omega^4] [-\omega^3] [-\omega^2] [-\omega]

 Generators Vector representationPerpendicular representation
10 [\alpha = C_{10z}] [\Gamma_{1}\oplus\Gamma_{7}\oplus\Gamma_{10}] [\Gamma_{8}\oplus\Gamma_{9}]
[{\bar{5}}] [\alpha = S_{5z}] [\Gamma_6\oplus\Gamma_8\oplus\Gamma_9] [\Gamma_7\oplus\Gamma_{10}]
[\overline{10}] [\alpha =S_{10z}] [\Gamma_2\oplus\Gamma_4\oplus\Gamma_6] [\Gamma_3\oplus\Gamma_5]
[10/m ] [\sim 10\times {\bb Z}_2] [\Gamma_{1u}\oplus\Gamma_{7u}\oplus\Gamma_{10u}] [\Gamma_{8u}\oplus\Gamma_{9u}]

(f) [D_{10}] [[\tau = (\sqrt{5}-1)/2]].

[D_{10}] [\varepsilon] [\alpha] [\alpha^{2}] [\alpha^{3}]
n1222
Order 1 10 5 10
[\Gamma_{1}] 1 1 1 1
[\Gamma_{2}] 1 1 1 1
[\Gamma_{3}] 1 [-1 ] 1 [-1]
[\Gamma_{4}] 1 [-1] 1 [-1]
[\Gamma_{5}] 2 [1+\tau] [\tau] [-\tau]
[\Gamma_{6}] 2 [\tau] [-1-\tau] [-1-\tau]
[\Gamma_{7}] 2 [-\tau] [-1-\tau] [1+\tau]
[\Gamma_{8}] 2 [-1-\tau] [\tau] [\tau]

[D_{10}][\alpha^{4}] [\alpha^{5}] [\beta] [\alpha \beta]
n2155
Order 5 2 2 2
[\Gamma_{1}] 1 1 1 1
[\Gamma_{2}] 1 1 [-1 ] [-1]
[\Gamma_{3}] 1 [-1] 1 [-1]
[\Gamma_{4}] 1 [-1 ] [-1] 1
[\Gamma_{5}] [-1-\tau] [-2 ] 0 0
[\Gamma_{6}] [\tau] 2 0 0
[\Gamma_{7}] [\tau ] [-2 ] 0 0
[\Gamma_{8}] [-1-\tau] 2 0 0

 GeneratorsVector representationPerpendicular representation
[10\,22] [\alpha = C_{10z}] [\Gamma_{2}\oplus\Gamma_{5}] [\Gamma_7]
  [\beta = C_{2x}]    
[10\,mm] [\alpha = C_{10z}] [\Gamma_{1}\oplus\Gamma_{5}] [\Gamma_7]
  [\beta = m_{x}]    
[\overline{10}\,2m] [\alpha = S_{10z}] [\Gamma_{4}\oplus\Gamma_{8}] [\Gamma_6]
  [\beta = C_{2x}]    
[\bar{5}m] [\alpha = S_{5z}] [\Gamma_{3}\oplus\Gamma_{7}] [\Gamma_5]
  [\beta = C_{2x}]    
[10/mmm] [\sim 10\,22\times{\bb Z}_{2}] [\Gamma_{2u}\oplus\Gamma_{5u}] [\Gamma_{7u}]

(g) [C_{12}] [[\omega = \exp (\pi i/6)]].

[C_{12}][\varepsilon] [\alpha][\alpha^{2}][\alpha^{3}][\alpha^4][\alpha^{5}]
n111111
Order11264312
[\Gamma_{1}] 1 1 1 1 1 1
[\Gamma_{2}] 1 [\omega] [\omega^2] i [\omega^4] [\omega^5]
[\Gamma_{3}] 1 [\omega^2] [\omega^4] [-1] [-\omega^2] [-\omega^4]
[\Gamma_4] 1 i [-1] [-i] 1 i
[\Gamma_{5}] 1 [\omega^4] [-\omega^2] [1] [\omega^4] [-\omega^2]
[\Gamma_{6}] 1 [\omega^5] [-\omega^4] i [-\omega^2] [\omega]
[\Gamma_{7}] 1 [-1] 1 [-1] 1 [-1]
[\Gamma_{8}] 1 [-\omega] [\omega^2] [-i] [\omega^4] [-\omega^5]
[\Gamma_{9}] 1 [-\omega^2] [\omega^4] [1] [-\omega^2] [\omega^4]
[\Gamma_{10}] 1 [-i] [-1] i 1 [-i]
[\Gamma_{11}] 1 [-\omega^4] [-\omega^2] [-1] [\omega^4] [\omega^2]
[\Gamma_{12}] 1 [-\omega^5] [-\omega^4] [-i] [-\omega^2] [-\omega]

[C_{12}][\alpha^{6}][\alpha^{7}][\alpha^{8}][\alpha^{9}][\alpha^{10}][\alpha^{11}]
n111111
Order21234612
[\Gamma_{1}] 1 1 1 1 1 1
[\Gamma_{2}] [-1] [-\omega] [-\omega^2] [-i] [-\omega^4] [-\omega^5]
[\Gamma_{3}] 1 [\omega^2] [\omega^4] [-1] [-\omega^2] [-\omega^4]
[\Gamma_4] [-1] [-i] 1 i [-1] [-i]
[\Gamma_{5}] 1 [\omega^4] [-\omega^2] [1] [\omega^4] [-\omega^2]
[\Gamma_{6}] [-1] [-\omega^5] [\omega^4] [-i] [\omega^2] [-\omega]
[\Gamma_{7}] 1 [-1] 1 [-1] 1 [-1]
[\Gamma_{8}] [-1] [\omega] [-\omega^2] i [-\omega^4] [\omega^5]
[\Gamma_{9}] 1 [-\omega^2] [\omega^4] [1] [-\omega^2] [\omega^4]
[\Gamma_{10}] [-1] i 1 [-i] [-1] i
[\Gamma_{11}] 1 [-\omega^4] [-\omega^2] [-1] [\omega^4] [\omega^2]
[\Gamma_{12}] [-1] [\omega^5] [\omega^4] i [\omega^2] [\omega]

 Generators Vector representationPerpendicular representation
12 [\alpha=C_{12z}] [\Gamma_{1}\oplus\Gamma_{2}\oplus\Gamma_{12}] [\Gamma_{6}\oplus\Gamma_{8}]
[\overline{12}] [\alpha=S_{12z}] [\Gamma_6\oplus\Gamma_7\oplus\Gamma_8] [\Gamma_2\oplus\Gamma_{12}]
[12/m ] [\sim 12\times {\bb Z}_2] [\Gamma_{1u}\oplus\Gamma_{2u}\oplus\Gamma_{12u}] [\Gamma_{6u}\oplus\Gamma_{8u}]

(h) [D_{12}]

[D_{12}][\varepsilon] [\alpha] [\alpha^{2}] [\alpha^{3}]
n1222
Order 1 12 6 4
[\Gamma_{1}] 1 1 1 1
[\Gamma_{2}] 1 1 1 1
[\Gamma_{3}] 1 [-1] 1 [-1 ]
[\Gamma_{4}] 1 [-1] 1 [-1]
[\Gamma_{5}] 2 [\sqrt{3}] 1 0
[\Gamma_{6}] 2 1 [-1 ] [-2]
[\Gamma_{7}] 2 0 [-2] 0
[\Gamma_{8}] 2 [-1] [-1 ] 2
[\Gamma_{9}] 2 [-\sqrt{3}] 1 0

[D_{12}][\alpha^{4}] [\alpha^{5}] [\alpha^{6}] [\beta] [\alpha \beta]
n22166
Order 3 12 2 2 2
[\Gamma_{1}] 1 1 1 1 1
[\Gamma_{2}] 1 1 1 [-1] [-1]
[\Gamma_{3}] 1 [-1] 1 1 [-1]
[\Gamma_{4}] 1 [-1] 1 [-1] 1
[\Gamma_{5}] [-1] [-\sqrt{3}] [-2 ] 0 0
[\Gamma_{6}] [-1] 1 2 0 0
[\Gamma_{7}] 2 0 [-2 ] 0 0
[\Gamma_{8}] [-1 ] [-1 ] 2 0 0
[\Gamma_{9}] [-1 ] [\sqrt{3}] [-2] 0 0

 GeneratorsVector representationPerpendicular representation
[12 \,22] [\alpha = C_{12z}] [\Gamma_{2}\oplus\Gamma_{5}] [\Gamma_9]
  [\beta = C_{2x}]    
[12\,mm] [\alpha = C_{12z}] [\Gamma_{1}\oplus\Gamma_{5}] [\Gamma_{9}]
  [\beta = m_{x}]    
[\overline{12}\,2m] [\alpha = S_{12z}] [\Gamma_{4}\oplus\Gamma_{9}] [\Gamma_5]
  [\beta = C_{2x}]    
[12/mmm] [\sim 12 \,22\times{\bb Z}_{2}] [\Gamma_{2u}\oplus\Gamma_{5u}] [\Gamma_{9u}]

(i) I [[\tau = (\sqrt 5 -1)/2]].

I[\varepsilon] [\alpha] [\alpha^{2}] [\beta] [\alpha \beta]
n112122015
Order 1 5 5 3 2
[\Gamma_{1}] 1 1 1 1 1
[\Gamma_{2}] 3 [1+\tau] [-\tau] 0 [-1]
[\Gamma_{3}] 3 [-\tau] [1+\tau] 0 [-1]
[\Gamma_{4}] 4 [-1] [-1] 1 0
[\Gamma_{5}] 5 0 0 [-1] 1

 GeneratorsVector representationPerpendicular representation
532 [\alpha =C_{5}] [\Gamma_2] [\Gamma_3]
  [\beta =C_{3d}]    
[\bar{5}\bar{3}m] [\sim 532 \times{\bb Z}_{2}] [\Gamma_{2u}] [\Gamma_{3u}]

In Table 1.10.5.2[link] the representation matrices for the irreducible representations in more than one dimension are given (one-dimensional representations are just the characters). For the cyclic groups there are only one-dimensional representations, for the dihedral groups there are one- and two-dimensional irreducible representations. There are four irreducible representations of I of dimension larger than one. The four- and five-dimensional ones are given as integer representations. They form crystallographic groups in 4D and 5D. The two three-dimensional representations have the same matrices. The elements, however, are connected by an outer automorphism. That means that the ith element [R_i] is represented by [\Gamma_2 (R_i)] in the representation [\Gamma_2], and by [\Gamma_3 (R_i)=\Gamma_2 (\varphi R_i)] in [\Gamma_3]. The element [\varphi R_i] is another element [R_j]. The corresponding j for each i is given in Table 1.10.5.3[link].

Table 1.10.5.2| top | pdf |
Matrices of the irreducible representations of dimension [d \geq 2] corresponding to the irreps of Table 1.10.5.1[link]

(a) [{D}_{5}]

Representation [D(\alpha^{p})][D(\beta)]
[\Gamma_{3}] [\pmatrix{\cos (2\pi p/5)& -\sin (2\pi p/5)\cr \sin (2\pi p/5)& \cos (2\pi p/5)\cr}] [\pmatrix{0 & 1 \cr 1 & 0 \cr}]
[\Gamma_{4}] [\pmatrix{\cos (4\pi p/5)& -\sin (4\pi p/5)\cr \sin (4\pi p/5)& \cos (4\pi p/5)\cr}] [\pmatrix{ 0 & 1 \cr 1 & 0\cr}]

(b) [{D}_{8}]

Representation [D(\alpha^{p})][D(\beta)]
[\Gamma_{5}] [\pmatrix{\cos (\pi p/4)& -\sin (\pi p/4)\cr \sin (\pi p/4)& \cos (\pi p/4)\cr}] [\pmatrix{0 & 1 \cr 1 & 0 \cr}]
[\Gamma_{6}] [\pmatrix{\cos (\pi p/2)& -\sin (\pi p/2)\cr \sin (\pi p/2)& \cos (\pi p/2) \cr}] [\pmatrix{0 & 1 \cr 1 & 0\cr}]
[\Gamma_{7}] [\pmatrix{\cos (3\pi p/4)& -\sin (3\pi p/4)\cr \sin (3\pi p/4)& \cos (3\pi p/4)\cr}] [\pmatrix{ 0 & 1 \cr1 & 0 \cr}]

(c) [{D}_{10}]

Representation [D(\alpha^{p})][D(\beta)]
[\Gamma_{5}] [\pmatrix{\cos (\pi p/5)& -\sin (\pi p/5)\cr \sin (\pi p/5)& \cos (\pi p/5)\cr}] [\pmatrix{0 & 1 \cr 1 & 0\cr}]
[\Gamma_{6}] [\pmatrix{\cos (2\pi p/5)& -\sin (2\pi p/5)\cr \sin (2\pi p/5)& \cos (2\pi p/5)\cr}] [\pmatrix{0 & 1 \cr 1 & 0\cr}]
[\Gamma_{7}] [\pmatrix{\cos (3\pi p/5)& -\sin (3\pi p/5)\cr \sin (3\pi p/5)& \cos (3\pi p/5)\cr}] [\pmatrix{0 & 1 \cr 1 & 0 \cr}]
[\Gamma_{8}] [\pmatrix{\cos (4\pi p/5)& -\sin (4\pi p/5)\cr \sin (4\pi p/5)& \cos (4\pi p/5)\cr}] [\pmatrix{0 & 1 \cr 1 & 0 \cr}]

(d) [{D}_{12}]

Representation [D(\alpha^{p})][D(\beta)]
[\Gamma_{5}] [\pmatrix{\cos (\pi p/6)& -\sin (\pi p/6)\cr \sin (\pi p/6)& \cos (\pi p/6)\cr} ] [\pmatrix{0 & 1 \cr 1 & 0\cr}]
[\Gamma_{6}] [\pmatrix{\cos (\pi p/3)& -\sin (\pi p/3)\cr \sin (\pi p/3)& \cos (\pi p/3)\cr} ] [\pmatrix{0 & 1 \cr 1 & 0\cr}]
[\Gamma_{7}] [\pmatrix{\cos (\pi p/2)& -\sin (\pi p/2)\cr \sin (\pi p/2)& \cos (\pi p/2)\cr} ] [\pmatrix{0 & 1 \cr 1 & 0 \cr}]
[\Gamma_{8}] [\pmatrix{\cos (2\pi p/3)& -\sin (2\pi p/3)\cr \sin (2\pi p/3)& \cos (2\pi p/3)\cr}] [\pmatrix{0 & 1 \cr 1 & 0\cr}]
[\Gamma_{9}] [\pmatrix{\cos (5\pi p/6)& -\sin (5\pi p/6)\cr \sin (5\pi p/6)& \cos (5\pi p/6)\cr}] [\pmatrix{0 & 1 \cr 1 & 0 \cr}]

(e) I. First column: numbering of the elements. [f=(1+\sqrt{5})/2, t=(\sqrt{5}-1)/2]. Horizontal rules separate conjugation classes.

No. Order[\Gamma_{2}][\Gamma_{4}][\Gamma_{5}]
1 1 [\pmatrix{1&0&0 \cr 0&1&0\cr 0&0&1 \cr}] [\pmatrix{1&0&0&0\cr 0&1&0&0 \cr 0&0&1&0 \cr 0&0&0&1 \cr}] [\pmatrix{1&0&0&0&0\cr 0&1&0&0&0\cr 0&0&1&0&0 \cr 0&0&0&1&0\cr 0&0&0&0&1 \cr}]
2 5 [\pmatrix{1/2&t/2&-f/2\cr t/2&f/2&1/2\cr f/2&-1/2&t/2 \cr}] [\pmatrix{0&0&0&-1\cr1&0&0&-1\cr 0&1&0&-1\cr 0&0&1&-1\cr}] [\pmatrix{1&0&0&0&-1\cr0&0&0&0&-1\cr0&1&0&0&-1\cr0&0&1&0&-1\cr 0&0&0&1&-1\cr}]
3 5 [\pmatrix{1/2&-t/2&f/2\cr-t/2&f/2&1/2\cr\-f/2&-1/2&t/2\cr}] [\pmatrix{0&0&1&-1\cr 1&0&0&-1\cr0&0&0&-1\cr0&1&0&-1\cr}] [\pmatrix{0&-1&1&0&0\cr0&-1&0&0&1\cr0&-1&0&0&0\cr 0&-1&0&1&0\cr 1&-1&0&0&0\cr}]
4 5 [\pmatrix{1/2&t/2&f/2\cr t/2&f/2&-1/2\cr -f/2&1/2&t/2\cr}] [\pmatrix{-1&1&0&0\cr -1&0&1&0\cr -1&0&0&1\cr -1&0&0&0\cr}] [\pmatrix{1&-1&0&0&0\cr 0&-1&1&0&0\cr 0&-1&0&1&0\cr 0&-1&0&0&1\cr 0&-1&0&0&0 \cr}]
5 5 [\pmatrix{t/2&-f/2&1/2\cr f/2&1/2&t/2\cr -1/2&t/2&f/2\cr}] [\pmatrix{0&-1&0&0\cr 0&-1&0&1\cr 1&-1&0&0\cr 0&-1&1&0\cr}] [\pmatrix{0&1&0&-1&0\cr 0&0&0&-1&1\cr 0&0&1&-1&0\cr 1&0&0&-1&0\cr 0&0&0&-1&0\cr}]
6 5 [\pmatrix{ f/2&-1/2&-t/2\cr 1/2&t/2&f/2\cr -t/2&-f/2&1/2\cr}] [\pmatrix{0&1&-1&0\cr 0&0&-1&0\cr 0&0&-1&1\cr 1&0&-1&0\cr} ] [\pmatrix{0&1&0&0&0\cr 0&0&0&1&0\cr 0&0&0&0&1\cr 0&0&1&0&0\cr 1&0&0&0&0\cr}]
7 5 [\pmatrix{ f/2&1/2&t/2\cr -1/2&t/2&f/2\cr t/2&-f/2&1/2\cr}] [\pmatrix{0&-1&0&1\cr 0&-1&1&0\cr 1&-1&0&0\cr 0&-1&0&0\cr}] [\pmatrix{-1&0&1&0&0\cr -1&1&0&0&0\cr -1&0&0&0&1\cr -1&0&0&0&0\cr -1&0&0&1&0\cr}]
8 5 [\pmatrix{ t/2&f/2&-1/2\cr -f/2&1/2&t/2\cr 1/2&t/2&f/2\cr}] [\pmatrix{-1&0&1&0\cr -1&0&0&0\cr -1&0&0&1\cr -1&1&0&0 \cr}] [\pmatrix{0&0&0&1&-1\cr 1&0&0&0&-1\cr 0&0&1&0&-1\cr 0&0&0&0&-1\cr 0&1&0&0&-1\cr}]
9 5 [\pmatrix{ t/2&f/2&1/2\cr -f/2&1/2&-t/2\cr -1/2&-t/2&f/2\cr}] [\pmatrix{0&0&-1&0\cr 0&0&-1&1\cr 0&1&-1&0\cr 1&0&-1&0\cr}] [\pmatrix{-1&0&0&1&0\cr -1&0&1&0&0\cr -1&0&0&0&0\cr -1&1&0&0&0\cr -1&0&0&0&1\cr}]
10 5 [\pmatrix{ f/2&1/2&-t/2\cr -1/2&t/2&-f/2\cr -t/2&f/2&1/2\cr}] [\pmatrix{0&-1&0&1\cr 1&-1&0&0\cr 0&-1&0&0\cr 0&-1&1&0\cr}] [\pmatrix{0&0&0&0&1\cr 1&0&0&0&0\cr 0&0&0&1&0\cr 0&1&0&0&0\cr 0&0&1&0&0\cr} ]
11 5 [\pmatrix{ 1/2&-t/2&-f/2\cr -t/2&f/2&-1/2\cr f/2&1/2&t/2\cr}] [\pmatrix{0&1&-1&0\cr 0&0&-1&1\cr 1&0&-1&0\cr 0&0&-1&0\cr}] [\pmatrix{0&0&-1&0&1\cr 0&0&-1&0&0\cr 1&0&-1&0&0\cr 0&0&-1&1&0\cr 0&1&-1&0&0\cr}]
12 5 [\pmatrix{ f/2&-1/2&t/1\cr 1/2&t/2&-f/2\cr t/2&f/2&1/2\cr}] [\pmatrix{0&0&1&-1\cr 0&0&0&-1\cr 0&1&0&-1\cr 1&0&0&-1\cr}] [\pmatrix{0&0&0&-1&0\cr 0&1&0&-1&0\cr 1&0&0&-1&0\cr 0&0&0&-1&1\cr 0&0&1&-1&0\cr}]
13 5 [\pmatrix{ t/2&-f/2&-1/2\cr f/2&1/2&-t/2\cr 1/2&-t/2&f/2\cr} ] [\pmatrix{-1&0&0&1\cr -1&0&1&0\cr -1&0&0&0\cr -1&1&0&0\cr}] [\pmatrix{0&0&-1&0&0\cr 0&0&-1&1&0\cr 0&1&-1&0&0\cr 1&0&-1&0&0\cr 0&0&-1&0&1\cr}]
14 5 [\pmatrix{ -t/2&f/2&-1/2\cr f/2&1/2&t/2\cr 1/2&-t/2&-f/2\cr} ] [\pmatrix{0&0&-1&1\cr 0&0&-1&0\cr 1&0&-1&0\cr 0&1&-1&0\cr}] [\pmatrix{1&0&0&-1&0\cr 0&0&0&-1&1\cr 0&0&0&-1&0\cr 0&1&0&-1&0\cr 0&0&1&-1&0\cr}]
15 5 [\pmatrix{ -t/2&f/2&1/2\cr f/2&1/2&-t/2\cr -1/2&t/2&-f/2\cr}] [\pmatrix{0&-1&1&0\cr 0&-1&0&1\cr 0&-1&0&0\cr 1&-1&0&0\cr}] [\pmatrix{1&0&-1&0&0\cr 0&0&-1&1&0\cr 0&0&-1&0&1\cr 0&0&-1&0&0\cr 0&1&-1&0&0\cr}]
16 5 [\pmatrix{ -f/2&1/2&-t/2\cr -1/2&-t/2&f/2\cr t/2&f/2&1/2\cr}] [\pmatrix{0&0&-1&1\cr 1&0&-1&0\cr 0&1&-1&0\cr 0&0&-1&0\cr} ] [\pmatrix{0&-1&0&0&0\cr 0&-1&0&1&0\cr 0&-1&1&0&0\cr 0&-1&0&0&1\cr 1&-1&0&0&0\cr} ]
17 5 [\pmatrix{ -t/2&-f/2&1/2\cr -f/2&1/2&t/2\cr -1/2&-t/2&-f/2\cr}] [\pmatrix{ 0 & -1 & 0 & 0\cr 0 & -1 & 1 & 0\cr 0 & -1 & 0 & 1\cr 1 & -1 & 0 & 0\cr} ] [\pmatrix{0&0&0&0&-1\cr 1&0&0&0&-1\cr 0&1&0&0&-1\cr 0&0&0&1&-1\cr 0&0&1&0&-1\cr}]
18 5 [\pmatrix{ -t/2&-f/2&-1/2\cr -f/2&1/2&-t/2\cr 1/2&t/2&-f/2\cr} ] [\pmatrix{-1&0&0&1\cr -1&0&0&0\cr -1&1&0&0\cr -1&0&1&0\cr}] [\pmatrix{-1&1&0&0&0\cr -1&0&1&0&0\cr -1&0&0&0&1\cr -1&0&0&1&0\cr -1&0&0&0&0\cr}]
19 5 [\pmatrix{ -f/2&-1/2&t/2\cr 1/2&-t/2&f/2\cr -t/2&f/2&1/2\cr}] [\pmatrix{0&1&0&-1\cr 0&0&1&-1\cr 0&0&0&-1\cr 1&0&0&-1\cr}] [\pmatrix{-1&0&0&0&1\cr -1&0&0&0&0\cr -1&0&1&0&0\cr -1&1&0&0&0\cr -1&0&0&1&0\cr}]
20 5 [\pmatrix{ -f/2&1/2&t/2\cr -1/2&-t/2&-f/2\cr -t/2&-f/2&1/2\cr}] [\pmatrix{0&-1&1&0\cr 1&-1&0&0\cr 0&-1&0&1\cr 0&-1&0&0\cr}] [\pmatrix{0&1&0&-1&0\cr 0&0&0&-1&0\cr 1&0&0&-1&0\cr 0&0&1&-1&0\cr 0&0&0&-1&1\cr}]
21 5 [\pmatrix{ 1/2&-t/2&f/2\cr t/2&-f/2&-1/2\cr f/2&1/2&-t/2\cr}] [\pmatrix{-1&1&0&0\cr -1&0&0&1\cr -1&0&0&0\cr -1&0&1&0\cr}] [\pmatrix{0&0&0&1&-1\cr 0&1&0&0&-1\cr 0&0&0&0&-1\cr 0&0&1&0&-1\cr 1&0&0&0&-1\cr}]
22 5 [\pmatrix{ 1/2&-t/2&-f/2\cr t/2&-f/2&1/2\cr -f/2&-1/2&-t/2\cr} ] [\pmatrix{0&0&0&-1\cr 0&0&1&-1\cr 1&0&0&-1\cr 0&1&0&-1\cr}] [\pmatrix{0&0&0&1&0\cr 0&0&1&0&0\cr 1&0&0&0&0\cr 0&0&0&0&1\cr 0&1&0&0&0\cr}]
23 5 [\pmatrix{ 1/2&t/2&f/2\cr -t/2&-f/2&1/2\cr f/2&-1/2&-t/2\cr} ] [\pmatrix{0&0&-1&0\cr 1&0&-1&0\cr 0&0&-1&1\cr 0&1&-1&0\cr}] [\pmatrix{0&0&-1&0&1\cr 0&1&-1&0&0\cr 0&0&-1&1&0\cr 1&0&-1&0&0\cr 0&0&-1&0&0\cr} ]
24 5 [\pmatrix{ 1/2&t/2&-f/2\cr -t/2&-f/2&-1/2\cr -f/2&1/2&-t/2\cr}] [\pmatrix{-1&0&1&0\cr -1&0&0&1\cr -1&1&0&0\cr -1&0&0&0\cr}] [\pmatrix{0&0&1&0&0\cr 0&0&0&0&1\cr 0&1&0&0&0\cr 1&0&0&0&0\cr 0&0&0&1&0\cr}]
25 5 [\pmatrix{ -f/2&-1/2&-t/2\cr 1/2&-t/2&-f/2\cr t/2&-f/2&1/2\cr}] [\pmatrix{0&1&0&-1\cr 0&0&0&-1\cr 1&0&0&-1\cr 0&0&1&-1\cr}] [\pmatrix{0&-1&1&0&0\cr 1&-1&0&0&0\cr 0&-1&0&1&0\cr 0&-1&0&0&0\cr 0&-1&0&0&1\cr}]
26 3 [\pmatrix{ -1/2&t/2&-f/2\cr -t/2&f/2&1/2\cr f/2&1/2&-t/2\cr}] [\pmatrix{1&-1&0&0\cr 0&-1&1&0\cr 0&-1&0&0\cr 0&-1&0&1\cr}] [\pmatrix{0&-1&0&1&0\cr 0&-1&0&0&1\cr 1&-1&0&0&0\cr 0&-1&1&0&0\cr 0&-1&0&0&0\cr} ]
27 3 [\pmatrix{ -1/2&-t/2&f/2\cr t/2&f/2&1/2\cr -f/2&1/2&-t/2\cr}] [\pmatrix{1&0&-1&0\cr 0&0&-1&0\cr 0&1&-1&0\cr 0&0&-1&1\cr} ] [\pmatrix{0&0&1&0&-1\cr 0&0&0&0&-1\cr 0&0&0&1&-1\cr 1&0&0&0&-1\cr 0&1&0&0&-1\cr}]
28 3 [\pmatrix{ -1/2&t/2&f/1\cr -t/2&f/2&-1/2\cr -f/2&-1/2&-t/2\cr} ] [\pmatrix{0&0&0&1\cr 0&1&0&0\cr 1&0&0&0\cr 0&0&1&0\cr}] [\pmatrix{0&0&-1&1&0\cr 0&0&-1&0&0\cr 0&1&-1&0&0\cr 0&0&-1&0&1\cr 1&0&-1&0&0\cr}]
29 3 [\pmatrix{ 0&0&1\cr 1&0&0\cr 0&1&0\cr}] [\pmatrix{0&0&0&1\cr 1&0&0&0\cr 0&0&1&0\cr 0&1&0&0\cr}] [\pmatrix{0&1&0&0&-1\cr 0&0&1&0&-1\cr 1&0&0&0&-1\cr 0&0&0&0&-1\cr 0&0&0&1&-1\cr}]
30 3 [\pmatrix{ -1/2&-t/2&-f/2\cr t/2&f/2&-1/2\cr f/2&-1/2&-t/2\cr} ] [\pmatrix{0&0&1&0\cr 0&1&0&0\cr 0&0&0&1\cr 1&0&0&0\cr}] [\pmatrix{0&-1&0&0&1\cr 0&-1&1&0&0\cr 0&-1&0&0&0\cr 1&-1&0&0&0\cr 0&-1&0&1&0\cr}]
31 3 [\pmatrix{ 0&0&-1\cr 1&0&0\cr 0&-1&0\cr} ] [\pmatrix{1&-1&0&0\cr 0&-1&0&1\cr 0&-1&1&0\cr 0&-1&0&0\cr} ] [\pmatrix{0&0&0&0&-1\cr 0&0&1&0&-1\cr 0&0&0&1&-1\cr 0&1&0&0&-1\cr 1&0&0&0&-1\cr}]
32 3 [\pmatrix{ f/2&1/2&-t/2\cr 1/2&-t/2&f/2\cr t/2&-f/2&-1/2\cr}] [\pmatrix{-1&0&1&0\cr -1&1&0&0\cr -1&0&0&0\cr -1&0&0&1\cr}] [\pmatrix{-1&1&0&0&0\cr -1&0&0&0&0\cr -1&0&0&1&0\cr -1&0&0&0&1\cr -1&0&1&0&0\cr}]
33 3 [\pmatrix{ 0&1&0\cr 0&0&1\cr 1&0&0\cr}] [\pmatrix{0&1&0&0\cr 0&0&0&1\cr 0&0&1&0\cr 1&0&0&0\cr}] [\pmatrix{0&0&1&-1&0\cr 1&0&0&-1&0\cr 0&1&0&-1&0\cr 0&0&0&-1&1\cr 0&0&0&-1&0\cr}]
34 3 [\pmatrix{ 0&0&-1\cr -1&0&0\cr 0&1&0\cr} ] [\pmatrix{0&0&0&-1\cr 0&1&0&-1\cr 0&0&1&-1\cr 1&0&0&-1\cr}] [\pmatrix{0&1&-1&0&0\cr 0&0&-1&0&1\cr 0&0&-1&1&0\cr 0&0&-1&0&0\cr 1&0&-1&0&0\cr} ]
35 3 [\pmatrix{ 0&-1&0\cr 0&0&1\cr -1&0&0\cr}] [\pmatrix{-1&0&0&1\cr -1&1&0&0\cr -1&0&1&0\cr -1&0&0&0\cr} ] [\pmatrix{0&0&0&-1&1\cr 1&0&0&-1&0\cr 0&0&0&-1&0\cr 0&0&1&-1&0\cr 0&1&0&-1&0\cr}]
36 3 [\pmatrix{ f/2&-1/2&t/2\cr -1/2&-t/2&f/2\cr -t/2&-f/2&-1/2\cr}] [\pmatrix{1&0&0&0\cr 0&0&0&1\cr 0&1&0&0\cr 0&0&1&0\cr}] [\pmatrix{0&-1&0&0&1\cr 0&-1&0&1&0\cr 1&-1&0&0&0\cr 0&-1&0&0&0\cr 0&-1&1&0&0\cr}]
37 3 [\pmatrix{ 0&0&1\cr -1&0&0\cr 0&-1&0\cr}] [\pmatrix{-1&1&0&0\cr -1&0&0&0\cr -1&0&1&0\cr -1&0&0&1\cr}] [\pmatrix{0&0&-1&0&0\cr 0&0&-1&0&1\cr 1&0&-1&0&0\cr 0&1&-1&0&0\cr 0&0&-1&1&0\cr}]
38 3 [\pmatrix{ 0&1&0\cr 0&0&-1\cr -1&0&0\cr} ] [\pmatrix{1&0&0&-1\cr 0&0&0&-1\cr 0&0&1&-1\cr 0&1&0&-1\cr}] [\pmatrix{-1&0&0&0&1\cr -1&0&0&1&0\cr -1&1&0&0&0\cr -1&0&1&0&0\cr -1&0&0&0&0\cr}]
39 3 [\pmatrix{ f/2&1/2&t/2\cr 1/2&-t/2&-f/2\cr -t/2&f/2&-1/2\cr}] [\pmatrix{0&0&-1&0\cr 0&1&-1&0\cr 1&0&-1&0\cr 0&0&-1&1\cr}] [\pmatrix{0&-1&0&0&0\cr 1&-1&0&0&0\cr 0&-1&0&0&1\cr 0&-1&1&0&0\cr 0&-1&0&1&0\cr}]
40 3 [\pmatrix{ -t/2&-f/2&1/2\cr f/2&-1/2&-t/2\cr 1/2&t/2&f/2\cr}] [\pmatrix{1&0&-1&0\cr 0&1&-1&0\cr 0&0&-1&1\cr 0&0&-1&0\cr}] [\pmatrix{-1&0&0&1&0\cr -1&0&0&0&1\cr -1&1&0&0&0\cr -1&0&0&0&0\cr -1&0&1&0&0\cr} ]
41 3 [\pmatrix{ -t/2&-f/2&-1/2\cr f/2&-1/2&t/2\cr -1/2&-t/2&f/2\cr} ] [\pmatrix{0&0&1&0\cr 1&0&0&0\cr 0&1&0&0\cr 0&0&0&1\cr}] [\pmatrix{0&0&-1&1&0\cr 1&0&-1&0&0\cr 0&0&-1&0&1\cr 0&1&-1&0&0\cr 0&0&-1&0&0\cr}]
42 3 [\pmatrix{ -t/2&f/2&1/2\cr -f/2&-1/2&t/2\cr 1/2&-t/2&f/2\cr}] [\pmatrix{1&0&0&-1\cr 0&1&0&-1\cr 0&0&0&-1\cr 0&0&1&-1\cr}] [\pmatrix{0&0&0&-1&0\cr 0&0&1&-1&0\cr 0&0&0&-1&1\cr 1&0&0&-1&0\cr 0&1&0&-1&0\cr}]
43 3 [\pmatrix{ -t/2&f/2&-1/2\cr -f/2&-1/2&-t/2\cr -1/2&t/2&f/2\cr}] [\pmatrix{0&1&0&0\cr 0&0&1&0\cr 1&0&0&0\cr 0&0&0&1\cr}] [\pmatrix{0&1&0&0&-1\cr 0&0&0&1&-1\cr 0&0&0&0&-1\cr 1&0&0&0&-1\cr 0&0&1&0&-1\cr} ]
44 3 [\pmatrix{ f/2&-1/2&-t/2\cr -1/2&-t/2&-f/2\cr t/2&f/2&-1/2\cr}] [\pmatrix{1&0&0&0\cr 0&0&1&0\cr 0&0&0&1\cr 0&1&0&0\cr}] [\pmatrix{0&0&1&-1&0\cr 0&0&0&-1&0\cr 0&0&0&-1&1\cr 0&1&0&-1&0\cr 1&0&0&-1&0\cr}]
45 3 [\pmatrix{ 0&-1&0\cr 0&0&-1\cr 1&0&0\cr}] [\pmatrix{0&-1&0&0\cr 1&-1&0&0\cr 0&-1&1&0\cr 0&-1&0&1\cr}] [\pmatrix{-1&0&1&0&0\cr -1&0&0&1&0\cr -1&0&0&0&0\cr -1&0&0&0&1\cr -1&1&0&0&0\cr}]
46 2 [\pmatrix{ -1&0&0\cr 0&1&0\cr 0&0&-1\cr}] [\pmatrix{0&1&0&-1\cr 1&0&0&-1\cr 0&0&1&-1\cr 0&0&0&-1\cr}] [\pmatrix{0&0&0&1&0\cr 0&1&0&0&0\cr 0&0&0&0&1\cr 1&0&0&0&0\cr 0&0&1&0&0\cr}]
47 2 [\pmatrix{ -f/2&1/2&t/2\cr 1/2&t/2&f/2\cr t/2&f/2&-1/2\cr}] [\pmatrix{-1&0&0&0\cr -1&1&0&0\cr -1&0&0&1\cr -1&0&1&0\cr}] [\pmatrix{0&0&1&0&0\cr 0&0&0&1&0\cr 1&0&0&0&0\cr 0&1&0&0&0\cr 0&0&0&0&1\cr}]
48 2 [\pmatrix{ -f/2&-1/2&-t/2\cr -1/2&t/2&f/2\cr -t/2&f/2&-1/2\cr}] [\pmatrix{0&0&1&0\cr 0&0&0&1\cr 1&0&0&0\cr 0&1&0&0\cr}] [\pmatrix{-1&0&0&0&0\cr -1&1&0&0&0\cr -1&0&0&1&0\cr -1&0&1&0&0\cr -1&0&0&0&1\cr}]
49 2 [\pmatrix{ -f/2&-1/2&t/2\cr -1/2&t/2&-f/2\cr t/2&-f/2&-1/2\cr}] [\pmatrix{1&0&-1&0\cr 0&0&-1&1\cr 0&0&-1&0\cr 0&1&-1&0\cr}] [\pmatrix{0&1&0&0&0\cr 1&0&0&0&0\cr 0&0&1&0&0\cr 0&0&0&0&1\cr 0&0&0&1&0\cr} ]
50 2 [\pmatrix{ -f/2&1/2&-t/2\cr 1/2&t/2&-f/2\cr -t/2&-f/2&-1/2\cr}] [\pmatrix{-1&0&0&0\cr -1&0&1&0\cr -1&1&0&0\cr -1&0&0&1\cr} ] [\pmatrix{0&0&0&-1&1\cr 0&1&0&-1&0\cr 0&0&1&-1&0\cr 0&0&0&-1&0\cr 1&0&0&-1&0\cr}]
51 2 [\pmatrix{ t/2&f/2&-1/2\cr f/2&-1/2&-t/2\cr -1/2&-t/2&-f/2\cr}] [\pmatrix{0&1&0&0\cr 1&0&0&0\cr 0&0&0&1\cr 0&0&1&0\cr} ] [\pmatrix{-1&0&0&0&0\cr -1&0&0&0&1\cr -1&0&1&0&0\cr -1&0&0&1&0\cr -1&1&0&0&0\cr}]
52 2 [\pmatrix{ t/2&f/2&1/2\cr f/2&-1/2&t/2\cr 1/2&t/2&-f/2\cr} ] [\pmatrix{1&0&0&-1\cr 0&0&1&-1\cr 0&1&0&-1\cr 0&0&0&-1\cr}] [\pmatrix{0&1&-1&0&0\cr 1&0&-1&0&0\cr 0&0&-1&0&0\cr 0&0&-1&1&0\cr 0&0&-1&0&1\cr}]
53 2 [\pmatrix{ -1/2&t/2&f/2\cr t/2&-f/2&1/2\cr f/2&1/2&t/2\cr}] [\pmatrix{0&-1&1&0\cr 0&-1&0&0\cr 1&-1&0&0\cr 0&-1&0&1\cr}] [\pmatrix{0&0&0&0&1\cr 0&0&1&0&0\cr 0&1&0&0&0\cr 0&0&0&1&0\cr 1&0&0&0&0\cr}]
54 2 [\pmatrix{ -1/2&t/2&-f/2\cr t/2&-f/2&-1/2\cr -f/2&-1/2&t/2\cr} ] [\pmatrix{0&0&-1&1\cr 0&1&-1&0\cr 0&0&-1&0\cr 1&0&-1&0\cr}] [\pmatrix{0&0&1&0&-1\cr 0&1&0&0&-1\cr 1&0&0&0&-1\cr 0&0&0&1&-1\cr 0&0&0&0&-1\cr} ]
55 2 [\pmatrix{ 1&0&0\cr 0&-1&0\cr 0&0&-1\cr}] [\pmatrix{0&-1&0&1\cr 0&-1&0&0\cr 0&-1&1&0\cr 1&-1&0&0\cr}] [\pmatrix{0&-1&0&1&0\cr 0&-1&0&0&0\cr 0&-1&1&0&0\cr 1&-1&0&0&0\cr 0&-1&0&0&1\cr}]
56 2 [\pmatrix{ -1&0&0\cr 0&-1&0\cr 0&0&1\cr}] [\pmatrix{-1&0&0&0\cr -1&0&0&1\cr -1&0&1&0\cr -1&1&0&0\cr} ] [\pmatrix{1&-1&0&0&0\cr 0&-1&0&0&0\cr 0&-1&0&0&1\cr 0&-1&0&1&0\cr 0&-1&1&0&0\cr}]
57 2 [\pmatrix{ -1/2&-t/2&-f/2\cr -t/2&-f/2&1/2\cr -f/2&1/2&t/2\cr} ] [\pmatrix{1&-1&0&0\cr 0&-1&0&0\cr 0&-1&0&1\cr 0&-1&1&0\cr} ] [\pmatrix{1&0&-1&0&0\cr 0&1&-1&0&0\cr 0&0&-1&0&0\cr 0&0&-1&0&1\cr 0&0&-1&1&0\cr}]
58 2 [\pmatrix{ t/2&-f/2&-1/2\cr -f/2&-1/2&t/2\cr -1/2&t/2&-f/2\cr}] [\pmatrix{0&1&-1&0\cr 1&0&-1&0\cr 0&0&-1&0\cr 0&0&-1&1\cr}] [\pmatrix{1&0&0&-1&0\cr 0&0&1&-1&0\cr 0&1&0&-1&0\cr 0&0&0&-1&0\cr 0&0&0&-1&1\cr}]
59 2 [\pmatrix{ t/2&-f/2&1/2\cr -f/2&-1/2&-t/2\cr 1/2&-t/2&-f/2\cr}] [\pmatrix{0&0&1&-1\cr 0&1&0&-1\cr 1&0&0&-1\cr 0&0&0&-1\cr}] [\pmatrix{1&0&0&0&-1\cr 0&0&0&1&-1\cr 0&0&1&0&-1\cr 0&1&0&0&-1\cr 0&0&0&0&-1\cr} ]
60 2 [\pmatrix{ -1/2&-t/2&f/2\cr -t/2&-f/2&-1/2\cr f/2&-1/2&t/2\cr}] [\pmatrix{0&0&0&1\cr 0&0&1&0\cr 0&1&0&0\cr 1&0&0&0\cr}] [\pmatrix{1&0&0&0&0\cr 0&0&0&0&1\cr 0&0&0&1&0\cr 0&0&1&0&0\cr 0&1&0&0&0\cr} ]

Table 1.10.5.3| top | pdf |
The representation matrices for [\Gamma_3]

The representation matrices for [\Gamma_3] are the same as for [\Gamma_2]. Correspondences are given as pairs i, j: [\Gamma_3(R_{i}) = \Gamma_2(R_{j})].

ijijijijijij
1 1 11 21 21 5 31 42 41 29 51 48
2 14 12 16 22 6 32 45 42 39 52 54
3 23 13 17 23 8 33 36 43 33 53 46
4 15 14 4 24 10 34 27 44 30 54 50
5 25 15 2 25 11 35 26 45 38 55 52
6 24 16 13 26 34 36 28 46 49 56 57
7 19 17 12 27 35 37 31 47 53 57 59
8 20 18 7 28 43 38 40 48 51 58 56
9 18 19 9 29 44 39 37 49 47 59 58
10 22 20 3 30 41 40 32 50 55 60 60








































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