Tables

Janssen, T.

doi:10.1107/97809553602060000637

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

pdf | chapter contents | chapter index | related articles

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

Section 1.10.5. Tables

T. Janssen^a ^*

^a Institute 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 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] $[[\omega =\exp (2\pi i/5)]]$ .

$[C_{5}]$	$[\varepsilon]$	$[\alpha]$	$[\alpha^{2}]$	$[\alpha^{3}]$	$[\alpha^{4}]$
n	1	1	1	1	1
Order	1	5	5	5	5
$[\Gamma_{1}]$	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 representation	Perpendicular representation
5	$[\alpha = C_{5z}]$	$[\Gamma_{1}\oplus\Gamma_{2}\oplus\Gamma_{5}]$	$[\Gamma_{3}\oplus\Gamma_{4}]$

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

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

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

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

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

(d) [D_8]

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

	Generators	Vector representation	Perpendicular representation
822	$[\alpha = C_{8z}]$	$[\Gamma_{2}\oplus\Gamma_{5}]$	$[\Gamma_{7}]$
	$[\beta = C_{2x}]$
	$[\alpha = C_{8z}]$	$[\Gamma_{1}\oplus\Gamma_{5}]$	$[\Gamma_{7}]$
	$[\beta = m_{x}]$
$[\bar{8}2m]$	$[\alpha = S_{8z}]$	$[\Gamma_{3}\oplus\Gamma_{7}]$	$[\Gamma_{5}]$
	$[\beta = C_{2x}]$
	$[\sim 822\times{\bb Z}_{2}]$	$[\Gamma_{2u}\oplus\Gamma_{5u}]$	$[\Gamma_{7u}]$

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

$[C_{10}]$	$[ \varepsilon]$	$[\alpha^2]$	$[\alpha^4]$	$[\alpha^6]$	$[\alpha^8]$
n	1	1	1	1	1
Order	1	5	5	5	5
$[\Gamma_1]$	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	$[\omega^4]$	$[\omega^3]$	$[\omega^2]$	$[\omega]$

$[C_{10}]$	$[\alpha^5]$	$[\alpha^7]$	$[\alpha^9]$	$[\alpha]$	$[\alpha^3]$
n	1	1	1	1	1
Order	2	10	10	10	10
$[\Gamma_1]$	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	$[\omega^4]$	$[\omega^3]$	$[\omega^2]$	$[\omega]$
$[\Gamma_6]$
$[\Gamma_7]$		$[-\omega]$	$[-\omega^2]$	$[-\omega^3]$	$[-\omega^4]$
$[\Gamma_8]$		$[-\omega^2]$	$[-\omega^4]$	$[-\omega]$	$[-\omega^3]$
$[\Gamma_9]$		$[-\omega^3]$	$[-\omega]$	$[-\omega^4]$	$[-\omega^2]$
$[\Gamma_{10}]$		$[-\omega^4]$	$[-\omega^3]$	$[-\omega^2]$	$[-\omega]$

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

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

$[D_{10}]$	$[\varepsilon]$	$[\alpha]$	$[\alpha^{2}]$	$[\alpha^{3}]$
n	1	2	2	2
Order	1	10	5	10
$[\Gamma_{1}]$	1	1	1	1
$[\Gamma_{2}]$	1	1	1	1
$[\Gamma_{3}]$	1		1
$[\Gamma_{4}]$	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	$[-1-\tau]$	$[\tau]$	$[\tau]$

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

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

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

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

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

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

(h) $[D_{12}]$

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

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

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

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

I	$[\varepsilon]$	$[\alpha]$	$[\alpha^{2}]$	$[\beta]$	$[\alpha \beta]$
n	1	12	12	20	15
Order	1	5	5	3	2
$[\Gamma_{1}]$	1	1	1	1	1
$[\Gamma_{2}]$	3	$[1+\tau]$	$[-\tau]$	0
$[\Gamma_{3}]$	3	$[-\tau]$	$[1+\tau]$	0
$[\Gamma_{4}]$	4			1	0
$[\Gamma_{5}]$	5	0	0		1

	Generators	Vector representation	Perpendicular 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 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.

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

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

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: $[\Gamma_3(R_{i}) = \Gamma_2(R_{j})]$ .

i	j	i	j	i	j	i	j	i	j	i	j
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

References

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