Symmetry operations for the point group Th (m-3)
1 : x,y,z           => 1                   

[[ 1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0.  1.]]

2 : -x,-y,z         => 2 [ 0 0 1 ]         

[[-1.  0.  0.]
 [ 0. -1.  0.]
 [ 0.  0.  1.]]

3 : -x,y,-z         => 2 [ 0 1 0 ]         

[[-1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0. -1.]]

4 : x,-y,-z         => 2 [ 1 0 0 ]         

[[ 1.  0.  0.]
 [ 0. -1.  0.]
 [ 0.  0. -1.]]

5 : z,x,y           => 3+ [ 1 1 1 ]        

[[ 0.  0.  1.]
 [ 1.  0.  0.]
 [ 0.  1.  0.]]

6 : z,-x,-y         => 3- [ 1 -1 1 ]       

[[ 0.  0.  1.]
 [-1.  0.  0.]
 [ 0. -1.  0.]]

7 : -z,-x,y         => 3+ [ 1 -1 -1 ]      

[[ 0.  0. -1.]
 [-1.  0.  0.]
 [ 0.  1.  0.]]

8 : -z,x,-y         => 3- [ 1 1 -1 ]       

[[ 0.  0. -1.]
 [ 1.  0.  0.]
 [ 0. -1.  0.]]

9 : y,z,x           => 3- [ 1 1 1 ]        

[[ 0.  1.  0.]
 [ 0.  0.  1.]
 [ 1.  0.  0.]]

10: -y,z,-x         => 3- [ 1 -1 -1 ]      

[[ 0. -1.  0.]
 [ 0.  0.  1.]
 [-1.  0.  0.]]

11: y,-z,-x         => 3+ [ 1 1 -1 ]       

[[ 0.  1.  0.]
 [ 0.  0. -1.]
 [-1.  0.  0.]]

12: -y,-z,x         => 3+ [ 1 -1 1 ]       

[[ 0. -1.  0.]
 [ 0.  0. -1.]
 [ 1.  0.  0.]]

13: -x,-y,-z        => -1                  

[[-1.  0.  0.]
 [ 0. -1.  0.]
 [ 0.  0. -1.]]

14: x,y,-z          => m [ 0 0 1 ]         

[[ 1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0. -1.]]

15: x,-y,z          => m [ 0 1 0 ]         

[[ 1.  0.  0.]
 [ 0. -1.  0.]
 [ 0.  0.  1.]]

16: -x,y,z          => m [ 1 0 0 ]         

[[-1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0.  1.]]

17: -z,-x,-y        => -3+ [ 1 1 1 ]       

[[ 0.  0. -1.]
 [-1.  0.  0.]
 [ 0. -1.  0.]]

18: -z,x,y          => -3- [ 1 -1 1 ]      

[[ 0.  0. -1.]
 [ 1.  0.  0.]
 [ 0.  1.  0.]]

19: z,x,-y          => -3+ [ 1 -1 -1 ]     

[[ 0.  0.  1.]
 [ 1.  0.  0.]
 [ 0. -1.  0.]]

20: z,-x,y          => -3- [ 1 1 -1 ]      

[[ 0.  0.  1.]
 [-1.  0.  0.]
 [ 0.  1.  0.]]

21: -y,-z,-x        => -3- [ 1 1 1 ]       

[[ 0. -1.  0.]
 [ 0.  0. -1.]
 [-1.  0.  0.]]

22: y,-z,x          => -3- [ 1 -1 -1 ]     

[[ 0.  1.  0.]
 [ 0.  0. -1.]
 [ 1.  0.  0.]]

23: -y,z,x          => -3+ [ 1 1 -1 ]      

[[ 0. -1.  0.]
 [ 0.  0.  1.]
 [ 1.  0.  0.]]

24: y,z,-x          => -3+ [ 1 -1 1 ]      

[[ 0.  1.  0.]
 [ 0.  0.  1.]
 [-1.  0.  0.]]

Irreducible representations for the point group Th (m-3) 
Irrep Ag ( dimension  1 )
1 : 1.0                 

2 : 1.0                 

3 : 1.0                 

4 : 1.0                 

5 : 1.0                 

6 : 1.0                 

7 : 1.0                 

8 : 1.0                 

9 : 1.0                 

10: 1.0                 

11: 1.0                 

12: 1.0                 

13: 1.0                 

14: 1.0                 

15: 1.0                 

16: 1.0                 

17: 1.0                 

18: 1.0                 

19: 1.0                 

20: 1.0                 

21: 1.0                 

22: 1.0                 

23: 1.0                 

24: 1.0                 

Irrep Au ( dimension  1 )
1 : 1.0                 

2 : 1.0                 

3 : 1.0                 

4 : 1.0                 

5 : 1.0                 

6 : 1.0                 

7 : 1.0                 

8 : 1.0                 

9 : 1.0                 

10: 1.0                 

11: 1.0                 

12: 1.0                 

13: -1.0                

14: -1.0                

15: -1.0                

16: -1.0                

17: -1.0                

18: -1.0                

19: -1.0                

20: -1.0                

21: -1.0                

22: -1.0                

23: -1.0                

24: -1.0                

Irrep Eg1 ( dimension  1 )
1 : 1.0                 

2 : 1.0                 

3 : 1.0                 

4 : 1.0                 

5 : (-0.5+0.86603j)     

6 : (-0.5+0.86603j)     

7 : (-0.5+0.86603j)     

8 : (-0.5+0.86603j)     

9 : (-0.5-0.86603j)     

10: (-0.5-0.86603j)     

11: (-0.5-0.86603j)     

12: (-0.5-0.86603j)     

13: 1.0                 

14: 1.0                 

15: 1.0                 

16: 1.0                 

17: (-0.5+0.86603j)     

18: (-0.5+0.86603j)     

19: (-0.5+0.86603j)     

20: (-0.5+0.86603j)     

21: (-0.5-0.86603j)     

22: (-0.5-0.86603j)     

23: (-0.5-0.86603j)     

24: (-0.5-0.86603j)     

Irrep Eu1 ( dimension  1 )
1 : 1.0                 

2 : 1.0                 

3 : 1.0                 

4 : 1.0                 

5 : (-0.5+0.86603j)     

6 : (-0.5+0.86603j)     

7 : (-0.5+0.86603j)     

8 : (-0.5+0.86603j)     

9 : (-0.5-0.86603j)     

10: (-0.5-0.86603j)     

11: (-0.5-0.86603j)     

12: (-0.5-0.86603j)     

13: -1.0                

14: -1.0                

15: -1.0                

16: -1.0                

17: (0.5-0.86603j)      

18: (0.5-0.86603j)      

19: (0.5-0.86603j)      

20: (0.5-0.86603j)      

21: (0.5+0.86603j)      

22: (0.5+0.86603j)      

23: (0.5+0.86603j)      

24: (0.5+0.86603j)      

Irrep Eg2 ( dimension  1 )
1 : 1.0                 

2 : 1.0                 

3 : 1.0                 

4 : 1.0                 

5 : (-0.5-0.86603j)     

6 : (-0.5-0.86603j)     

7 : (-0.5-0.86603j)     

8 : (-0.5-0.86603j)     

9 : (-0.5+0.86603j)     

10: (-0.5+0.86603j)     

11: (-0.5+0.86603j)     

12: (-0.5+0.86603j)     

13: 1.0                 

14: 1.0                 

15: 1.0                 

16: 1.0                 

17: (-0.5-0.86603j)     

18: (-0.5-0.86603j)     

19: (-0.5-0.86603j)     

20: (-0.5-0.86603j)     

21: (-0.5+0.86603j)     

22: (-0.5+0.86603j)     

23: (-0.5+0.86603j)     

24: (-0.5+0.86603j)     

Irrep Eu2 ( dimension  1 )
1 : 1.0                 

2 : 1.0                 

3 : 1.0                 

4 : 1.0                 

5 : (-0.5-0.86603j)     

6 : (-0.5-0.86603j)     

7 : (-0.5-0.86603j)     

8 : (-0.5-0.86603j)     

9 : (-0.5+0.86603j)     

10: (-0.5+0.86603j)     

11: (-0.5+0.86603j)     

12: (-0.5+0.86603j)     

13: -1.0                

14: -1.0                

15: -1.0                

16: -1.0                

17: (0.5+0.86603j)      

18: (0.5+0.86603j)      

19: (0.5+0.86603j)      

20: (0.5+0.86603j)      

21: (0.5-0.86603j)      

22: (0.5-0.86603j)      

23: (0.5-0.86603j)      

24: (0.5-0.86603j)      

Irrep Tu ( dimension  3 )
1 :
[[ 1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0.  1.]]

2 :
[[ 1.  0.  0.]
 [ 0. -1.  0.]
 [ 0.  0. -1.]]

3 :
[[-1.  0.  0.]
 [ 0. -1.  0.]
 [ 0.  0.  1.]]

4 :
[[-1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0. -1.]]

5 :
[[ 0.  0.  1.]
 [ 1.  0.  0.]
 [ 0.  1.  0.]]

6 :
[[ 0.  0. -1.]
 [ 1.  0.  0.]
 [ 0. -1.  0.]]

7 :
[[ 0.  0.  1.]
 [-1.  0.  0.]
 [ 0. -1.  0.]]

8 :
[[ 0.  0. -1.]
 [-1.  0.  0.]
 [ 0.  1.  0.]]

9 :
[[ 0.  1.  0.]
 [ 0.  0.  1.]
 [ 1.  0.  0.]]

10:
[[ 0. -1.  0.]
 [ 0.  0. -1.]
 [ 1.  0.  0.]]

11:
[[ 0. -1.  0.]
 [ 0.  0.  1.]
 [-1.  0.  0.]]

12:
[[ 0.  1.  0.]
 [ 0.  0. -1.]
 [-1.  0.  0.]]

13:
[[-1.  0.  0.]
 [ 0. -1.  0.]
 [ 0.  0. -1.]]

14:
[[-1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0.  1.]]

15:
[[ 1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0. -1.]]

16:
[[ 1.  0.  0.]
 [ 0. -1.  0.]
 [ 0.  0.  1.]]

17:
[[ 0.  0. -1.]
 [-1.  0.  0.]
 [ 0. -1.  0.]]

18:
[[ 0.  0.  1.]
 [-1.  0.  0.]
 [ 0.  1.  0.]]

19:
[[ 0.  0. -1.]
 [ 1.  0.  0.]
 [ 0.  1.  0.]]

20:
[[ 0.  0.  1.]
 [ 1.  0.  0.]
 [ 0. -1.  0.]]

21:
[[ 0. -1.  0.]
 [ 0.  0. -1.]
 [-1.  0.  0.]]

22:
[[ 0.  1.  0.]
 [ 0.  0.  1.]
 [-1.  0.  0.]]

23:
[[ 0.  1.  0.]
 [ 0.  0. -1.]
 [ 1.  0.  0.]]

24:
[[ 0. -1.  0.]
 [ 0.  0.  1.]
 [ 1.  0.  0.]]

Irrep Tg ( dimension  3 )
1 :
[[ 1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0.  1.]]

2 :
[[ 1.  0.  0.]
 [ 0. -1.  0.]
 [ 0.  0. -1.]]

3 :
[[-1.  0.  0.]
 [ 0. -1.  0.]
 [ 0.  0.  1.]]

4 :
[[-1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0. -1.]]

5 :
[[ 0.  0.  1.]
 [ 1.  0.  0.]
 [ 0.  1.  0.]]

6 :
[[ 0.  0. -1.]
 [ 1.  0.  0.]
 [ 0. -1.  0.]]

7 :
[[ 0.  0.  1.]
 [-1.  0.  0.]
 [ 0. -1.  0.]]

8 :
[[ 0.  0. -1.]
 [-1.  0.  0.]
 [ 0.  1.  0.]]

9 :
[[ 0.  1.  0.]
 [ 0.  0.  1.]
 [ 1.  0.  0.]]

10:
[[ 0. -1.  0.]
 [ 0.  0. -1.]
 [ 1.  0.  0.]]

11:
[[ 0. -1.  0.]
 [ 0.  0.  1.]
 [-1.  0.  0.]]

12:
[[ 0.  1.  0.]
 [ 0.  0. -1.]
 [-1.  0.  0.]]

13:
[[ 1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0.  1.]]

14:
[[ 1.  0.  0.]
 [ 0. -1.  0.]
 [ 0.  0. -1.]]

15:
[[-1.  0.  0.]
 [ 0. -1.  0.]
 [ 0.  0.  1.]]

16:
[[-1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0. -1.]]

17:
[[ 0.  0.  1.]
 [ 1.  0.  0.]
 [ 0.  1.  0.]]

18:
[[ 0.  0. -1.]
 [ 1.  0.  0.]
 [ 0. -1.  0.]]

19:
[[ 0.  0.  1.]
 [-1.  0.  0.]
 [ 0. -1.  0.]]

20:
[[ 0.  0. -1.]
 [-1.  0.  0.]
 [ 0.  1.  0.]]

21:
[[ 0.  1.  0.]
 [ 0.  0.  1.]
 [ 1.  0.  0.]]

22:
[[ 0. -1.  0.]
 [ 0.  0. -1.]
 [ 1.  0.  0.]]

23:
[[ 0. -1.  0.]
 [ 0.  0.  1.]
 [-1.  0.  0.]]

24:
[[ 0.  1.  0.]
 [ 0.  0. -1.]
 [-1.  0.  0.]]

