N  Matrix presentation  Seitz symbol  GM_{1}  GM_{2}GM_{3}  GM_{4}  GM_{5}  GM_{6}GM_{7} 
1    1       1  0  0  0  0  1  0  0  0  0  1  0  0  0  0  1 
 

2    2_{001}       i  0  0  0  0  i  0  0  0  0  i  0  0  0  0  i 
 

3    2_{010}       0  1  0  0  1  0  0  0  0  0  0  1  0  0  1  0 
 

4    2_{100}       0  i  0  0  i  0  0  0  0  0  0  i  0  0  i  0 
 

5    (1i)/2  (1+i)/2  (1i)/2  (1+i)/2 
 
 3^{+}_{111}      e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{iπ/4}√2/2 
 
  e^{i5π/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{i11π/12}√2/2  0  0  0  0  e^{i5π/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{i11π/12}√2/2 
 

6    (1+i)/2  (1i)/2  (1+i)/2  (1i)/2 
 
 3^{+}_{111}      e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{iπ/4}√2/2 
 
  e^{i11π/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{i5π/12}√2/2  0  0  0  0  e^{i11π/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{i5π/12}√2/2 
 

7    (1+i)/2  (1i)/2  (1+i)/2  (1i)/2 
 
 3^{+}_{111}      e^{iπ/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2 
 
  e^{i11π/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{i5π/12}√2/2  0  0  0  0  e^{i11π/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{i5π/12}√2/2 
 

8    (1i)/2  (1+i)/2  (1i)/2  (1+i)/2 
 
 3^{+}_{111}      e^{iπ/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2 
 
  e^{i5π/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{i11π/12}√2/2  0  0  0  0  e^{i5π/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{i11π/12}√2/2 
 

9    (1+i)/2  (1+i)/2  (1i)/2  (1i)/2 
 
 3^{}_{111}      e^{iπ/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2 
 
  e^{i5π/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{i11π/12}√2/2  0  0  0  0  e^{i5π/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{i11π/12}√2/2 
 

10    (1i)/2  (1i)/2  (1+i)/2  (1+i)/2 
 
 3^{}_{111}      e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{iπ/4}√2/2 
 
  e^{i11π/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{i5π/12}√2/2  0  0  0  0  e^{i11π/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{i5π/12}√2/2 
 

11    (1+i)/2  (1+i)/2  (1i)/2  (1i)/2 
 
 3^{}_{111}      e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{iπ/4}√2/2 
 
  e^{i5π/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{i11π/12}√2/2  0  0  0  0  e^{i5π/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{i11π/12}√2/2 
 

12    (1i)/2  (1i)/2  (1+i)/2  (1+i)/2 
 
 3^{}_{111}      e^{iπ/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2 
 
  e^{i11π/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{i5π/12}√2/2  0  0  0  0  e^{i11π/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{i5π/12}√2/2 
 

13    ^{d}1_{}       1  0  0  0  0  1  0  0  0  0  1  0  0  0  0  1 
 

14    ^{d}2_{001}       i  0  0  0  0  i  0  0  0  0  i  0  0  0  0  i 
 

15    ^{d}2_{010}       0  1  0  0  1  0  0  0  0  0  0  1  0  0  1  0 
 

16    ^{d}2_{100}       0  i  0  0  i  0  0  0  0  0  0  i  0  0  i  0 
 

17    (1i)/2  (1+i)/2  (1i)/2  (1+i)/2 
 
 ^{d}3^{+}_{111}      e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{i3π/4}√2/2 
 
  e^{i7π/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{iπ/12}√2/2  0  0  0  0  e^{i7π/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{iπ/12}√2/2 
 

18    (1+i)/2  (1i)/2  (1+i)/2  (1i)/2 
 
 ^{d}3^{+}_{111}      e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{i3π/4}√2/2 
 
  e^{iπ/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{i7π/12}√2/2  0  0  0  0  e^{iπ/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{i7π/12}√2/2 
 

19    (1+i)/2  (1i)/2  (1+i)/2  (1i)/2 
 
 ^{d}3^{+}_{111}      e^{i3π/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2 
 
  e^{iπ/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{i7π/12}√2/2  0  0  0  0  e^{iπ/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{i7π/12}√2/2 
 

20    (1i)/2  (1+i)/2  (1i)/2  (1+i)/2 
 
 ^{d}3^{+}_{111}      e^{i3π/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2 
 
  e^{i7π/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{iπ/12}√2/2  0  0  0  0  e^{i7π/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{iπ/12}√2/2 
 

21    (1+i)/2  (1+i)/2  (1i)/2  (1i)/2 
 
 ^{d}3^{}_{111}      e^{i3π/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2 
 
  e^{i7π/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{iπ/12}√2/2  0  0  0  0  e^{i7π/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{iπ/12}√2/2 
 

22    (1i)/2  (1i)/2  (1+i)/2  (1+i)/2 
 
 ^{d}3^{}_{111}      e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{i3π/4}√2/2 
 
  e^{iπ/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{i7π/12}√2/2  0  0  0  0  e^{iπ/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{i7π/12}√2/2 
 

23    (1+i)/2  (1+i)/2  (1i)/2  (1i)/2 
 
 ^{d}3^{}_{111}      e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{i3π/4}√2/2 
 
  e^{i7π/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{iπ/12}√2/2  0  0  0  0  e^{i7π/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{iπ/12}√2/2 
 

24    (1i)/2  (1i)/2  (1+i)/2  (1+i)/2 
 
 ^{d}3^{}_{111}      e^{i3π/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2 
 
  e^{iπ/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{i7π/12}√2/2  0  0  0  0  e^{iπ/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{i7π/12}√2/2 
 

25    1'_{}       0  0  1  0  0  0  0  1  1  0  0  0  0  1  0  0 
 

26    m'_{001}       0  0  i  0  0  0  0  i  i  0  0  0  0  i  0  0 
 

27    m'_{010}       0  0  0  1  0  0  1  0  0  1  0  0  1  0  0  0 
 

28    m'_{100}       0  0  0  i  0  0  i  0  0  i  0  0  i  0  0  0 
 

29    (1i)/2  (1+i)/2  (1i)/2  (1+i)/2 
 
 3^{'+}_{111}      e^{iπ/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2 
 
  0  0  e^{i7π/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{iπ/12}√2/2  e^{i5π/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{i11π/12}√2/2  0  0 
 

30    (1+i)/2  (1i)/2  (1+i)/2  (1i)/2 
 
 3^{'+}_{111}      e^{iπ/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2 
 
  0  0  e^{iπ/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{i7π/12}√2/2  e^{i11π/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{i5π/12}√2/2  0  0 
 

31    (1+i)/2  (1i)/2  (1+i)/2  (1i)/2 
 
 3^{'+}_{111}      e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{i3π/4}√2/2 
 
  0  0  e^{iπ/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{i7π/12}√2/2  e^{i11π/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{i5π/12}√2/2  0  0 
 

32    (1i)/2  (1+i)/2  (1i)/2  (1+i)/2 
 
 3^{'+}_{111}      e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{i3π/4}√2/2 
 
  0  0  e^{i7π/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{iπ/12}√2/2  e^{i5π/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{i11π/12}√2/2  0  0 
 

33    (1+i)/2  (1+i)/2  (1i)/2  (1i)/2 
 
 3^{'}_{111}      e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{i3π/4}√2/2 
 
  0  0  e^{i7π/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{iπ/12}√2/2  e^{i5π/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{i11π/12}√2/2  0  0 
 

34    (1i)/2  (1i)/2  (1+i)/2  (1+i)/2 
 
 3^{'}_{111}      e^{iπ/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2 
 
  0  0  e^{iπ/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{i7π/12}√2/2  e^{i11π/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{i5π/12}√2/2  0  0 
 

35    (1+i)/2  (1+i)/2  (1i)/2  (1i)/2 
 
 3^{'}_{111}      e^{iπ/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2 
 
  0  0  e^{i7π/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{iπ/12}√2/2  e^{i5π/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{i11π/12}√2/2  0  0 
 

36    (1i)/2  (1i)/2  (1+i)/2  (1+i)/2 
 
 3^{'}_{111}      e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{i3π/4}√2/2 
 
  0  0  e^{iπ/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{i7π/12}√2/2  e^{i11π/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{i5π/12}√2/2  0  0 
 

37    ^{d}1'_{}       0  0  1  0  0  0  0  1  1  0  0  0  0  1  0  0 
 

38    ^{d}m'_{001}       0  0  i  0  0  0  0  i  i  0  0  0  0  i  0  0 
 

39    ^{d}m'_{010}       0  0  0  1  0  0  1  0  0  1  0  0  1  0  0  0 
 

40    ^{d}m'_{100}       0  0  0  i  0  0  i  0  0  i  0  0  i  0  0  0 
 

41    (1i)/2  (1+i)/2  (1i)/2  (1+i)/2 
 
 ^{d}3^{'+}_{111}      e^{i3π/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2 
 
  0  0  e^{i5π/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{i11π/12}√2/2  e^{i7π/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{iπ/12}√2/2  0  0 
 

42    (1+i)/2  (1i)/2  (1+i)/2  (1i)/2 
 
 ^{d}3^{'+}_{111}      e^{i3π/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2 
 
  0  0  e^{i11π/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{i5π/12}√2/2  e^{iπ/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{i7π/12}√2/2  0  0 
 

43    (1+i)/2  (1i)/2  (1+i)/2  (1i)/2 
 
 ^{d}3^{'+}_{111}      e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{iπ/4}√2/2 
 
  0  0  e^{i11π/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{i5π/12}√2/2  e^{iπ/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{i7π/12}√2/2  0  0 
 

44    (1i)/2  (1+i)/2  (1i)/2  (1+i)/2 
 
 ^{d}3^{'+}_{111}      e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{iπ/4}√2/2 
 
  0  0  e^{i5π/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{i11π/12}√2/2  e^{i7π/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{iπ/12}√2/2  0  0 
 

45    (1+i)/2  (1+i)/2  (1i)/2  (1i)/2 
 
 ^{d}3^{'}_{111}      e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{iπ/4}√2/2 
 
  0  0  e^{i5π/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{i11π/12}√2/2  e^{i7π/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{iπ/12}√2/2  0  0 
 

46    (1i)/2  (1i)/2  (1+i)/2  (1+i)/2 
 
 ^{d}3^{'}_{111}      e^{i3π/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2 
 
  0  0  e^{i11π/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{i5π/12}√2/2  e^{iπ/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{i7π/12}√2/2  0  0 
 

47    (1+i)/2  (1+i)/2  (1i)/2  (1i)/2 
 
 ^{d}3^{'}_{111}      e^{i3π/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{i3π/4}√2/2 
 
  0  0  e^{i5π/12}√2/2  e^{i7π/12}√2/2  0  0  e^{i11π/12}√2/2  e^{i11π/12}√2/2  e^{i7π/12}√2/2  e^{i5π/12}√2/2  0  0  e^{iπ/12}√2/2  e^{iπ/12}√2/2  0  0 
 

48    (1i)/2  (1i)/2  (1+i)/2  (1+i)/2 
 
 ^{d}3^{'}_{111}      e^{iπ/4}√2/2  e^{i3π/4}√2/2  e^{iπ/4}√2/2  e^{iπ/4}√2/2 
 
  0  0  e^{i11π/12}√2/2  e^{i11π/12}√2/2  0  0  e^{i7π/12}√2/2  e^{i5π/12}√2/2  e^{iπ/12}√2/2  e^{iπ/12}√2/2  0  0  e^{i5π/12}√2/2  e^{i7π/12}√2/2  0  0 
 
