Direct sum of irreducible representation

A1	A2	B1	B2	E1	E2	E3	E4	E5
3	0	0	2	2	2	2	2	2

Properties of derivatives and isotopomers by single substitution, h(D_6d)=24

Atom Set*	Site Symmetry**	h(Site Symmetry)	Identical Atoms***	Element	Chrial	Polar	Isotopomer
							Isotope	Mass	Abundance****
1	D6d	24	1	Cr	no	no	53Cr 50Cr 54Cr	209.0345 206.0399 210.0328	8.3045 3.7978 2.0672
2	Cs	2	12	C	no	yes	13C	209.0378	9.7748
3	Cs	2	12	H	no	yes	2H	209.0407	0.1371
Total Number of Atoms:			25	✅ Correct Number of Atoms found

*Atom Orbit
**Subgroup of point group D_6d
***Calculated as h( D_6d)/h(Site Symmetry)
****Natural Abundance of single substituted Isotopomer in %

Numbers of isomers by substitution

Replacement	Pattern	Achiral Isomers	Chiral Isomer Pairs
Single	X	3	0
Double	X2	10	11
Double	XY	10	20
Triple	X3	30	85
Triple	X2Y	40	271
Triple	XYZ	30	560
Quadruple	X4	76	509
Quadruple	X3Y	110	2.054
Quadruple	X2Y2	160	3.121
Quadruple	X2YZ	130	6.260
Quadruple	WXYZ	60	12.620
Quintuple	X5	164	2.151
Quintuple	VWXYZ	60	265.620
Sextuple	X6	310	7.291
Sextuple	UVWXYZ	0	5.313.000

Further Reading

• P.W. Fowler, J. Chem. Soc. Faraday Trans. 91(15) 2241 (1995)
  Isomer Counting using Point Group Symmetry
  
  

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Representation Γ_3N