character table
TRANSCRIPT
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Character table for Cspoint group
E hlinear,
rotationsquadratic
A' 1 1 x, y, R z x2, y2, z2, xy
A'' 1 -1 z, R x, Ry yz, xz
Character table for C2point group
E C2linear,
rotationsquadratic
A 1 1 z, R z x2, y2, z2, xy
B 1 -1 x, y, R x, Ry yz, xz
Character table for Cipoint group
E ilinear,
rotationsquadratic
Ag 1 1 Rx, Ry, Rz x2, y2, z2, xy, xz, yz
Au 1 -1 x, y, z
Character table for C2point group
E C2linear,
rotationsquadratic
A 1 1 z, R z x2, y2, z2, xy
B 1 -1 x, y, R x, Ry yz, xz
Character table for C3point group
E C3 (C3)2 linear,
rotationsquadratic
A 1 1 1 z, R z x2
+y2
, z2
E 1
1
e
e*e*
e
x+iy; Rx+iRyx-iy; Rx-iRy
(x2-y2, xy) (yz, xz)
e = exp(2i/3)
Character table for C4point group
E C4 C2 (C4)3 linear,
rotationsquadratic
A 1 1 1 1 z, R z x2
+y2
, z2
B 1 -1 1 -1 x2-y2, xy
E 1
1
i
-i
-1
-1
-i
i
x+iy; Rx+iRyx-iy; Rx-iRy
(yz, xz)
Character table for Cpoint group
E C (C)2 (C)
3 (C)4 linear,
rotations
quadratic
A 1 1 1 1 1 z, R z x2+y2, z2
E!1
1
e
e*e2
e2*e2*
e2e*
e
x+iy, Rx+iRy
x-iy, Rx-iRy(yz, xz)
E21
1
e2
e2*e*
e
e
e*e2*
e2 (x2-y2, xy)
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e = exp(2i/5)
Character table for C"point group
E C" C3 C2 (C3)2 (C")
linear,
rotationsquadratic
A 1 1 1 1 1 1 z, R z x2+y2, z2
B 1 -1 1 -1 1 -1
E!1
1
e
e*-e*
-e
-1
-1
-e
-e*e*
e
x+iy; Rx+iRyx-iy; Rx-iRy
(xz, yz)
E21
1
-e*
-e
-e
-e*1
1
-e*
-e
-e
-e* (x2-y2, xy)
e = exp(i/3)
Character table for C2#point group
E C2($) #(%$) #(&$)linear,
rotationsquadratic
A! 1 1 1 1 z x2, y2, z2
A2 1 1 -1 -1 R z xy
B! 1 -1 1 -1 x, R y xz
B2 1 -1 -1 1 y, R x yz
Character table for C3#point group
E 2C3($) 3#linear,
rotationsquadratic
A! 1 1 1 z x2+y2, z2
A2 1 1 -1 R z
E 2 -1 0 (x, y) (R x, Ry) (x2-y2, xy) (xz, yz)
Character table for C4#point group
E 2C4($) C2 2# 2dlinear,
rotationsquadratic
A! 1 1 1 1 1 z x2+y2, z2
A2 1 1 1 -1 -1 R z
B! 1 -1 1 1 -1 x2-y2
B2 1 -1 1 -1 1 xy
E 2 0 -2 0 0 (x, y) (R x, Ry) (xz, yz)
Character table for C#point group
E 2C($) 2(C)2 #
linear,
rotationsquadratic
A! 1 1 1 1 z x2+y2, z2
A2 1 1 1 -1 R z
E! 2 2cos(2/5) 2cos(4/5) 0 (x, y) (Rx, Ry) (xz, yz)
E2 2 2cos(4/5) 2cos(2/5) 0 (x2-y2, xy)
Character table for C"#point group
E 2C"($) 2C3($) C2($) 3# 3dlinear,
rotationsquadratic
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A! 1 1 1 1 1 1 z x2+y2, z2
A2 1 1 1 1 -1 -1 R z
B! 1 -1 1 -1 1 -1
B2 1 -1 1 -1 -1 1
E! 2 1 -1 -2 0 0 (x, y) (R x, Ry) (xz, yz)
E2 2 -1 -1 2 0 0 (x2-y2, xy)
Character table for C2hpoint group
E C2($) i hlinear,
rotationsquadratic
Ag 1 1 1 1 R z x2, y2, z2, xy
Bg 1 -1 1 -1 R x, Ry xz, yz
Au 1 1 -1 -1 z
Bu 1 -1 -1 1 x, y
Character table for C3hpoint group
E C3($) (C3)2 h 3 (3)
linear functions,
rotations
quadratic
functions
A' 1 1 1 1 1 1 R z x2+y2, z2
E' 1
1
e
e
*
e*
e
1
1
e
e
*
e*
e
x+iy
x-iy
(x2-y2, xy)
A'' 1 1 1 -1 -1 -1 z
E'' 1
1
e
e*e*
e
-1
-1
-e
-e*-e*
-e
Rx+iRyRx-iRy
(xz, yz)
e = exp(2i/3)
Character table for C4hpoint group
E C4($) C2 (C4)3 i (4)
3 h 4linear,
rotationsquadratic
Ag 1 1 1 1 1 1 1 1 R z x2+y2, z2
Bg 1 -1 1 -1 1 -1 1 -1 x2-y2, xy
Eg1
1
i
-i
-1
-1
-i
i
1
1
i
-i
-1
-1
-i
i
Rx+iRy
Rx-iRy(xz, yz)
Au 1 1 1 1 -1 -1 -1 -1 z
Bu 1 -1 1 -1 -1 1 -1 1
Eu1
1
i
-i
-1
-1
-i
i
-1
-1
-i
i
1
1
i
-i
x+iy
x-iy
Character table for Chpoint group
EC
(C)2
(C)3
(C)4
h
()
()3
()
linear,
rotation
s
quadrati
c
A' 1 1 1 1 1 1 1 1 1 1 R z x2+y2, z2
E'!1
1
e
e*e2
e2*e2*
e2e*
e
1
1
e
e*e2
e2*e2*
e2e*
e
x+iy
x-iy
E'21
1
e2
e2
*
e*
e
e
e*e2*
e21
1
e2
e2
*
e*
e
e
e*e2*
e2(x2-y2,
xy)
A'' 1 1 1 1 1 -1 -1 -1 -1 -1 z
E''
!
1
1
e
e*e2
e2*e2*
e2e*
e
-
1
-
1
-e
-
e*
-e2
-e2*-e2*
-e2-e*
-e
Rx+iRyRx-iRy
(xz, yz)
E'' 1 e2 e* e e2* - - -e* -e -e2*
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2 1 e2
* e e* e2
1
-
1
e2
-
e2
*
-e -e* -e2
e = exp(2i/5)
Character table for C"hpoint group
EC6(z
)
C
3
C
2
(C3)2
(C6)5 i
(S3)5
(S6)5
!
S
6
S3
"i#e$%,
%o&$&io#
s
'$%$&i
c
1 1 1 1 1 1 1 1 1 1 1 R
z
x2+y2,
z2
1 -1 1 -1 1 -1 1 -1 1 -
1 1 -1
E1
1
1
e
e*
-
e*
-e
-1
-1
-e
-e*e*
e
1
1
e
e*-e*
-e
-
1
-
1
-e
-
e*
e*
e
Rx+iRyRx-iRy
(xz, yz)
E2
1
1
-e*
-e
-e
-
1
1
-e*
-e
-e
-e*1 -e*
-e
-e
-e*1
1
-
e*-e
-
(x2-y2,
xy)
e* 1 -e e*
1 1 1 1 1 1 -
1 -1 -1
-
1
-
1 -1 z
1 -1 1 -1 1 -1 -
1 1 -1 1
-
1 1
E1
1
1
e
e*
-
e*
-e
-1
-1
-e
-e*e*
e
-1
-
1
-e
-e*e*
e
1
1
e
e*
-
e*
-e
x+iy
x-iy
E2
1
1
-e*
-e
-e
-
e*
1
1
-e*
-e
-e
-e*
-
1
-
1
e*
e
e
e*
-
1
-
1
e*
e
e
e*
e = exp(i/3)
Character table for *2point group
E C2($) C2(&) C2(%)linear,
rotationsquadratic
A 1 1 1 1 x2, y2, z2
B! 1 1 -1 -1 z, R z xyB2 1 -1 1 -1 y, R y xz
B3 1 -1 -1 1 x, R x yz
Character table for *3point group
E 2C3($) 3C'2linear,
rotationsquadratic
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A! 1 1 1 x2+y2, z2
A2 1 1 -1 z, R z
E 2 -1 0 (x, y) (R x, Ry) (x2-y2, xy) (xz, yz)
Character table for *4point group
E 2C4($) C2($) 2C'2 2C''2linear functions,
rotations
quadratic
functionsA! 1 1 1 1 1 x
2+y2, z2
A2 1 1 1 -1 -1 z, R z
B! 1 -1 1 1 -1 x2-y2
B2 1 -1 1 -1 1 xy
E 2 0 -2 0 0 (x, y) (R x, Ry) (xz, yz)
Character table for *point group
E 2C($) 2(C)2 C'2
linear,
rotationsquadratic
A! 1 1 1 1 x2+y2, z2
A2 1 1 1 -1 z, R z
E! 2 2cos(2/5) 2cos(4/5) 0 (x, y) (Rx, Ry) (xz, yz)
E2 2 2cos(4/5) 2cos(2/5) 0
Character table for *"point group
E 2C6(z) 2C3(z) C2(z) 3C2 3C2"i#e$%,
%o&$&io#s '$%$&ic
1 1 1 1 1 1 1 x2+y2, z2
2 1 1 1 1 -1 -1 z, R z
1 1 -1 1 -1 1 -1
2 1 -1 1 -1 -1 1
E1 2 1 -1 -2 0 0 (x, y) (R x, Ry) (xz, yz)
E2 2 -1 -1 2 0 0 (x2-y2, xy)
Character table for *2hpoint group
E C2($)
C2(&)
C2(%)
i (%&)
(%$)
(&$)
linear,rotation
s
quadratic
Ag 1 1 1 1 1 1 1 1 x2, y2, z2
B!g 1 1 -1 -1 1 1 -1 -1 R z xy
B2g 1 -1 1 -1 1 -1 1 -1 R y xz
B3g 1 -1 -1 1 1 -1 -1 1 R x yz
Au 1 1 1 1 -
1 -1 -1 -1
B!u 1 1 -1 -1 -
1 -1 1 1 z
B2u 1 -1 1 -1 -
1 1 -1 1 y
B3u 1 -1 -1 1 -
1 1 1 -1 x
Character table for *3hpoint group
E 2C3 3C'2 h 23 3#linear,
rotationsquadratic
A'! 1 1 1 1 1 1 x2+y2, z2
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A'2 1 1 -1 1 1 -1 R z
E' 2 -1 0 2 -1 0 (x, y) (x2-y2, xy)
A''! 1 1 1 -1 -1 -1
A''2 1 1 -1 -1 -1 1 z
E'' 2 -1 0 -2 1 0 (R x, Ry) (xz, yz)
Character table for *4hpoint group
E2C4($
)
C
2
2C'
2
2C''
2
i 24 h2
#
2d
linears,
rotation
s
quadrati
c
A!g 1 1 1 1 1 1 1 1 1 1 x2+y2, z2
A2g 1 1 1 -1 -1 1 1 1 -1 -1 R z
B!g 1 -1 1 1 -1 1 -1 1 1 -1 x2-y2
B2g 1 -1 1 -1 1 1 -1 1 -1 1 xy
Eg 2 0 -2 0 0 2 0 -2 0 0 (R x, Ry) (xz, yz)
A!u 1 1 1 1 1 -
1 -1 -1 -1 -1
A2u 1 1 1 -1 -1 -
1 -1 -1 1 1 z
B!u 1 -1 1 1 -1 -
1 1 -1 -1 1
B2u 1 -1 1 -1 1 -
1 1 -1 1 -1
Eu 2 0 -2 0 0 -
2 0 2 0 0 (x, y)
Character table for *hpoint group
E 2C 2(C)2 C
'2
h
2 2()3
#
linear,
rotatio
ns
quadra
tic
A'
!
1 1 1 1 1 1 1 1 x2+y2,
z2
A'
2 1 1 1 -1 1 1 1 -1 R z
E'
!
22cos(2
/5)
2cos(4
/5) 0 2
2cos(2
/5)
2cos(4
/5) 0 (x, y)
E'
2
22cos(4
/5)
2cos(2
/5) 0 2
2cos(4
/5)
2cos(2
/5) 0
(x2-y2,
xy)
A'
'!1 1 1 1
-
1 -1 -1 -1
A'
'2 1 1 1 -1 -
1 -1 -1 1 z
E'
'!2
2cos(2
/5)
2cos(4
/5) 0
-
2
-
2cos(2
/5)
-
2cos(4
/5)
0 (Rx,
Ry) (xz, yz)
E'
'22
2cos(4
/5)
2cos(2
/5) 0
-
2
-
2cos(4
/5)
-
2cos(2
/5)
0
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Character table for *"hpoint group
E2C
"
2C
3
C
2
3C'
2
3C''
2
i2
3
2
"
h
3
d
3
#
+inear,
rotatio
ns
uadrat
ic
A!
g
1 1 1 1 1 1 1 1 1 1 1 1 x2+y2, z2
A2
g
1 1 1 1 -1 -1 1 1 1 1 -1 -1 R z
B!g1 -1 1 -1 1 -1 1 -1 1 -
1 1 -1
B2g1 -1 1 -1 -1 1 1 -1 1 -
1 -1 1
E!g2 1 -1 -2 0 0 2 1 -1 -
2 0 0 (R x, Ry) (xz, yz)
E2g2 -1 -1 2 0 0 2 -1 -1 2 0 0 (x2-y2,
xy)
A!
u
1 1 1 1 1 1 -
1 -1 -1
-
1 -1 -1
A2
u
1 1 1 1 -1 -1 -
1 -1 -1
-
1 1 1 z
B!
u
1 -1 1 -1 1 -1 -
1 1 -1 1 -1 1
B2
u
1 -1 1 -1 -1 1 -
1 1 -1 1 1 -1
E!
u
2 1 -1 -2 0 0 -
2 -1 1 2 0 0 (x, y)
E2
u
2 -1 -1 2 0 0 -
2 1 1
-
2 0 0
Character table for *2dpoint group
E 24 C2($) 2C'2 2dlinear,
rotationsquadratic
A! 1 1 1 1 1 x2+y2, z2
A2 1 1 1 -1 -1 R z
B! 1 -1 1 1 -1 x2-y2
B2 1 -1 1 -1 1 z xy
E 2 0 -2 0 0 (x, y) (R x, Ry) (xz, yz)
Character table for *3dpoint group
E 2C3 3C'2 i 2" 3dlinear,
rotationsquadratic
A!g 1 1 1 1 1 1 x2
+y2
, z2
A2g 1 1 -1 1 1 -1 R z
Eg 2 -1 0 2 -1 0 (R x, Ry) (x2-y2, xy) (xz, yz)
A!u 1 1 1 -1 -1 -1
A2u 1 1 -1 -1 -1 1 z
Eu 2 -1 0 -2 1 0 (x, y)
Character table for *4dpoint group
E 2- 2C4 2(-)3 C2 4C'2 4d
linear,
rotationsquadratic
A! 1 1 1 1 1 1 1 x2+y2, z2
A2 1 1 1 1 1 -1 -1 R z
B! 1 -1 1 -1 1 1 -1
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B2 1 -1 1 -1 1 -1 1 z
E! 2 (2)1/2 0 -(2)1/2 -2 0 0 (x, y)
E2 2 0 -2 0 2 0 0 (x2-y2, xy)
E3 2 -(2)1/2 0 (2)1/2 -2 0 0 (R x, Ry) (xz, yz)
Character table for *dpoint group
E 2C 2(C)2 C
'2i 2(!.)
3 2!.
d
linear,rotatio
ns
quadra
tic
1
1 1 1 1 1 1 1 1 x2+y2,
z2
2
1 1 1 -1 1 1 1 -1 R z
E1
2
2cos(2
/5)
2cos(4
/5) 0 2
2cos(2
/5)
2cos(4
/5) 0
(Rx,
Ry) (xz, yz)
E2
22cos(4
/5)
2cos(2
/5) 0 2
2cos(4
/5)
2cos(2
/5) 0
(x2-y2,
xy)
1
1 1 1 1 -
1 -1 -1 -1
2
1 1 1 -1 -
1 -1 -1 1 z
E1
22cos(2/5)
2cos(4/5)
0 -2
-
2cos(2
/5)
-
2cos(4
/5)
0 (x, y)
E2
22cos(4
/5)
2cos(2
/5) 0
-
2
-
2cos(4
/5)
-
2cos(2
/5)
0
Character table for *"dpoint group
E 2!2 2C" 24 2C3 2(!2) C2 "C'2 "d
linear,
rotationsquadratic
A! 1 1 1 1 1 1 1 1 1 x2+y2, z2
A2 1 1 1 1 1 1 1 -1 -1 R z
B! 1 -1 1 -1 1 -1 1 1 -1
B2 1 -1 1 -1 1 -1 1 -1 1 z
E! 2 (3)1/2 1 0 -1 -(3)1/2 -2 0 0 (x, y)
E2 2 1 -1 -2 -1 1 2 0 0 (x2-y2, xy)
E3 2 0 -2 0 2 0 -2 0 0
E4 2 -1 -1 2 -1 -1 2 0 0
E 2 -(3)1/2 1 0 -1 (3)1/2 -2 0 0 (R x, Ry) (xz, yz)
Character table for 4point group
E 4 C2 (4)
3 linear,
rotations quadratic
A 1 1 1 1 R z x2+y2, z2
B 1 -1 1 -1 z x2-y2, xy
E 1
1
i
-i
-1
-1
-i
i
x+iy; Rx+iRy
x-iy; Rx-iRy(xz, yz)
Character table for "point group
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E C3($) (C3)2 i (")
"linear,
rotationsquadratic
Ag 1 1 1 1 1 1 R z x2+y2, z2
Eg1
1
e
e*e*
e
1
1
e
e*e*
e
Rx+iRyRx-iRy
(x2-y2, xy) (xz, yz)
Au 1 1 1 -1 -1 -1 z
Eu1
1
e
e*e*
e
-1
-1
-e
-e*-e*
-e
x+iy
x-iy
e = exp(2i/3)
Character table for -point group
E - C4($) (-)3 C2 (-)
(C4)3 (-)
linear,
rotationsquadratic
A 1 1 1 1 1 1 1 1 R z x2+y2, z2
B 1 -1 1 -1 1 -1 1 -1 z
E!1
1
e
e*i
-i
-e*
-e
-1
-1
-e
-e*-i
i
e*
e
x+iy
x-iy
E21
1
i
-i
-1
-1
-i
i
1
1
i
-i
-1
-1
-i
i (x2-y2, xy)
E31
1
-e
-e*i
-i
e*
e
-1
-1
e
e*-i
i
-e*
-e
Rx+iRyRx-iRy
(xz, yz)
e = exp(i/4)
Character table for !.point group
EC
(C)2
(C)3
(C)4 i
(!.)
(!.) !.
(!.)3
linear,
rotatio
ns
quadrat
ic
Ag 1 1 1 1 1 1 1 1 1 1 R z z2, x2+y2
E! 1 e e2 e2* e* + +e +e2 +e2 +e* Rx+iRy (xz, yz)
g 1 e* e2* e2 e
1
1 e* e2*
*
e2 e+ Rx-iRy
E2
g
1
1
e2
e2
*
e*
e
e
e*e2*
e21
1
+e2
e2*+e*
e+
+e
e*+e2*
e2(x2-y2,
xy)
Au 1 1 1 1 1 -1 -1 -1 -1 -1 z
E!
u
1
1
e
e*e2
e2*e2*
e2e*
e
-1
-1
-e
-e*-e2
-e2*-e2*
-e2-e*
e-
x+iy
x-iy
E2
u
1
1
e2
e2
*
e*
e
e
e*e2*
e2-1
-1
-e2
-e2*-e*
-e
-e
-e*-e2*
-e2
e = exp(2i/5)
Character table for /dpoint group
E -C3 3C2 "4 "dlinear,
rotations quadratic
A! 1 1 1 1 1 x2+y2+z2
A2 1 1 1 -1 -1
E 2 -1 2 0 0 (2z2-x2-y2, x2-y2)
/! 3 0 -1 1 -1 (R x, Ry, Rz)
/2 3 0 -1 -1 1 (x, y, z) (xy, xz, yz)
Character table for 0hpoint group
E-C
3
"C
2
"C
4
3C21(C4)2 i
"
4
-
"
3
h
"
d
linear,
rotation
s
quadrati
c
A!
g
1 1 1 1 1 1 1 1 1 1 x2+y2+z2
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A2
g
1 1 -1 -1 1 1 -1 1 1 -1
Eg 2 -1 0 0 2 2 0 -1 2 0 (2z2-x2-
y2, x2-y2)
/!g 3 0 -1 1 -1 3 1 0 -1 -1 (Rx, Ry,
Rz)
/2g 3 0 1 -1 -1 3 -1 0 -1 1 (xz, yz,
xy)
A!
u
1 1 1 1 1 -
1 -1 -1 -1 -1
A2
u
1 1 -1 -1 1 -
1 1 -1 -1 1
Eu 2 -1 0 0 2 -
2 0 1 -2 0
/!u3 0 -1 1 -1 -
3 -1 0 1 1 (x, y, z)
Character table for C#point group
E 2C sig5a#linear,
rotationsquadratic
A!167 1 1 ... 1 z x2+y2, z2
A2168
1 1 ... -1 R z
E!19 2 2cos() ... 0 (x, y) (R x, Ry) (xz, yz)
E21: 2 2cos(2) ... 0 (x2-y2, xy)
E31; 2 2cos(3) ... 0
... ... ... ...
Character table for *hpoint group
E 2C
#
i 2
C
'2
linear
functions,
rotations
quadrati
c
A!g167
g 1 1 ..
. 1 1 1
..
. 1 x2+y2, z2
A2g168g 1 1 ..
. -1 1 1 ..
. -1 Rz
E!g19g 2 2cos() ..
. 0 2
-
2cos()
..
. 0 (Rx, Ry) (xz, yz)
E2g1:g 2 2cos(2
)
..
. 0 2
2cos(2
)
..
. 0
(x2-y2,
xy)
-
7/24/2019 Character Table
11/11
E3g1;g 2 2cos(3
)
..
. 0 2
-
2cos(3
)
..
. 0
..
. ...
..
. ...
..
. ...
..
. ...
A!u
167u1 1
..
. 1
-
1 -1
..
. -1 z
A2u168u 1 1
..
. -1
-
1 -1
..
. 1
E!u19u 2 2cos() ..
. 0
-
2 2cos()
..
. 0 (x, y)
E2u1:u 2 2cos(2
)
..
. 0
-
2
-
2cos(2
)
..
. 0
E3u
1;u 2
2cos(3
)
..
. 0
-
2
2cos(3
)
..
. 0
..
. ...
..
. ...
..
. ...
..
. ...