Space group
|
Euclidean normalizer
|
Additional generators of
|
Index of in
|
No.
|
Hermann–Mauguin symbol
|
Symbol
|
Basis vectors
|
Translations
|
Inversion through a centre at
|
Further generators
|
75
|
P
4
|
|
|
;
|
0, 0, 0
|
y
, x, z
|
|
76
|
|
|
|
;
|
/
|
|
|
77
|
|
|
|
;
|
0, 0, 0
|
y
, x, z
|
|
78
|
|
|
|
;
|
/
|
|
|
79
|
I
4
|
|
|
0, 0, t
|
0, 0, 0
|
y
, x, z
|
|
80
|
|
|
|
0, 0, t
|
|
|
|
81
|
|
|
|
;
|
0, 0, 0
|
y
, x, z
|
|
82
|
|
|
|
|
0, 0, 0
|
y
, x, z
|
|
83
|
|
|
|
;
|
|
y
, x, z
|
|
84
|
|
|
|
;
|
|
y
, x, z
|
|
85
|
|
|
|
;
|
|
y
, x, z
|
|
85
|
|
|
|
;
|
|
y
, x, z
|
|
86
|
|
|
|
;
|
|
y
, x, z
|
|
86
|
|
|
|
;
|
|
y
, x, z
|
|
87
|
|
|
|
|
|
y
, x, z
|
|
88
|
|
|
|
|
|
|
|
88
|
|
|
|
|
|
, ,
|
|
89
|
P
422
|
|
|
;
|
0, 0, 0
|
|
|
90
|
|
|
|
;
|
0, 0, 0
|
|
|
91
|
|
|
|
;
|
/
|
|
|
92
|
|
|
|
;
|
/
|
|
|
93
|
|
|
|
;
|
0, 0, 0
|
|
|
94
|
|
|
|
;
|
0, 0, 0
|
|
|
95
|
|
|
|
;
|
/
|
|
|
96
|
|
|
|
;
|
/
|
|
|
97
|
I
422
|
|
|
|
0, 0, 0
|
|
|
98
|
|
|
|
|
|
|
|
99
|
P
4mm
|
|
|
;
|
0, 0, 0
|
|
|
100
|
P
4bm
|
|
|
;
|
0, 0, 0
|
|
|
101
|
|
|
|
;
|
0, 0, 0
|
|
|
102
|
|
|
|
;
|
0, 0, 0
|
|
|
103
|
P
4cc
|
|
|
;
|
0, 0, 0
|
|
|
104
|
P
4nc
|
|
|
;
|
0, 0, 0
|
|
|
105
|
|
|
|
;
|
0, 0, 0
|
|
|
106
|
|
|
|
;
|
0, 0, 0
|
|
|
107
|
I
4mm
|
|
|
0, 0, t
|
0, 0, 0
|
|
|
108
|
I
4cm
|
|
|
0, 0, t
|
0, 0, 0
|
|
|
109
|
|
|
|
0, 0, t
|
|
|
|
110
|
|
|
|
0, 0, t
|
|
|
|
111
|
|
|
|
;
|
0, 0, 0
|
|
|
112
|
|
|
|
;
|
0, 0, 0
|
|
|
113
|
|
|
|
;
|
0, 0, 0
|
|
|
114
|
|
|
|
;
|
0, 0, 0
|
|
|
115
|
|
|
|
;
|
0, 0, 0
|
|
|
116
|
|
|
|
;
|
0, 0, 0
|
|
|
117
|
|
|
|
;
|
0, 0, 0
|
|
|
118
|
|
|
|
;
|
0, 0, 0
|
|
|
119
|
|
|
|
|
0, 0, 0
|
|
|
120
|
|
|
|
|
0, 0, 0
|
|
|
121
|
|
|
|
|
0, 0, 0
|
|
|
122
|
|
|
|
|
|
|
|
123
|
|
|
|
;
|
|
|
|
124
|
|
|
|
;
|
|
|
|
125
|
|
|
|
;
|
|
|
|
125
|
|
|
|
;
|
|
|
|
126
|
|
|
|
;
|
|
|
|
126
|
|
|
|
;
|
|
|
|
127
|
|
|
|
;
|
|
|
|
128
|
|
|
|
;
|
|
|
|
129
|
|
|
|
;
|
|
|
|
129
|
|
|
|
;
|
|
|
|
130
|
|
|
|
;
|
|
|
|
130
|
|
|
|
;
|
|
|
|
131
|
|
|
|
;
|
|
|
|
132
|
|
|
|
;
|
|
|
|
133
|
|
|
|
;
|
|
|
|
133
|
|
|
|
;
|
|
|
|
134
|
|
|
|
;
|
|
|
|
134
|
|
|
|
;
|
|
|
|
135
|
|
|
|
;
|
|
|
|
136
|
|
|
|
;
|
|
|
|
137
|
|
|
|
;
|
|
|
|
137
|
|
|
|
;
|
|
|
|
138
|
|
|
|
;
|
|
|
|
138
|
|
|
|
;
|
|
|
|
139
|
|
|
|
|
|
|
|
140
|
|
|
|
|
|
|
|
141
|
|
|
|
|
|
|
|
141
|
|
|
|
|
|
|
|
142
|
|
|
|
|
|
|
|
142
|
|
|
|
|
|
|
|
143
|
P
3
|
|
|
;
|
0, 0, 0
|
;
|
|
144
|
|
|
|
;
|
/
|
;
|
|
145
|
|
|
|
;
|
/
|
;
|
|
146
|
R
3 (hexag.)
|
|
|
0, 0, t
|
0, 0, 0
|
|
|
146
|
R
3 (rhomboh.)
|
|
,
|
r
, r, r
|
0, 0, 0
|
y
, x, z
|
|
147
|
|
|
, ,
|
|
|
;
|
|
148
|
(hexag.)
|
(hexag.)
|
,
|
|
|
|
|
148
|
(rhomboh.)
|
(rhomboh.)
|
, ,
|
|
|
y
, x, z
|
|
149
|
P
312
|
|
,
|
;
|
0, 0, 0
|
|
|
150
|
P
321
|
|
, ,
|
|
0, 0, 0
|
|
|
151
|
|
|
,
|
;
|
/
|
|
|
152
|
|
|
,
|
|
/
|
|
|
153
|
|
|
,
|
;
|
/
|
|
|
154
|
|
|
,
|
|
/
|
|
|
155
|
R
32 (hexag.)
|
(hexag.)
|
,
|
|
0, 0, 0
|
|
|
155
|
R
32 (rhomboh.)
|
(rhomboh.)
|
, ,
|
|
0, 0, 0
|
|
|
156
|
P
3m1
|
|
|
;
|
0, 0, 0
|
|
|
157
|
P
31m
|
|
, ,
|
0, 0, t
|
0, 0, 0
|
|
|
158
|
P
3c1
|
|
|
;
|
0, 0, 0
|
|
|
159
|
P
31c
|
|
,
|
0, 0, t
|
0, 0, 0
|
|
|
160
|
R
3m (hexag.)
|
|
|
0, 0, t
|
0, 0, 0
|
|
|
160
|
R
3m (rhomboh.)
|
|
, ,
|
r
, r, r
|
0, 0, 0
|
|
|
161
|
R
3c (hexag.)
|
|
|
0, 0, t
|
0, 0, 0
|
|
|
161
|
R
3c (rhomboh.)
|
|
, ,
|
r
, r, r
|
0, 0, 0
|
|
|
162
|
|
|
, ,
|
|
|
|
|
163
|
|
|
, ,
|
|
|
|
|
164
|
|
|
, ,
|
|
|
|
|
165
|
|
|
, ,
|
|
|
|
|
166
|
(hexag.)
|
(hexag.)
|
,
|
|
|
|
|
166
|
(rhomboh.)
|
(rhomboh.)
|
, ,
|
|
|
|
|
167
|
(hexag.)
|
(hexag.)
|
,
|
|
|
|
|
167
|
(rhomboh.)
|
(rhomboh.)
|
, ,
|
|
|
|
|
168
|
P
6
|
|
,
|
0, 0, t
|
0, 0, 0
|
y
, x, z
|
|
169
|
|
|
,
|
0, 0, t
|
/
|
|
|
170
|
|
|
,
|
0, 0, t
|
/
|
|
|
171
|
|
|
,
|
0, 0, t
|
/
|
|
|
172
|
|
|
,
|
0, 0, t
|
/
|
|
|
173
|
|
|
,
|
0, 0, t
|
0, 0, 0
|
y
, x, z
|
|
174
|
|
|
,
|
;
|
0, 0, 0
|
y
, x, z
|
|
175
|
|
|
, ,
|
|
|
y
, x, z
|
|
176
|
|
|
, ,
|
|
|
y
, x, z
|
|
177
|
P
622
|
|
, ,
|
|
0, 0, 0
|
|
|
178
|
|
|
, ,
|
|
/
|
|
|
179
|
|
|
, ,
|
|
/
|
|
|
180
|
|
|
, ,
|
|
/
|
|
|
181
|
|
|
, ,
|
|
/
|
|
|
182
|
|
|
, ,
|
|
0, 0, 0
|
|
|
183
|
P
6mm
|
|
,
|
0, 0, t
|
0, 0, 0
|
|
|
184
|
P
6cc
|
|
,
|
0, 0, t
|
0, 0, 0
|
|
|
185
|
|
|
,
|
0, 0, t
|
0, 0, 0
|
|
|
186
|
|
|
,
|
0, 0, t
|
0, 0, 0
|
|
|
187
|
|
|
,
|
;
|
0, 0, 0
|
|
|
188
|
|
|
,
|
;
|
0, 0, 0
|
|
|
189
|
|
|
, ,
|
|
0, 0, 0
|
|
|
190
|
|
|
, ,
|
|
0, 0, 0
|
|
|
191
|
|
|
, ,
|
|
|
|
|
192
|
|
|
, ,
|
|
|
|
|
193
|
|
|
, ,
|
|
|
|
|
194
|
|
|
, ,
|
|
|
|
|
195
|
P
23
|
|
a
, b, c
|
|
0, 0, 0
|
y
, x, z
|
|
196
|
F
23
|
|
, ,
|
|
0, 0, 0
|
y
, x, z
|
|
197
|
I
23
|
|
a
, b, c
|
|
0, 0, 0
|
y
, x, z
|
|
198
|
|
|
a
, b, c
|
|
0, 0, 0
|
, ,
|
|
199
|
|
|
a
, b, c
|
|
0, 0, 0
|
, ,
|
|
200
|
|
|
a
, b, c
|
|
|
y
, x, z
|
|
201
|
|
|
a
, b, c
|
|
|
y
, x, z
|
|
201
|
|
|
a
, b, c
|
|
|
y
, x, z
|
|
202
|
|
|
, ,
|
|
|
y
, x, z
|
|
203
|
|
|
, ,
|
|
|
y
, x, z
|
|
203
|
|
|
, ,
|
|
|
y
, x, z
|
|
204
|
|
|
a
, b, c
|
|
|
y
, x, z
|
|
205
|
|
|
a
, b, c
|
|
|
|
|
206
|
|
|
a
, b, c
|
|
|
, ,
|
|
207
|
P
432
|
|
a
, b, c
|
|
0, 0, 0
|
|
|
208
|
|
|
a
, b, c
|
|
0, 0, 0
|
|
|
209
|
F
432
|
|
, ,
|
|
0, 0, 0
|
|
|
210
|
|
|
, ,
|
|
|
|
|
211
|
I
432
|
|
a
, b, c
|
|
0, 0, 0
|
|
|
212
|
|
|
a
, b, c
|
|
/
|
|
|
213
|
|
|
a
, b, c
|
|
/
|
|
|
214
|
|
|
a
, b, c
|
|
0, 0, 0
|
|
|
215
|
|
|
a
, b, c
|
|
0, 0, 0
|
|
|
216
|
|
|
, ,
|
|
0, 0, 0
|
|
|
217
|
|
|
a
, b, c
|
|
0, 0, 0
|
|
|
218
|
|
|
a
, b, c
|
|
0, 0, 0
|
|
|
219
|
|
|
, ,
|
|
0, 0, 0
|
|
|
220
|
|
|
a
, b, c
|
|
0, 0, 0
|
|
|
221
|
|
|
a
, b, c
|
|
|
|
|
222
|
|
|
a
, b, c
|
|
|
|
|
222
|
|
|
a
, b, c
|
|
|
|
|
223
|
|
|
a
, b, c
|
|
|
|
|
224
|
|
|
a
, b, c
|
|
|
|
|
224
|
|
|
a
, b, c
|
|
|
|
|
225
|
|
|
, ,
|
|
|
|
|
226
|
|
|
, ,
|
|
|
|
|
227
|
|
|
, ,
|
|
|
|
|
227
|
|
|
, ,
|
|
|
|
|
228
|
|
|
, ,
|
|
|
|
|
228
|
|
|
, ,
|
|
|
|
|
229
|
|
|
a
, b, c
|
|
|
|
|
230
|
|
|
a
, b, c
|
|
|
|
|
|