In geometry, the [6,3], (*632) symmetry group is bounded by mirrors meeting with angles of 30, 60, and 90 degrees. There are a number of small index subgroups constructed by mirror removal and alternation. h[6,3] = [1+,6,3] creates [3[3]], (*333) symmetry, shown as red mirror lines. Removing mirrors at the order-3 point creates [6,3+], 3*3 symmetry, index 2. Removing all mirrors creates [6,3]+ (632) subgroup, index 2. The communtator subgroup is [1+,6,3+], (333) symmetry, index 4. An index 6 subgroup constructed as [6,3*], also becomes (*333), shown in blue mirror lines, and which has its own (333) rotational symmetry, index 12.
| Small index subgroups [6,3] (*632)
|
| Index
|
1
|
2
|
3
|
6
|
| Diagram
|
File:632 symmetry lines.png
|
File:632 symmetry lines-b.png
|
File:632 symmetry lines-c.png
|
File:632 symmetry lines-lambda.png
|
File:632 symmetry lines-lambda-2.png
|
File:632 symmetry lines-a.png
|
Intl (orb.)
Coxeter
|
p6m (*632)
[6,3] = File:CDel node c1.pngFile:CDel 6.pngFile:CDel node c2.pngFile:CDel 3.pngFile:CDel node c2.png = File:CDel node c2.pngFile:CDel split1-63.pngFile:CDel branch c1-2.pngFile:CDel label2.png
|
p3m1 (*333)
[1+,6,3] = File:CDel node h0.pngFile:CDel 6.pngFile:CDel node c2.pngFile:CDel 3.pngFile:CDel node c2.png = File:CDel branch c2.pngFile:CDel split2.pngFile:CDel node c2.png
|
p31m (3*3)
[6,3+] = File:CDel node c1.pngFile:CDel 6.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.png
|
cmm (2*22)
|
pmm (*2222)
|
p3m1 (*333)
[6,3*] = File:CDel node c1.pngFile:CDel 6.pngFile:CDel node g.pngFile:CDel 3sg.pngFile:CDel node g.png = File:CDel node c1.pngFile:CDel split1.pngFile:CDel branch c1.png
|
| Direct subgroups
|
| Index
|
2
|
4
|
6
|
12
|
| Diagram
|
File:632 symmetry alternated.png
|
File:632 symmetry lines-b2.png
|
File:632 symmetry lines-delta.png
|
File:632 symmetry lines-delta-2.png
|
File:632 symmetry lines-a2.png
|
Intl (orb.)
Coxeter
|
p6 (632)
[6,3]+ = File:CDel node h2.pngFile:CDel 6.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.png = File:CDel node h2.pngFile:CDel split1-63.pngFile:CDel branch h2h2.pngFile:CDel label2.png
|
p3 (333)
[1+,6,3+] = File:CDel node h0.pngFile:CDel 6.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.png = File:CDel branch h2h2.pngFile:CDel split2.pngFile:CDel node h2.png
|
p2 (2222)
|
p2 (2222)
|
p3 (333)
[1+,6,3*] = File:CDel node h2.pngFile:CDel 6.pngFile:CDel node g.pngFile:CDel 3sg.pngFile:CDel node g.png = File:CDel node h2.pngFile:CDel split1.pngFile:CDel branch h2h2.png
|
Wallpaper subgroup relationships
| Subgroup relationships among 14 wallpaper group[1]
|
|
|
o |
2222 |
×× |
** |
*× |
22× |
22* |
*2222 |
2*22 |
333 |
*333 |
3*3 |
632 |
*632
|
|
|
p1 |
p2 |
pg |
pm |
cm |
pgg |
pmg |
pmm |
cmm |
p3 |
p3m1 |
p31m |
p6 |
p6m
|
| o |
p1
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
| 2222 |
p2
|
2 |
2 |
2 |
|
|
|
|
|
|
|
|
|
|
|
| 333 |
p3
|
3 |
|
|
|
|
|
|
|
|
3 |
|
|
|
|
| 632 |
p6
|
6 |
3 |
|
|
|
|
|
|
|
2 |
|
|
4 |
|
| ×× |
pg
|
2 |
|
2 |
|
|
|
|
|
|
|
|
|
|
|
| ** |
pm
|
2 |
|
2 |
2 |
2 |
|
|
|
|
|
|
|
|
|
| *× |
cm
|
2 |
|
2 |
2 |
3 |
|
|
|
|
|
|
|
|
|
| 22× |
pgg
|
4 |
2 |
2 |
|
|
3 |
|
|
|
|
|
|
|
|
| 22* |
pmg
|
4 |
2 |
2 |
2 |
4 |
2 |
3 |
|
|
|
|
|
|
|
| *2222 |
pmm
|
4 |
2 |
4 |
2 |
4 |
4 |
2 |
2 |
2 |
|
|
|
|
|
| 2*22 |
cmm
|
4 |
2 |
4 |
4 |
2 |
2 |
2 |
2 |
4 |
|
|
|
|
|
| *333 |
p3m1
|
6 |
|
6 |
6 |
3 |
|
|
|
|
2 |
4 |
3 |
|
|
| 3*3 |
p31m
|
6 |
|
6 |
6 |
3 |
|
|
|
|
2 |
3 |
4 |
|
|
| *632 |
p6m
|
12 |
6 |
12 |
12 |
6 |
6 |
6 |
6 |
3 |
4 |
2 |
2 |
2 |
3
|
References
- ↑ Coxeter, (1980), The 17 plane groups, Table 4
- Coxeter, H. S. M. & Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9.
