Template:632 symmetry table

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
Intl (orb.)
Coxeter
p6m (*632)
[6,3] = =
p3m1 (*333)
[1+,6,3] = =
p31m (3*3)
[6,3+] =
cmm (2*22) pmm (*2222) p3m1 (*333)
[6,3*] = =
Direct subgroups
Index 2 4 6 12
Diagram
Intl (orb.)
Coxeter
p6 (632)
[6,3]+ = =
p3 (333)
[1+,6,3+] = =
p2 (2222) p2 (2222) p3 (333)
[1+,6,3*] = =

Wallpaper subgroup relationships

edit
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

edit
  1. ^ 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.