updated 12-04-2004
The group Td of the symmetries of the regular tetrahedra is isomorphic to S4, symmetric group of order 4 or group of the permutations of a set of four elements (the four vertices of the tetrahedron). The subgroup of the half-turns is a Klein's group (orange corner); the sub-group T of the twelve rotations (orange and red corner) is isomorphic to A4, alternate group of order 4.
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regular tetrahedron
ABCD 1 = (ABCD) |
rab = BA DC half turn
ra = (A) CDB ra' = (A) DBC 120° rotations, opposite directions |
mab = (AB) DC plane reflection
c1 = BCDA c1' = DABC inverse cycles c2 = BDAC c2' = CADB c3 = CDBA c3' = DCAB |
| Td | 1 | rab | rac | rad | ra | ra' | rb | rb' | rc | rc' | rd | rd' | mab | mac | mad | mbc | mbd | mcd | c1 | c1' | c2 | c2' | c3 | c3' |
| 1 | 1 | rab | rac | rad | ra | ra' | rb | rb' | rc | rc' | rd | rd' | mab | mac | mad | mbc | mbd | mcd | c1 | c1' | c2 | c2' | c3 | c3' |
| rab | rab | 1 | rad | rac | rd' | rc' | rd | rc | rb' | ra' | rb | ra | mcd | c1' | c2' | c2 | c1 | mab | mbd | mac | mbc | mad | c3' | c3 |
| rac | rac | rad | 1 | rab | rb' | rd | rc' | ra | rd' | rb | ra' | rc | c3' | mbd | c2 | c2' | mac | c3 | c1' | c1 | mad | mbc | mcd | mab |
| rad | rad | rac | rab | 1 | rc | rb | ra' | rd' | ra | rd | rc' | rb' | c3 | c1 | mbc | mad | c1' | c3' | mac | mbd | c2' | c2 | mab | mcd |
| ra | ra | rc | rd' | rb' | ra' | 1 | rac | rc' | rd | rad | rab | rb | mac | mad | mab | c3' | c2' | c1 | c2 | mbc | mcd | c3 | mbd | c1' |
| ra' | ra' | rd | rb | rc' | 1 | ra | rd' | rad | rab | rb' | rc | rac | mad | mab | mac | c1' | c3 | c2 | mcd | c3' | c1 | mbd | c2' | mbc |
| rb | rb | rc' | ra' | rd | rad | rc | rb' | 1 | rac | rd' | ra | rab | mbc | c3 | c1 | mbd | mab | c2' | c3' | mcd | mac | c1' | c2 | mad |
| rb' | rb' | rd' | rc | ra | rd | rac | 1 | rb | ra' | rab | rad | rc' | mbd | c2 | c3' | mab | mbc | c1' | mad | c2' | c3 | mcd | mac | c1 |
| rc | rc | ra | rb' | rd' | rb | rad | rab | rd | rc' | 1 | rac | ra' | c2' | mcd | c1' | mac | c3' | mbc | mab | c3 | mbd | c1 | mad | c2 |
| rc' | rc' | rb | rd | ra' | rab | rd' | ra | rac | 1 | rc | rb' | rad | c1 | mbc | c3 | mcd | c2 | mac | c2' | mad | c3' | mab | c1' | mbd |
| rd | rd | ra' | rc' | rb | rac | rb' | rc | rab | rad | ra | rd' | 1 | c1' | c2' | mcd | c3 | mad | mbd | mbc | c2 | mab | c3' | c1 | mac |
| rd' | rd' | rb' | ra | rc | rc' | rab | rad | ra' | rb | rac | 1 | rd | c2 | c3' | mbd | c1 | mcd | mad | c3 | mab | c1' | mac | mbc | c2' |
| mab | mab | mcd | c3 | c3' | mad | mac | mbd | mbc | c1' | c2 | c2' | c1 | 1 | ra | ra' | rb | rb' | rab | rd' | rc | rc' | rd | rac | rad |
| mac | mac | c1 | mbd | c1' | mab | mad | c2' | c3' | mbc | mcd | c3 | c2 | ra' | 1 | ra | rc | rac | rc' | rab | rad | rd' | rb' | rd | rb |
| mad | mad | c2 | c2' | mbc | mac | mab | c3 | c1' | c3' | c1 | mbd | mcd | ra | ra' | 1 | rad | rd | rd' | rc' | rb | rab | rac | rb' | rc |
| mbc | mbc | c2' | c2 | mad | c1 | c3 | mab | mbd | mcd | mac | c1' | c3' | rb' | rc' | rad | 1 | rb | rc | ra' | rd | rac | rab | ra | rd' |
| mbd | mbd | c1' | mac | c1 | c3' | c2 | mbc | mab | c2' | c3 | mcd | mad | rb | rac | rd' | rb' | 1 | rd | rad | rab | ra | rc | rc' | ra' |
| mcd | mcd | mab | c3' | c3 | c2' | c1' | c1 | c2 | mac | mbc | mad | mbd | rab | rc | rd | rc' | rd' | 1 | rb' | ra | rb | ra' | rad | rac |
| c1 | c1 | mac | c1' | mbd | c3 | mbc | c2 | mcd | mad | c3' | mab | c2' | rc' | rad | rb' | rd' | rab | ra' | rac | 1 | rc | ra | rb | rd |
| c1' | c1' | mbd | c1 | mac | mcd | c2' | mad | c3 | c2 | mab | c3' | mbc | rd | rab | rc | ra | rad | rb | 1 | rac | rb' | rd' | ra' | rc' |
| c2 | c2 | mad | mbc | c2' | mbd | c3' | mcd | c1 | mab | c1' | mac | c3 | rd' | rb | rac | rab | rc' | ra | rd | ra' | rad | 1 | rc | rb' |
| c2' | c2' | mbc | mad | c2 | c1' | mcd | c3' | mac | c3 | mbd | c1 | mab | rc | rd | rab | rac | ra' | rb' | rb | rc' | 1 | rad | rd' | ra |
| c3 | c3 | c3' | mab | mcd | mbc | c1 | c1' | mad | mbd | c2' | c2 | mac | rad | rb' | rc' | rd | ra | rac | rc | rd' | ra' | rb | rab | 1 |
| c3' | c3' | c3 | mcd | mab | c2 | mbd | mac | c2' | c1 | mad | mbc | c1' | rac | rd' | rb | ra' | rc | rad | ra | rb' | rd | rc' | 1 | rab |
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1(x,y,z) = (x,y,z)
i(x,y,z) = (-x,-y,-z) |
rx(x,y,z) = (x,-y,-z) half turn around Ox
r1 and r1' 120° rotations (opposite directions) |
mx(x,y,z) = (-x,y,z) = rx[i] = i[rx] plane reflection wrt (yOz)
f1 = r1[i] = i[r1] and f1' = r1'[i] = i[r1'] |
| Th | 1 | rx | ry | rz | r1 | r1' | r2 | r2' | r3 | r3' | r4 | r4' | i | mx | my | mz | f1 | f1' | f2 | f2' | f3 | f3' | f4 | f4' |
| 1 | 1 | rx | ry | rz | r1 | r1' | r2 | r2' | r3 | r3' | r4 | r4' | i | mx | my | mz | f1 | f1' | f2 | f2' | f3 | f3' | f4 | f4' |
| rx | rx | 1 | rz | ry | r4' | r3' | r4 | r3 | r2' | r1' | r2 | r1 | mx | i | mz | my | f4' | f3' | f4 | f3 | f2' | f1' | f2 | f1 |
| ry | ry | rz | 1 | rx | r2' | r4 | r3' | r1 | r4' | r2 | r1' | r3 | my | mz | i | mx | f2' | f4 | f3' | f1 | f4' | f2 | f1' | f3 |
| rz | rz | ry | rx | 1 | r3 | r2 | r1' | r4' | r1 | r4 | r3' | r2' | mz | my | mx | i | f3 | f2 | f1' | f4' | f1 | f4 | f3' | f2' |
| r1 | r1 | r3 | r4' | r2' | r1' | 1 | ry | r3' | r4 | rz | rx | r2 | f1 | f3 | f4' | f2' | f1' | i | my | f3' | f4 | mz | mx | f2 |
| r1' | r1' | r4 | r2 | r3' | 1 | r1 | r4' | rz | rx | r2' | r3 | ry | f1' | f4 | f2 | f3' | i | f1 | f4' | mz | mx | f2 | f3 | my |
| r2 | r2 | r3' | r1' | r4 | rz | r3 | r2' | 1 | ry | r4' | r1 | rx | f2 | f3' | f1' | f4 | mz | f3 | f2' | i | my | f4' | f1 | mx |
| r2' | r2' | r4' | r3 | r1 | r4 | ry | 1 | r2 | r1' | rx | rz | r3' | f2' | f4' | f3 | f1 | f4 | my | i | f2 | f1' | mx | mz | f3' |
| r3 | r3 | r1 | r2' | r4' | r2 | rz | rx | r4 | r3' | 1 | ry | r1' | f3 | f1 | f2' | f4' | f2 | mz | mx | f4 | f3' | i | my | f1' |
| r3' | r3' | r2 | r4 | r1' | rx | r4' | r1 | ry | 1 | r3 | r2' | rz | f3' | f2 | f4 | f1' | mx | f4' | f1 | mx | i | f3 | f2' | mz |
| r4 | r4 | r1' | r3' | r2 | ry | r2' | r3 | rx | rz | r1 | r4' | 1 | f4 | f1' | f3' | f2 | my | f2' | f3 | mx | mz | f1 | f4' | i |
| r4' | r4' | r2' | r1 | r3 | r3' | rx | rz | r1' | r2 | ry | 1 | r4 | f4' | f2' | f1 | f3 | f3' | mx | mz | f1' | f2 | my | i | f4 |
| i | i | mx | my | mz | f1 | f1' | f2 | f2' | f3 | f3' | f4 | f4' | 1 | rx | ry | rz | r1 | r1' | r2 | r2' | r3 | r3' | r4 | r4' |
| mx | mx | i | mz | my | f4' | f3' | f4 | f3 | f2' | f1' | f2 | f1 | rx | 1 | rz | ry | r4' | r3' | r4 | r3 | r2 | r1' | r2 | r1 |
| my | my | mz | i | mx | f2' | f4 | f3 | f1 | f4' | f2 | f1' | f3 | ry | rz | 1 | rx | r2' | r4 | r3 | r1 | r4' | r2 | r1' | r3 |
| mz | mz | my | mx | i | f3 | f2 | f1' | f4' | f1 | f4 | f3' | f2' | rz | ry | rx | 1 | r3 | r2 | r1' | r4' | r1 | r4 | r3' | r2' |
| f1 | f1 | f3 | f4' | f2' | f1' | i | my | f3' | f4 | mz | mx | f2 | r1 | r3 | r4' | r2' | r1' | 1 | ry | r3' | r4 | rz | rx | r2 |
| f1' | f1' | f4 | f2 | f3' | i | f1 | f4' | mz | mx | f2' | f3 | my | r1' | r4 | r2 | r3' | 1 | r1 | r4' | rz | rx | r2' | r3 | ry |
| f2 | f2 | f3' | f1' | f4 | mz | f3 | f2' | i | my | f4' | f1 | mx | r2 | r3' | r1' | r4 | rz | r3 | r2' | 1 | ry | r4' | r1 | rx |
| f2' | f2' | f4' | f3 | f1 | f4 | my | i | f2 | f1' | mx | mz | f3' | r2' | r4' | r3 | r1 | r4 | ry | 1 | r2 | r1' | rx | rz | r3' |
| f3 | f3 | f1 | f2' | f4' | f2 | mz | mx | f4 | f3' | i | my | f1' | r3 | r1 | r2' | r4' | r2 | rz | rx | r4 | r3' | 1 | ry | r1' |
| f3' | f3' | f2 | f4 | f1' | mx | f4' | f1 | my | i | f3 | f2' | mz | r3' | r2 | r4 | r1' | rx | r4' | r1 | ry | 1 | r3 | r2' | rz |
| f4 | f4 | f1' | f3' | f2 | my | f2' | f3 | mx | mz | f1 | f4' | i | r4 | r1' | r3' | r2 | ry | r2' | r3 | rx | rz | r1 | r4' | 1 |
| f4' | f4' | f2' | f1 | f3 | f3' | mx | mz | f1' | f2 | my | i | f4 | r4' | r2 | r1 | r3 | r3' | rx | rz | r1' | r2 | ry | 1 | r4 |
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convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects | April 2004 updated 12-04-2004 |