the two groups of tetrahedral symmetries

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.

f g
 f   f2=f[f]  g[f]
g f[g]  g2=g[g] 
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

 
The group Th  also contains 24 isometries and the same rotations, but its structure is different: it is isomorphic to  A4xC2  (the 12 indirect isometries are the combinations of the rotations with the central inversion).
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 et 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]   et   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