In geometry , this family of uniform 4-polytopes has diploid hexadecachoric symmetry ,[1] [4,3,3], of order 24*16=384: 4!=24 permutations of the four axes, 24 =16 for reflection in each axis. There are 3 small index subgroups, with the first two generate uniform 4-polytopes which are also repeated in other families, [1+ ,4,3,3], [4,(3,3)+ ], and [4,3,3]+ , all order 192.
B4 symmetry polytopes
Name
tesseract
rectified tesseract
truncated tesseract
cantellated tesseract
runcinated tesseract
bitruncated tesseract
cantitruncated tesseract
runcitruncated tesseract
omnitruncated tesseract
Coxeter diagram
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Schläfli symbol
{4,3,3}
t1 {4,3,3} r{4,3,3}
t0,1 {4,3,3} t{4,3,3}
t0,2 {4,3,3} rr{4,3,3}
t0,3 {4,3,3}
t1,2 {4,3,3} 2t{4,3,3}
t0,1,2 {4,3,3} tr{4,3,3}
t0,1,3 {4,3,3}
t0,1,2,3 {4,3,3}
Schlegel diagram
B4
Name
16-cell
rectified 16-cell
truncated 16-cell
cantellated 16-cell
runcinated 16-cell
bitruncated 16-cell
cantitruncated 16-cell
runcitruncated 16-cell
omnitruncated 16-cell
Coxeter diagram
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Schläfli symbol
{3,3,4}
t1 {3,3,4} r{3,3,4}
t0,1 {3,3,4} t{3,3,4}
t0,2 {3,3,4} rr{3,3,4}
t0,3 {3,3,4}
t1,2 {3,3,4} 2t{3,3,4}
t0,1,2 {3,3,4} tr{3,3,4}
t0,1,3 {3,3,4}
t0,1,2,3 {3,3,4}
Schlegel diagram
B4
↑ Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups