Part of the Atlas of Small Regular Polytopes

Polytope of Type {21,8}

Atlas Canonical Name {21,8}*1344

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1344,6454)
Rank
3
Schläfli Type
{21,8}
Vertices, edges, …
84, 336, 32
Order of s0s1s2
42
Order of s0s1s2s1
8
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

4-fold

7-fold

8-fold

16-fold

28-fold

48-fold

56-fold

112-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s1*s0*(s1*s2)^3*s1*s0*s2*s1> of order 2

16 facets

56 vertex figures

P/N, where N=<(s0*(s2*s1)^2)^2*s2> of order 2

16 facets

42 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2> of order 4

8 facets

42 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 4

8 facets

28 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2, (s0*(s2*s1)^2)^2*s2> of order 4

8 facets

28 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  3,  4)(  5,  6)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 97)( 18, 98)( 19,100)( 20, 99)( 21,102)( 22,101)( 23,103)( 24,104)( 25,109)( 26,110)( 27,112)( 28,111)( 29,105)( 30,106)( 31,108)( 32,107)( 33, 81)( 34, 82)( 35, 84)( 36, 83)( 37, 86)( 38, 85)( 39, 87)( 40, 88)( 41, 93)( 42, 94)( 43, 96)( 44, 95)( 45, 89)( 46, 90)( 47, 92)( 48, 91)( 49, 65)( 50, 66)( 51, 68)( 52, 67)( 53, 70)( 54, 69)( 55, 71)( 56, 72)( 57, 77)( 58, 78)( 59, 80)( 60, 79)( 61, 73)( 62, 74)( 63, 76)( 64, 75);;
s1 := (  1, 17)(  2, 20)(  3, 19)(  4, 18)(  5, 30)(  6, 31)(  7, 32)(  8, 29)(  9, 25)( 10, 28)( 11, 27)( 12, 26)( 13, 24)( 14, 21)( 15, 22)( 16, 23)( 33, 97)( 34,100)( 35, 99)( 36, 98)( 37,110)( 38,111)( 39,112)( 40,109)( 41,105)( 42,108)( 43,107)( 44,106)( 45,104)( 46,101)( 47,102)( 48,103)( 49, 81)( 50, 84)( 51, 83)( 52, 82)( 53, 94)( 54, 95)( 55, 96)( 56, 93)( 57, 89)( 58, 92)( 59, 91)( 60, 90)( 61, 88)( 62, 85)( 63, 86)( 64, 87)( 66, 68)( 69, 78)( 70, 79)( 71, 80)( 72, 77)( 74, 76);;
s2 := (  1,  7)(  2,  8)(  3,  5)(  4,  6)(  9, 13)( 10, 14)( 11, 15)( 12, 16)( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 33, 39)( 34, 40)( 35, 37)( 36, 38)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 57, 61)( 58, 62)( 59, 63)( 60, 64)( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 81, 87)( 82, 88)( 83, 85)( 84, 86)( 89, 93)( 90, 94)( 91, 95)( 92, 96)( 97,103)( 98,104)( 99,101)(100,102)(105,109)(106,110)(107,111)(108,112);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s0, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(112)!(  3,  4)(  5,  6)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 97)( 18, 98)( 19,100)( 20, 99)( 21,102)( 22,101)( 23,103)( 24,104)( 25,109)( 26,110)( 27,112)( 28,111)( 29,105)( 30,106)( 31,108)( 32,107)( 33, 81)( 34, 82)( 35, 84)( 36, 83)( 37, 86)( 38, 85)( 39, 87)( 40, 88)( 41, 93)( 42, 94)( 43, 96)( 44, 95)( 45, 89)( 46, 90)( 47, 92)( 48, 91)( 49, 65)( 50, 66)( 51, 68)( 52, 67)( 53, 70)( 54, 69)( 55, 71)( 56, 72)( 57, 77)( 58, 78)( 59, 80)( 60, 79)( 61, 73)( 62, 74)( 63, 76)( 64, 75);
s1 := Sym(112)!(  1, 17)(  2, 20)(  3, 19)(  4, 18)(  5, 30)(  6, 31)(  7, 32)(  8, 29)(  9, 25)( 10, 28)( 11, 27)( 12, 26)( 13, 24)( 14, 21)( 15, 22)( 16, 23)( 33, 97)( 34,100)( 35, 99)( 36, 98)( 37,110)( 38,111)( 39,112)( 40,109)( 41,105)( 42,108)( 43,107)( 44,106)( 45,104)( 46,101)( 47,102)( 48,103)( 49, 81)( 50, 84)( 51, 83)( 52, 82)( 53, 94)( 54, 95)( 55, 96)( 56, 93)( 57, 89)( 58, 92)( 59, 91)( 60, 90)( 61, 88)( 62, 85)( 63, 86)( 64, 87)( 66, 68)( 69, 78)( 70, 79)( 71, 80)( 72, 77)( 74, 76);
s2 := Sym(112)!(  1,  7)(  2,  8)(  3,  5)(  4,  6)(  9, 13)( 10, 14)( 11, 15)( 12, 16)( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 33, 39)( 34, 40)( 35, 37)( 36, 38)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 57, 61)( 58, 62)( 59, 63)( 60, 64)( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 81, 87)( 82, 88)( 83, 85)( 84, 86)( 89, 93)( 90, 94)( 91, 95)( 92, 96)( 97,103)( 98,104)( 99,101)(100,102)(105,109)(106,110)(107,111)(108,112);
poly := sub<Sym(112)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s0, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References

None.

to this polytope.

Twisty Puzzle