Part of the Atlas of Small Regular Polytopes

Polytope of Type {21,6}

Atlas Canonical Name {21,6}*1344

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1344,6320)
Rank
3
Schläfli Type
{21,6}
Vertices, edges, …
112, 336, 32
Order of s0s1s2
56
Order of s0s1s2s1
6
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

28-fold

48-fold

56-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=<(s0*s1)^2*(s0*s2*s1)^4*s0*s1> of order 2

16 facets

56 vertex figures

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

8 facets

28 vertex figures

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

8 facets

28 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle