Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,21}

Atlas Canonical Name {6,21}*1344

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1344,6320)
Rank
3
Schläfli Type
{6,21}
Vertices, edges, …
32, 336, 112
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=<s1*(s2*s1*s0)^4*(s2*s1)^2*s2> of order 2

56 facets

16 vertex figures

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

28 facets

8 vertex figures

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

28 facets

8 vertex figures

Representations

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

References

None.

to this polytope.

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