Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,20}

Atlas Canonical Name {4,20}*1280c

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1280,1116447)
Rank
3
Schläfli Type
{4,20}
Vertices, edges, …
32, 320, 160
Order of s0s1s2
20
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

16-fold

32-fold

64-fold

80-fold

160-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*s2*s1*s0*s1*s2> of order 2

96 facets

16 vertex figures

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

80 facets

16 vertex figures

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

80 facets

16 vertex figures

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

80 facets

16 vertex figures

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

80 facets

16 vertex figures

P/N, where N=<s1*s0*(s2*s1)^3*s0*(s2*s1)^6*s2> of order 2

80 facets

16 vertex figures

P/N, where N=<(s1*s0*s2)^2*s1*s0*(s2*s1)^7*s2> of order 2

80 facets

16 vertex figures

P/N, where N=<s0*s1*(s2*s1*s0)^2*(s2*s1)^7*s2> of order 2

80 facets

16 vertex figures

P/N, where N=<s0*s1*(s2*s1*s0)^3*(s2*s1)^6*s2> of order 2

80 facets

16 vertex figures

P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4

48 facets

8 vertex figures

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

40 facets

8 vertex figures

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

48 facets

8 vertex figures

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

48 facets

8 vertex figures

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

40 facets

8 vertex figures

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

40 facets

8 vertex figures

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

40 facets

8 vertex figures

P/N, where N=<(s0*s1)^2, (s0*s2*s1)^3*s0*(s1*s2)^3> of order 4

48 facets

8 vertex figures

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

40 facets

8 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2, s0*(s2*s1)^2*s0*s2*s1*s0*(s2*s1)^3*s2> of order 4

40 facets

8 vertex figures

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

48 facets

8 vertex figures

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

56 facets

8 vertex figures

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

40 facets

8 vertex figures

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

40 facets

8 vertex figures

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

48 facets

8 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2, s0*s1*(s2*s1*s0)^2*(s2*s1)^6> of order 4

40 facets

8 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2, s0*s1*(s2*s1*s0)^2*(s2*s1)^7*s2> of order 4

40 facets

8 vertex figures

P/N, where N=<(s0*s2*s1)^3*s0*(s1*s2)^3, s0*s1*s0*(s2*s1)^3*s0*(s2*s1)^6*s2> of order 4

40 facets

8 vertex figures

P/N, where N=<s1*s0*s1*(s2*s1*s0)^3*(s1*s2)^3, s0*s1*s0*(s2*s1)^3*s0*(s2*s1)^6*s2> of order 4

40 facets

8 vertex figures

P/N, where N=<(s0*s2*s1)^3*s0*(s1*s2)^3, s1*s0*(s2*s1)^3*s0*(s2*s1)^6*s2> of order 4

40 facets

8 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2, s0*(s1*s2)^2*s1*s0*(s2*s1)^7*s2> of order 4

40 facets

8 vertex figures

P/N, where N=<s0*s1*(s2*s1*s0)^2*(s1*s2)^2*s1, s0*s1*s0*(s2*s1)^3*s0*(s2*s1)^6*s2> of order 4

40 facets

8 vertex figures

P/N, where N=<s0*s1*(s2*s1*s0)^2*(s1*s2)^2*s1, s1*s0*s2*s1*s0*(s2*s1)^2*s0*(s2*s1)^6*s2> of order 4

40 facets

8 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1)^2*s0*(s1*s2)^2, s1*s0*(s2*s1)^3*s0*(s2*s1)^6*s2> of order 4

40 facets

8 vertex figures

P/N, where N=<(s1*s0*s2)^2*s1*s0*(s1*s2)^2*s1, s0*s1*s0*(s2*s1)^3*s0*(s2*s1)^6*s2> of order 4

40 facets

8 vertex figures

P/N, where N=<(s0*s1)^2*s2*s1*s0*s1*s2, s0*s1*(s2*s1*s0)^2*(s2*s1)^7*s2> of order 4

40 facets

8 vertex figures

P/N, where N=<(s1*s0*s2)^2*s1*s0*(s1*s2)^2*s1, s1*s0*(s2*s1)^3*s0*(s2*s1)^6*s2> of order 4

40 facets

8 vertex figures

P/N, where N=<(s0*s2*s1)^2*s0*(s1*s2)^2, s1*s0*(s2*s1)^3*s0*(s2*s1)^6*s2> of order 4

40 facets

8 vertex figures

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

48 facets

8 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s0*(s1*s0*s2)^2*s1*s0*(s2*s1)^7*s2> of order 4

40 facets

8 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s0*s1*s0*(s2*s1)^2*s0*(s2*s1)^7*s2> of order 4

40 facets

8 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s0*(s1*s2)^2*s1*s0*(s2*s1)^7*s2> of order 4

40 facets

8 vertex figures

P/N, where N=<(s0*s1)^2, s0*s1*(s2*s1*s0)^3*(s2*s1)^6*s2> of order 4

48 facets

8 vertex figures

P/N, where N=<s0*s2*s1*s0*s1*s2, s0*s1*(s2*s1*s0)^2*(s2*s1)^7*s2> of order 4

48 facets

8 vertex figures

P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s1*s0*(s2*s1)^3*s0*(s2*s1)^6*s2> of order 4

48 facets

8 vertex figures

P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*(s2*s1)^2*s0*(s2*s1)^7*s2> of order 4

48 facets

8 vertex figures

P/N, where N=<s1*(s2*s1*s0)^2*(s1*s2)^2*s1, s1*s0*(s2*s1)^3*s0*(s2*s1)^6*s2> of order 4

48 facets

8 vertex figures

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

56 facets

8 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, (s1*s0*s2)^2*s1*s0*(s2*s1)^7*s2> of order 4

40 facets

8 vertex figures

P/N, where N=<(s0*s1)^2, s0*s2*s1*s0*s1*s2, s1*(s2*s1*s0)^2*(s1*s2)^2*s1> of order 8

32 facets

4 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s0*s1*s0*s2*s1*s0*s1*s2*s1, (s0*s2*s1)^2*s0*(s1*s2)^2> of order 8

24 facets

4 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, (s1*s0*s1*s2)^2, (s0*s2*s1)^2*s0*(s1*s2)^2> of order 8

20 facets

4 vertex figures

P/N, where N=<s0*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1, (s0*s1)^2*(s2*s1)^2*s0*(s1*s2)^2> of order 8

28 facets

4 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s1*(s2*s1*s0)^2*(s1*s2)^2*s1, s0*s1*s2*s1*s0*(s2*s1)^8> of order 8

24 facets

4 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s1*(s2*s1*s0)^2*(s1*s2)^2*s1, s0*s1*s0*s2*s1*s0*(s2*s1)^8> of order 8

24 facets

4 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s0*s1*s0*s2*s1*s0*(s2*s1)^3> of order 8

24 facets

4 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s0*s1*s2*s1*s0*(s2*s1)^3> of order 8

24 facets

4 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s1*s0*s2*s1*s0*s1*s2*s1, s0*(s1*s2)^2*s1*s0*(s2*s1)^7*s2> of order 8

24 facets

4 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, (s1*s0*s1*s2)^2, s0*s1*s0*(s2*s1)^2*s0*(s2*s1)^7*s2> of order 8

20 facets

4 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, (s0*s2*s1)^2*s0*(s1*s2)^2, s0*s1*s2*s1*s0*(s2*s1)^8*s2> of order 8

20 facets

4 vertex figures

P/N, where N=<s0*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*(s2*s1)^2*s0*(s2*s1)^7*s2> of order 8

28 facets

4 vertex figures

P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1, s1*(s2*s1*s0)^2*(s1*s2)^2*s1, s1*s0*(s2*s1)^3*s0*(s2*s1)^6*s2> of order 8

24 facets

4 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, (s1*s0*s2)^2*s1*s0*(s1*s2)^2*s1, s0*s1*s2*s1*s0*(s2*s1)^8*s2> of order 8

20 facets

4 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s1*(s2*s1*s0)^2*(s1*s2)^2*s1, s0*s1*s2*s1*s0*(s2*s1)^8*s2> of order 8

24 facets

4 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s1*(s2*s1*s0)^2*(s1*s2)^2*s1, s0*s1*s0*s2*s1*s0*(s2*s1)^8*s2> of order 8

24 facets

4 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2, (s1*s0*s2)^2*s1*s0*(s1*s2)^2*s1, s0*(s1*s2)^2*s1*s0*(s2*s1)^7*s2> of order 8

20 facets

4 vertex figures

P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, (s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2, s0*(s1*s2)^2*s1*s0*(s2*s1)^7*s2> of order 8

24 facets

4 vertex figures

P/N, where N=<(s0*s2*s1)^2*s0*(s1*s2)^2, (s1*s0*s2)^2*s1*s0*(s1*s2)^2*s1, s0*s1*s2*s1*s0*(s2*s1)^8*s2> of order 8

20 facets

4 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s1*s0*s2*s1*s0*s1*s2*s1, s0*(s2*s1)^2*s0*(s1*s2)^2> of order 8

24 facets

4 vertex figures

P/N, where N=<s1*(s2*s1*s0)^2*(s1*s2)^2*s1, (s0*s1)^2*(s2*s1*s0)^2*s1*s2*s1, s0*s1*s2*s1*s0*(s2*s1)^8> of order 8

24 facets

4 vertex figures

P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*s1*s2, s2*s1*s0*(s2*s1)^2*s0*(s2*s1)^4*s2> of order 8

24 facets

4 vertex figures

P/N, where N=<(s0*s1)^2, s0*s2*s1*s0*s1*s2, (s0*s1*s2*s1)^2> of order 8

32 facets

4 vertex figures

Representations

Permutation Representation (GAP)
s0 := (  1, 61)(  2, 62)(  3, 63)(  4, 64)(  5, 57)(  6, 58)(  7, 59)(  8, 60)(  9, 53)( 10, 54)( 11, 55)( 12, 56)( 13, 49)( 14, 50)( 15, 51)( 16, 52)( 17, 45)( 18, 46)( 19, 47)( 20, 48)( 21, 41)( 22, 42)( 23, 43)( 24, 44)( 25, 37)( 26, 38)( 27, 39)( 28, 40)( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 65,125)( 66,126)( 67,127)( 68,128)( 69,121)( 70,122)( 71,123)( 72,124)( 73,117)( 74,118)( 75,119)( 76,120)( 77,113)( 78,114)( 79,115)( 80,116)( 81,109)( 82,110)( 83,111)( 84,112)( 85,105)( 86,106)( 87,107)( 88,108)( 89,101)( 90,102)( 91,103)( 92,104)( 93, 97)( 94, 98)( 95, 99)( 96,100);;
s1 := (  3,  4)(  7,  8)(  9, 97)( 10, 98)( 11,100)( 12, 99)( 13,101)( 14,102)( 15,104)( 16,103)( 17, 57)( 18, 58)( 19, 60)( 20, 59)( 21, 61)( 22, 62)( 23, 64)( 24, 63)( 25, 89)( 26, 90)( 27, 92)( 28, 91)( 29, 93)( 30, 94)( 31, 96)( 32, 95)( 33, 73)( 34, 74)( 35, 76)( 36, 75)( 37, 77)( 38, 78)( 39, 80)( 40, 79)( 43, 44)( 47, 48)( 49,113)( 50,114)( 51,116)( 52,115)( 53,117)( 54,118)( 55,120)( 56,119)( 67, 68)( 71, 72)( 81,121)( 82,122)( 83,124)( 84,123)( 85,125)( 86,126)( 87,128)( 88,127)(107,108)(111,112);;
s2 := (  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)( 17, 28)( 18, 27)( 19, 26)( 20, 25)( 21, 32)( 22, 31)( 23, 30)( 24, 29)( 33, 44)( 34, 43)( 35, 42)( 36, 41)( 37, 48)( 38, 47)( 39, 46)( 40, 45)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)( 65,124)( 66,123)( 67,122)( 68,121)( 69,128)( 70,127)( 71,126)( 72,125)( 73,116)( 74,115)( 75,114)( 76,113)( 77,120)( 78,119)( 79,118)( 80,117)( 81,100)( 82, 99)( 83, 98)( 84, 97)( 85,104)( 86,103)( 87,102)( 88,101)( 89,108)( 90,107)( 91,106)( 92,105)( 93,112)( 94,111)( 95,110)( 96,109);;
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(128)!(  1, 61)(  2, 62)(  3, 63)(  4, 64)(  5, 57)(  6, 58)(  7, 59)(  8, 60)(  9, 53)( 10, 54)( 11, 55)( 12, 56)( 13, 49)( 14, 50)( 15, 51)( 16, 52)( 17, 45)( 18, 46)( 19, 47)( 20, 48)( 21, 41)( 22, 42)( 23, 43)( 24, 44)( 25, 37)( 26, 38)( 27, 39)( 28, 40)( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 65,125)( 66,126)( 67,127)( 68,128)( 69,121)( 70,122)( 71,123)( 72,124)( 73,117)( 74,118)( 75,119)( 76,120)( 77,113)( 78,114)( 79,115)( 80,116)( 81,109)( 82,110)( 83,111)( 84,112)( 85,105)( 86,106)( 87,107)( 88,108)( 89,101)( 90,102)( 91,103)( 92,104)( 93, 97)( 94, 98)( 95, 99)( 96,100);
s1 := Sym(128)!(  3,  4)(  7,  8)(  9, 97)( 10, 98)( 11,100)( 12, 99)( 13,101)( 14,102)( 15,104)( 16,103)( 17, 57)( 18, 58)( 19, 60)( 20, 59)( 21, 61)( 22, 62)( 23, 64)( 24, 63)( 25, 89)( 26, 90)( 27, 92)( 28, 91)( 29, 93)( 30, 94)( 31, 96)( 32, 95)( 33, 73)( 34, 74)( 35, 76)( 36, 75)( 37, 77)( 38, 78)( 39, 80)( 40, 79)( 43, 44)( 47, 48)( 49,113)( 50,114)( 51,116)( 52,115)( 53,117)( 54,118)( 55,120)( 56,119)( 67, 68)( 71, 72)( 81,121)( 82,122)( 83,124)( 84,123)( 85,125)( 86,126)( 87,128)( 88,127)(107,108)(111,112);
s2 := Sym(128)!(  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)( 17, 28)( 18, 27)( 19, 26)( 20, 25)( 21, 32)( 22, 31)( 23, 30)( 24, 29)( 33, 44)( 34, 43)( 35, 42)( 36, 41)( 37, 48)( 38, 47)( 39, 46)( 40, 45)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)( 65,124)( 66,123)( 67,122)( 68,121)( 69,128)( 70,127)( 71,126)( 72,125)( 73,116)( 74,115)( 75,114)( 76,113)( 77,120)( 78,119)( 79,118)( 80,117)( 81,100)( 82, 99)( 83, 98)( 84, 97)( 85,104)( 86,103)( 87,102)( 88,101)( 89,108)( 90,107)( 91,106)( 92,105)( 93,112)( 94,111)( 95,110)( 96,109);
poly := sub<Sym(128)|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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 

References

None.

to this polytope.

Twisty Puzzle