Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,20}

Atlas Canonical Name {4,20}*1280a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1280,90280)
Rank
3
Schläfli Type
{4,20}
Vertices, edges, …
32, 320, 160
Order of s0s1s2
40
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

2-fold

4-fold

5-fold

8-fold

10-fold

16-fold

20-fold

32-fold

40-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*s1*(s2*s1*s0)^2*s1*s2*s1*s0*s2*s1> of order 2

80 facets

16 vertex figures

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

90 facets

16 vertex figures

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

80 facets

16 vertex figures

P/N, where N=<(s1*s2)^10> of order 2

80 facets

18 vertex figures

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

50 facets

8 vertex figures

P/N, where N=<s0*s1*(s2*s1*s0)^2*s1*s2*s1*s0*s2*s1, s0*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*(s2*s1*s0)^2*s1*s2*s1*s0*s2*s1, (s1*s2)^10> of order 4

40 facets

10 vertex figures

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

45 facets

8 vertex figures

P/N, where N=<s0*(s2*s1*s0*s1)^3*s2*s1, s1*s0*(s2*s1)^2*s0*(s2*s1)^7*s2> of order 4

40 facets

9 vertex figures

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

40 facets

8 vertex figures

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

25 facets

5 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 11, 16)( 12, 17)( 13, 18)( 14, 19)( 15, 20)( 31, 36)( 32, 37)( 33, 38)( 34, 39)( 35, 40)( 41, 76)( 42, 77)( 43, 78)( 44, 79)( 45, 80)( 46, 71)( 47, 72)( 48, 73)( 49, 74)( 50, 75)( 51, 61)( 52, 62)( 53, 63)( 54, 64)( 55, 65)( 56, 66)( 57, 67)( 58, 68)( 59, 69)( 60, 70)( 91, 96)( 92, 97)( 93, 98)( 94, 99)( 95,100)(111,116)(112,117)(113,118)(114,119)(115,120)(121,156)(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)(130,155)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150);;
s1 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)( 21, 31)( 22, 35)( 23, 34)( 24, 33)( 25, 32)( 26, 36)( 27, 40)( 28, 39)( 29, 38)( 30, 37)( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)( 58, 59)( 61, 71)( 62, 75)( 63, 74)( 64, 73)( 65, 72)( 66, 76)( 67, 80)( 68, 79)( 69, 78)( 70, 77)( 81,121)( 82,125)( 83,124)( 84,123)( 85,122)( 86,126)( 87,130)( 88,129)( 89,128)( 90,127)( 91,131)( 92,135)( 93,134)( 94,133)( 95,132)( 96,136)( 97,140)( 98,139)( 99,138)(100,137)(101,151)(102,155)(103,154)(104,153)(105,152)(106,156)(107,160)(108,159)(109,158)(110,157)(111,141)(112,145)(113,144)(114,143)(115,142)(116,146)(117,150)(118,149)(119,148)(120,147);;
s2 := (  1, 83)(  2, 82)(  3, 81)(  4, 85)(  5, 84)(  6, 88)(  7, 87)(  8, 86)(  9, 90)( 10, 89)( 11, 98)( 12, 97)( 13, 96)( 14,100)( 15, 99)( 16, 93)( 17, 92)( 18, 91)( 19, 95)( 20, 94)( 21,108)( 22,107)( 23,106)( 24,110)( 25,109)( 26,103)( 27,102)( 28,101)( 29,105)( 30,104)( 31,113)( 32,112)( 33,111)( 34,115)( 35,114)( 36,118)( 37,117)( 38,116)( 39,120)( 40,119)( 41,123)( 42,122)( 43,121)( 44,125)( 45,124)( 46,128)( 47,127)( 48,126)( 49,130)( 50,129)( 51,138)( 52,137)( 53,136)( 54,140)( 55,139)( 56,133)( 57,132)( 58,131)( 59,135)( 60,134)( 61,148)( 62,147)( 63,146)( 64,150)( 65,149)( 66,143)( 67,142)( 68,141)( 69,145)( 70,144)( 71,153)( 72,152)( 73,151)( 74,155)( 75,154)( 76,158)( 77,157)( 78,156)( 79,160)( 80,159);;
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*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(160)!( 11, 16)( 12, 17)( 13, 18)( 14, 19)( 15, 20)( 31, 36)( 32, 37)( 33, 38)( 34, 39)( 35, 40)( 41, 76)( 42, 77)( 43, 78)( 44, 79)( 45, 80)( 46, 71)( 47, 72)( 48, 73)( 49, 74)( 50, 75)( 51, 61)( 52, 62)( 53, 63)( 54, 64)( 55, 65)( 56, 66)( 57, 67)( 58, 68)( 59, 69)( 60, 70)( 91, 96)( 92, 97)( 93, 98)( 94, 99)( 95,100)(111,116)(112,117)(113,118)(114,119)(115,120)(121,156)(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)(130,155)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150);
s1 := Sym(160)!(  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)( 21, 31)( 22, 35)( 23, 34)( 24, 33)( 25, 32)( 26, 36)( 27, 40)( 28, 39)( 29, 38)( 30, 37)( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)( 58, 59)( 61, 71)( 62, 75)( 63, 74)( 64, 73)( 65, 72)( 66, 76)( 67, 80)( 68, 79)( 69, 78)( 70, 77)( 81,121)( 82,125)( 83,124)( 84,123)( 85,122)( 86,126)( 87,130)( 88,129)( 89,128)( 90,127)( 91,131)( 92,135)( 93,134)( 94,133)( 95,132)( 96,136)( 97,140)( 98,139)( 99,138)(100,137)(101,151)(102,155)(103,154)(104,153)(105,152)(106,156)(107,160)(108,159)(109,158)(110,157)(111,141)(112,145)(113,144)(114,143)(115,142)(116,146)(117,150)(118,149)(119,148)(120,147);
s2 := Sym(160)!(  1, 83)(  2, 82)(  3, 81)(  4, 85)(  5, 84)(  6, 88)(  7, 87)(  8, 86)(  9, 90)( 10, 89)( 11, 98)( 12, 97)( 13, 96)( 14,100)( 15, 99)( 16, 93)( 17, 92)( 18, 91)( 19, 95)( 20, 94)( 21,108)( 22,107)( 23,106)( 24,110)( 25,109)( 26,103)( 27,102)( 28,101)( 29,105)( 30,104)( 31,113)( 32,112)( 33,111)( 34,115)( 35,114)( 36,118)( 37,117)( 38,116)( 39,120)( 40,119)( 41,123)( 42,122)( 43,121)( 44,125)( 45,124)( 46,128)( 47,127)( 48,126)( 49,130)( 50,129)( 51,138)( 52,137)( 53,136)( 54,140)( 55,139)( 56,133)( 57,132)( 58,131)( 59,135)( 60,134)( 61,148)( 62,147)( 63,146)( 64,150)( 65,149)( 66,143)( 67,142)( 68,141)( 69,145)( 70,144)( 71,153)( 72,152)( 73,151)( 74,155)( 75,154)( 76,158)( 77,157)( 78,156)( 79,160)( 80,159);
poly := sub<Sym(160)|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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*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