Part of the Atlas of Small Regular Polytopes

Polytope of Type {20,4}

Atlas Canonical Name {20,4}*1280a

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Overview

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

16 facets

80 vertex figures

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

16 facets

90 vertex figures

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

16 facets

80 vertex figures

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

18 facets

80 vertex figures

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

8 facets

50 vertex figures

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

8 facets

40 vertex figures

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

10 facets

40 vertex figures

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

8 facets

45 vertex figures

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

9 facets

40 vertex figures

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

8 facets

40 vertex figures

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

5 facets

25 vertex figures

Representations

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

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