Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,20}

Atlas Canonical Name {4,20}*1280e

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1280,1116457)
Rank
3
Schläfli Type
{4,20}
Vertices, edges, …
32, 320, 160
Order of s0s1s2
20
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

8-fold

32-fold

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

80 facets

16 vertex figures

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

80 facets

16 vertex figures

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

80 facets

16 vertex figures

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

80 facets

16 vertex figures

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

40 facets

8 vertex figures

Representations

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

References

None.

to this polytope.

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