Part of the Atlas of Small Regular Polytopes

Polytope of Type {20,4}

Atlas Canonical Name {20,4}*2000b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(2000,919)
Rank
3
Schläfli Type
{20,4}
Vertices, edges, …
250, 500, 50
Order of s0s1s2
10
Order of s0s1s2s1
10
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

5-fold

10-fold

50-fold

100-fold

250-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)^5> of order 2

25 facets

125 vertex figures

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

25 facets

125 vertex figures

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

26 facets

130 vertex figures

P/N, where N=<(s0*s1)^10, s0*s1*s0*s2*(s1*s0)^2*(s2*s1*s0*s1)^3*s2> of order 4

13 facets

65 vertex figures

P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*(s1*s2)^2> of order 5

10 facets

50 vertex figures

P/N, where N=<(s0*s1)^4*s0*s2*(s1*s0)^2*s2*s1*s0*s1> of order 5

10 facets

50 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2> of order 5

10 facets

50 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^3*s1*s2*s1*s0*s2*s1*s2> of order 5

10 facets

50 vertex figures

P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2> of order 5

10 facets

50 vertex figures

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

6 facets

30 vertex figures

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

5 facets

25 vertex figures

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

6 facets

30 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle