Part of the Atlas of Small Regular Polytopes

Polytope of Type {40,4}

Atlas Canonical Name {40,4}*1280f

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1280,1116431)
Rank
3
Schläfli Type
{40,4}
Vertices, edges, …
160, 320, 16
Order of s0s1s2
40
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable
  • Self-Petrie

Quotients maximal quotients in bold

2-fold

4-fold

8-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*s2> of order 2

8 facets

96 vertex figures

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

8 facets

80 vertex figures

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

8 facets

80 vertex figures

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

4 facets

56 vertex figures

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

4 facets

48 vertex figures

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

4 facets

48 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle