Part of the Atlas of Small Regular Polytopes

Polytope of Type {60,6}

Atlas Canonical Name {60,6}*1080a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1080,283)
Rank
3
Schläfli Type
{60,6}
Vertices, edges, …
90, 270, 9
Order of s0s1s2
60
Order of s0s1s2s1
6
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

3-fold

5-fold

15-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

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

References

None.

to this polytope.

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