Polytope of Type {30,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,6}*1620a
if this polytope has a name.
Group : SmallGroup(1620,135)
Rank : 3
Schlafli Type : {30,6}
Number of vertices, edges, etc : 135, 405, 27
Order of s0s1s2 : 45
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {30,6}*540
   5-fold quotients : {6,6}*324a
   15-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2> of order 3.
      15 facets:
         9 of {10}*20
         6 of {30}*60
      45 vertex figures:
         45 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 3.
      9 facets:
         9 of {30}*60
      45 vertex figures:
         45 of {6}*12

Permutation Representation (GAP) : 
s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 37)( 11, 39)( 12, 38)( 13, 43)( 14, 45)( 15, 44)( 16, 40)( 17, 42)( 18, 41)( 19, 28)( 20, 30)( 21, 29)( 22, 34)( 23, 36)( 24, 35)( 25, 31)( 26, 33)( 27, 32)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)( 62, 87)( 63, 86)( 64, 73)( 65, 75)( 66, 74)( 67, 79)( 68, 81)( 69, 80)( 70, 76)( 71, 78)( 72, 77)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(100,127)(101,129)(102,128)(103,133)(104,135)(105,134)(106,130)(107,132)(108,131)(109,118)(110,120)(111,119)(112,124)(113,126)(114,125)(115,121)(116,123)(117,122);;
s1 := (  1, 10)(  2, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)(  8, 18)(  9, 17)( 19, 37)( 20, 39)( 21, 38)( 22, 40)( 23, 42)( 24, 41)( 25, 43)( 26, 45)( 27, 44)( 29, 30)( 32, 33)( 35, 36)( 46,107)( 47,106)( 48,108)( 49,101)( 50,100)( 51,102)( 52,104)( 53,103)( 54,105)( 55, 98)( 56, 97)( 57, 99)( 58, 92)( 59, 91)( 60, 93)( 61, 95)( 62, 94)( 63, 96)( 64,134)( 65,133)( 66,135)( 67,128)( 68,127)( 69,129)( 70,131)( 71,130)( 72,132)( 73,125)( 74,124)( 75,126)( 76,119)( 77,118)( 78,120)( 79,122)( 80,121)( 81,123)( 82,116)( 83,115)( 84,117)( 85,110)( 86,109)( 87,111)( 88,113)( 89,112)( 90,114);;
s2 := (  1, 46)(  2, 47)(  3, 48)(  4, 53)(  5, 54)(  6, 52)(  7, 51)(  8, 49)(  9, 50)( 10, 55)( 11, 56)( 12, 57)( 13, 62)( 14, 63)( 15, 61)( 16, 60)( 17, 58)( 18, 59)( 19, 64)( 20, 65)( 21, 66)( 22, 71)( 23, 72)( 24, 70)( 25, 69)( 26, 67)( 27, 68)( 28, 73)( 29, 74)( 30, 75)( 31, 80)( 32, 81)( 33, 79)( 34, 78)( 35, 76)( 36, 77)( 37, 82)( 38, 83)( 39, 84)( 40, 89)( 41, 90)( 42, 88)( 43, 87)( 44, 85)( 45, 86)( 94, 98)( 95, 99)( 96, 97)(103,107)(104,108)(105,106)(112,116)(113,117)(114,115)(121,125)(122,126)(123,124)(130,134)(131,135)(132,133);;
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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(135)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 37)( 11, 39)( 12, 38)( 13, 43)( 14, 45)( 15, 44)( 16, 40)( 17, 42)( 18, 41)( 19, 28)( 20, 30)( 21, 29)( 22, 34)( 23, 36)( 24, 35)( 25, 31)( 26, 33)( 27, 32)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)( 62, 87)( 63, 86)( 64, 73)( 65, 75)( 66, 74)( 67, 79)( 68, 81)( 69, 80)( 70, 76)( 71, 78)( 72, 77)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(100,127)(101,129)(102,128)(103,133)(104,135)(105,134)(106,130)(107,132)(108,131)(109,118)(110,120)(111,119)(112,124)(113,126)(114,125)(115,121)(116,123)(117,122);
s1 := Sym(135)!(  1, 10)(  2, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)(  8, 18)(  9, 17)( 19, 37)( 20, 39)( 21, 38)( 22, 40)( 23, 42)( 24, 41)( 25, 43)( 26, 45)( 27, 44)( 29, 30)( 32, 33)( 35, 36)( 46,107)( 47,106)( 48,108)( 49,101)( 50,100)( 51,102)( 52,104)( 53,103)( 54,105)( 55, 98)( 56, 97)( 57, 99)( 58, 92)( 59, 91)( 60, 93)( 61, 95)( 62, 94)( 63, 96)( 64,134)( 65,133)( 66,135)( 67,128)( 68,127)( 69,129)( 70,131)( 71,130)( 72,132)( 73,125)( 74,124)( 75,126)( 76,119)( 77,118)( 78,120)( 79,122)( 80,121)( 81,123)( 82,116)( 83,115)( 84,117)( 85,110)( 86,109)( 87,111)( 88,113)( 89,112)( 90,114);
s2 := Sym(135)!(  1, 46)(  2, 47)(  3, 48)(  4, 53)(  5, 54)(  6, 52)(  7, 51)(  8, 49)(  9, 50)( 10, 55)( 11, 56)( 12, 57)( 13, 62)( 14, 63)( 15, 61)( 16, 60)( 17, 58)( 18, 59)( 19, 64)( 20, 65)( 21, 66)( 22, 71)( 23, 72)( 24, 70)( 25, 69)( 26, 67)( 27, 68)( 28, 73)( 29, 74)( 30, 75)( 31, 80)( 32, 81)( 33, 79)( 34, 78)( 35, 76)( 36, 77)( 37, 82)( 38, 83)( 39, 84)( 40, 89)( 41, 90)( 42, 88)( 43, 87)( 44, 85)( 45, 86)( 94, 98)( 95, 99)( 96, 97)(103,107)(104,108)(105,106)(112,116)(113,117)(114,115)(121,125)(122,126)(123,124)(130,134)(131,135)(132,133);
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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1 >;
References : None.
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