Polytope of Type {6,45}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,45}*1620c
if this polytope has a name.
Group : SmallGroup(1620,136)
Rank : 3
Schlafli Type : {6,45}
Number of vertices, edges, etc : 18, 405, 135
Order of s0s1s2 : 30
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,15}*540
   5-fold quotients : {6,9}*324b
   9-fold quotients : {6,15}*180
   15-fold quotients : {6,3}*108
   27-fold quotients : {2,15}*60
   45-fold quotients : {6,3}*36
   81-fold quotients : {2,5}*20
   135-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1> of order 3.
      75 facets:
         45 of {2}*4
         30 of {6}*12
      6 vertex figures:
         6 of {45}*90
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 3.
      45 facets:
         45 of {6}*12
      6 vertex figures:
         6 of {45}*90

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