Polytope of Type {18,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6}*648d
if this polytope has a name.
Group : SmallGroup(648,299)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 54, 162, 18
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {18,6,2} of size 1296
Vertex Figure Of :
   {2,18,6} of size 1296
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {9,6}*324b
   3-fold quotients : {6,6}*216c
   6-fold quotients : {3,6}*108
   9-fold quotients : {6,6}*72c
   18-fold quotients : {3,6}*36
   27-fold quotients : {6,2}*24
   54-fold quotients : {3,2}*12
   81-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,6}*1296c, {18,12}*1296g
   3-fold covers : {18,6}*1944a, {18,6}*1944d, {18,6}*1944f, {18,6}*1944h, {18,18}*1944ab, {18,6}*1944q
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 3.
      6 facets:
         6 of {18}*36
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 3.
      6 facets:
         6 of {18}*36
      30 vertex figures:
         12 of {6}*12
         18 of {2}*4

Permutation Representation (GAP) : 
s0 := (  2,  3)(  4,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 55)( 29, 57)( 30, 56)( 31, 60)( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 37, 73)( 38, 75)( 39, 74)( 40, 78)( 41, 77)( 42, 76)( 43, 80)( 44, 79)( 45, 81)( 46, 64)( 47, 66)( 48, 65)( 49, 69)( 50, 68)( 51, 67)( 52, 71)( 53, 70)( 54, 72)( 83, 84)( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)( 96,103)( 97,107)( 98,106)( 99,108)(109,136)(110,138)(111,137)(112,141)(113,140)(114,139)(115,143)(116,142)(117,144)(118,154)(119,156)(120,155)(121,159)(122,158)(123,157)(124,161)(125,160)(126,162)(127,145)(128,147)(129,146)(130,150)(131,149)(132,148)(133,152)(134,151)(135,153);;
s1 := (  1,148)(  2,150)(  3,149)(  4,151)(  5,153)(  6,152)(  7,145)(  8,147)(  9,146)( 10,142)( 11,144)( 12,143)( 13,136)( 14,138)( 15,137)( 16,139)( 17,141)( 18,140)( 19,156)( 20,155)( 21,154)( 22,159)( 23,158)( 24,157)( 25,162)( 26,161)( 27,160)( 28,121)( 29,123)( 30,122)( 31,124)( 32,126)( 33,125)( 34,118)( 35,120)( 36,119)( 37,115)( 38,117)( 39,116)( 40,109)( 41,111)( 42,110)( 43,112)( 44,114)( 45,113)( 46,129)( 47,128)( 48,127)( 49,132)( 50,131)( 51,130)( 52,135)( 53,134)( 54,133)( 55, 94)( 56, 96)( 57, 95)( 58, 97)( 59, 99)( 60, 98)( 61, 91)( 62, 93)( 63, 92)( 64, 88)( 65, 90)( 66, 89)( 67, 82)( 68, 84)( 69, 83)( 70, 85)( 71, 87)( 72, 86)( 73,102)( 74,101)( 75,100)( 76,105)( 77,104)( 78,103)( 79,108)( 80,107)( 81,106);;
s2 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 55)( 29, 57)( 30, 56)( 31, 61)( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)( 39, 65)( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)( 47, 75)( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)(130,160)(131,162)(132,161)(133,157)(134,159)(135,158);;
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*s2*s0*s1*s0*s1*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(162)!(  2,  3)(  4,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 55)( 29, 57)( 30, 56)( 31, 60)( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 37, 73)( 38, 75)( 39, 74)( 40, 78)( 41, 77)( 42, 76)( 43, 80)( 44, 79)( 45, 81)( 46, 64)( 47, 66)( 48, 65)( 49, 69)( 50, 68)( 51, 67)( 52, 71)( 53, 70)( 54, 72)( 83, 84)( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)( 96,103)( 97,107)( 98,106)( 99,108)(109,136)(110,138)(111,137)(112,141)(113,140)(114,139)(115,143)(116,142)(117,144)(118,154)(119,156)(120,155)(121,159)(122,158)(123,157)(124,161)(125,160)(126,162)(127,145)(128,147)(129,146)(130,150)(131,149)(132,148)(133,152)(134,151)(135,153);
s1 := Sym(162)!(  1,148)(  2,150)(  3,149)(  4,151)(  5,153)(  6,152)(  7,145)(  8,147)(  9,146)( 10,142)( 11,144)( 12,143)( 13,136)( 14,138)( 15,137)( 16,139)( 17,141)( 18,140)( 19,156)( 20,155)( 21,154)( 22,159)( 23,158)( 24,157)( 25,162)( 26,161)( 27,160)( 28,121)( 29,123)( 30,122)( 31,124)( 32,126)( 33,125)( 34,118)( 35,120)( 36,119)( 37,115)( 38,117)( 39,116)( 40,109)( 41,111)( 42,110)( 43,112)( 44,114)( 45,113)( 46,129)( 47,128)( 48,127)( 49,132)( 50,131)( 51,130)( 52,135)( 53,134)( 54,133)( 55, 94)( 56, 96)( 57, 95)( 58, 97)( 59, 99)( 60, 98)( 61, 91)( 62, 93)( 63, 92)( 64, 88)( 65, 90)( 66, 89)( 67, 82)( 68, 84)( 69, 83)( 70, 85)( 71, 87)( 72, 86)( 73,102)( 74,101)( 75,100)( 76,105)( 77,104)( 78,103)( 79,108)( 80,107)( 81,106);
s2 := Sym(162)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 55)( 29, 57)( 30, 56)( 31, 61)( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)( 39, 65)( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)( 47, 75)( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)(130,160)(131,162)(132,161)(133,157)(134,159)(135,158);
poly := sub<Sym(162)|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*s2*s0*s1*s0*s1*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.
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