Polytope of Type {6,18}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18}*648b
if this polytope has a name.
Group : SmallGroup(648,297)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 18, 162, 54
Order of s0s1s2 : 18
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,18,2} of size 1296
Vertex Figure Of :
   {2,6,18} of size 1296
   {3,6,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,18}*324a
   3-fold quotients : {6,18}*216a, {6,6}*216b
   6-fold quotients : {6,6}*108
   9-fold quotients : {2,18}*72, {6,6}*72a
   18-fold quotients : {2,9}*36
   27-fold quotients : {2,6}*24, {6,2}*24
   54-fold quotients : {2,3}*12, {3,2}*12
   81-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,18}*1296a, {6,36}*1296b
   3-fold covers : {18,18}*1944c, {6,54}*1944b, {6,18}*1944g, {18,18}*1944v, {18,18}*1944z, {6,18}*1944j, {6,54}*1944d, {6,54}*1944f, {6,18}*1944n
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {6}*12
      6 vertex figures:
         6 of {18}*36
   P/N, where N=<s0*s1*s0*s1> of order 3.
      30 facets:
         18 of {2}*4
         12 of {6}*12
      6 vertex figures:
         6 of {18}*36

Permutation Representation (GAP) : 
s0 := (  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161);;
s1 := (  1, 10)(  2, 11)(  3, 12)(  4, 16)(  5, 17)(  6, 18)(  7, 13)(  8, 14)(  9, 15)( 22, 25)( 23, 26)( 24, 27)( 28, 70)( 29, 71)( 30, 72)( 31, 67)( 32, 68)( 33, 69)( 34, 64)( 35, 65)( 36, 66)( 37, 61)( 38, 62)( 39, 63)( 40, 58)( 41, 59)( 42, 60)( 43, 55)( 44, 56)( 45, 57)( 46, 79)( 47, 80)( 48, 81)( 49, 76)( 50, 77)( 51, 78)( 52, 73)( 53, 74)( 54, 75)( 82, 91)( 83, 92)( 84, 93)( 85, 97)( 86, 98)( 87, 99)( 88, 94)( 89, 95)( 90, 96)(103,106)(104,107)(105,108)(109,151)(110,152)(111,153)(112,148)(113,149)(114,150)(115,145)(116,146)(117,147)(118,142)(119,143)(120,144)(121,139)(122,140)(123,141)(124,136)(125,137)(126,138)(127,160)(128,161)(129,162)(130,157)(131,158)(132,159)(133,154)(134,155)(135,156);;
s2 := (  1,109)(  2,111)(  3,110)(  4,115)(  5,117)(  6,116)(  7,112)(  8,114)(  9,113)( 10,119)( 11,118)( 12,120)( 13,125)( 14,124)( 15,126)( 16,122)( 17,121)( 18,123)( 19,129)( 20,128)( 21,127)( 22,135)( 23,134)( 24,133)( 25,132)( 26,131)( 27,130)( 28, 82)( 29, 84)( 30, 83)( 31, 88)( 32, 90)( 33, 89)( 34, 85)( 35, 87)( 36, 86)( 37, 92)( 38, 91)( 39, 93)( 40, 98)( 41, 97)( 42, 99)( 43, 95)( 44, 94)( 45, 96)( 46,102)( 47,101)( 48,100)( 49,108)( 50,107)( 51,106)( 52,105)( 53,104)( 54,103)( 55,142)( 56,144)( 57,143)( 58,139)( 59,141)( 60,140)( 61,136)( 62,138)( 63,137)( 64,152)( 65,151)( 66,153)( 67,149)( 68,148)( 69,150)( 70,146)( 71,145)( 72,147)( 73,162)( 74,161)( 75,160)( 76,159)( 77,158)( 78,157)( 79,156)( 80,155)( 81,154);;
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*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(162)!(  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161);
s1 := Sym(162)!(  1, 10)(  2, 11)(  3, 12)(  4, 16)(  5, 17)(  6, 18)(  7, 13)(  8, 14)(  9, 15)( 22, 25)( 23, 26)( 24, 27)( 28, 70)( 29, 71)( 30, 72)( 31, 67)( 32, 68)( 33, 69)( 34, 64)( 35, 65)( 36, 66)( 37, 61)( 38, 62)( 39, 63)( 40, 58)( 41, 59)( 42, 60)( 43, 55)( 44, 56)( 45, 57)( 46, 79)( 47, 80)( 48, 81)( 49, 76)( 50, 77)( 51, 78)( 52, 73)( 53, 74)( 54, 75)( 82, 91)( 83, 92)( 84, 93)( 85, 97)( 86, 98)( 87, 99)( 88, 94)( 89, 95)( 90, 96)(103,106)(104,107)(105,108)(109,151)(110,152)(111,153)(112,148)(113,149)(114,150)(115,145)(116,146)(117,147)(118,142)(119,143)(120,144)(121,139)(122,140)(123,141)(124,136)(125,137)(126,138)(127,160)(128,161)(129,162)(130,157)(131,158)(132,159)(133,154)(134,155)(135,156);
s2 := Sym(162)!(  1,109)(  2,111)(  3,110)(  4,115)(  5,117)(  6,116)(  7,112)(  8,114)(  9,113)( 10,119)( 11,118)( 12,120)( 13,125)( 14,124)( 15,126)( 16,122)( 17,121)( 18,123)( 19,129)( 20,128)( 21,127)( 22,135)( 23,134)( 24,133)( 25,132)( 26,131)( 27,130)( 28, 82)( 29, 84)( 30, 83)( 31, 88)( 32, 90)( 33, 89)( 34, 85)( 35, 87)( 36, 86)( 37, 92)( 38, 91)( 39, 93)( 40, 98)( 41, 97)( 42, 99)( 43, 95)( 44, 94)( 45, 96)( 46,102)( 47,101)( 48,100)( 49,108)( 50,107)( 51,106)( 52,105)( 53,104)( 54,103)( 55,142)( 56,144)( 57,143)( 58,139)( 59,141)( 60,140)( 61,136)( 62,138)( 63,137)( 64,152)( 65,151)( 66,153)( 67,149)( 68,148)( 69,150)( 70,146)( 71,145)( 72,147)( 73,162)( 74,161)( 75,160)( 76,159)( 77,158)( 78,157)( 79,156)( 80,155)( 81,154);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope

Twisty Puzzle