Polytope of Type {6,6}

Play with this polytope as a twisty puzzle

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

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