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}*1728b
if this polytope has a name.
Group : SmallGroup(1728,46100)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 144, 432, 48
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 :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,6}*576e
   16-fold quotients : {9,6}*108
   48-fold quotients : {9,2}*36, {3,6}*36
   144-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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> of order 2.
      24 facets:
         24 of {18}*36
      72 vertex figures:
         72 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2> of order 2.
      28 facets:
         8 of {9}*18
         20 of {18}*36
      72 vertex figures:
         72 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2> of order 4.
      14 facets:
         4 of {9}*18
         10 of {18}*36
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 4.
      12 facets:
         12 of {18}*36
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 4.
      12 facets:
         12 of {18}*36
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1> of order 4.
      16 facets:
         8 of {9}*18
         8 of {18}*36
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
      18 facets:
         12 of {9}*18
         6 of {18}*36
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 8.
      8 facets:
         4 of {9}*18
         4 of {18}*36
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2> of order 8.
      10 facets:
         8 of {9}*18
         2 of {18}*36
      18 vertex figures:
         18 of {6}*12

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