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}*1728c
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 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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}*576c
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*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 2.
      24 facets:
         24 of {18}*36
      72 vertex figures:
         72 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1> of order 4.
      12 facets:
         12 of {18}*36
      36 vertex figures:
         36 of {6}*12

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {18,6}

Twisty Puzzle