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}*1728b
if this polytope has a name.
Group : SmallGroup(1728,46100)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 48, 432, 144
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
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.
      72 facets:
         72 of {6}*12
      24 vertex figures:
         24 of {18}*36
   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.
      36 facets:
         36 of {6}*12
      12 vertex figures:
         12 of {18}*36

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {6,18}

Twisty Puzzle