Polytope of Type {9,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,12249)
Rank : 3
Schlafli Type : {9,6}
Number of vertices, edges, etc : 144, 432, 96
Order of s0s1s2 : 72
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 : {3,6}*576
   4-fold quotients : {9,6}*432
   9-fold quotients : {3,6}*192
   12-fold quotients : {3,6}*144
   16-fold quotients : {9,6}*108
   36-fold quotients : {3,6}*48
   48-fold quotients : {9,2}*36, {3,6}*36
   72-fold quotients : {3,3}*24
   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*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      48 facets:
         48 of {9}*18
      72 vertex figures:
         72 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 3.
      32 facets:
         32 of {9}*18
      54 vertex figures:
         45 of {6}*12
         9 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 4.
      24 facets:
         24 of {9}*18
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 4.
      24 facets:
         24 of {9}*18
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      24 facets:
         24 of {9}*18
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 4.
      24 facets:
         24 of {9}*18
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 8.
      12 facets:
         12 of {9}*18
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 8.
      12 facets:
         12 of {9}*18
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 8.
      12 facets:
         12 of {9}*18
      18 vertex figures:
         18 of {6}*12

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