Polytope of Type {8,9}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,9}*576
if this polytope has a name.
Group : SmallGroup(576,1381)
Rank : 3
Schlafli Type : {8,9}
Number of vertices, edges, etc : 32, 144, 36
Order of s0s1s2 : 18
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,9,2} of size 1152
Vertex Figure Of :
   {2,8,9} of size 1152
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {8,3}*192
   4-fold quotients : {4,9}*144
   8-fold quotients : {4,9}*72
   12-fold quotients : {4,3}*48
   16-fold quotients : {2,9}*36
   24-fold quotients : {4,3}*24
   48-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,9}*1152, {8,18}*1152a, {8,18}*1152d
   3-fold covers : {8,27}*1728, {24,9}*1728
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      24 facets:
         12 of {4}*8
         12 of {8}*16
      16 vertex figures:
         16 of {9}*18
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1> of order 2.
      18 facets:
         18 of {8}*16
      16 vertex figures:
         16 of {9}*18
   P/N, where N=<s0*s1*s0*s1> of order 4.
      18 facets:
         12 of {2}*4
         6 of {8}*16
      8 vertex figures:
         8 of {9}*18
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 4.
      12 facets:
         6 of {4}*8
         6 of {8}*16
      8 vertex figures:
         8 of {9}*18
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s2*s1> of order 4.
      12 facets:
         6 of {4}*8
         6 of {8}*16
      8 vertex figures:
         8 of {9}*18

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