Polytope of Type {36,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,4}*1152a
if this polytope has a name.
Group : SmallGroup(1152,32532)
Rank : 3
Schlafli Type : {36,4}
Number of vertices, edges, etc : 144, 288, 16
Order of s0s1s2 : 72
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {36,4}*576a
   3-fold quotients : {12,4}*384a
   4-fold quotients : {36,4}*288a
   6-fold quotients : {12,4}*192a
   8-fold quotients : {36,2}*144, {18,4}*144a
   9-fold quotients : {4,4}*128
   12-fold quotients : {12,4}*96a
   16-fold quotients : {18,2}*72
   18-fold quotients : {4,4}*64
   24-fold quotients : {12,2}*48, {6,4}*48a
   32-fold quotients : {9,2}*36
   36-fold quotients : {4,4}*32
   48-fold quotients : {6,2}*24
   72-fold quotients : {2,4}*16, {4,2}*16
   96-fold quotients : {3,2}*12
   144-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s2*s1*s2> of order 2.
      8 facets:
         8 of {36}*72
      90 vertex figures:
         36 of {2}*4
         54 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      8 facets:
         8 of {36}*72
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 2.
      8 facets:
         8 of {36}*72
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      10 facets:
         4 of {18}*36
         6 of {36}*72
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 2.
      8 facets:
         8 of {36}*72
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
      4 facets:
         4 of {36}*72
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2> of order 4.
      4 facets:
         4 of {36}*72
      45 vertex figures:
         18 of {2}*4
         27 of {4}*8
   P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s0*s2> of order 4.
      4 facets:
         4 of {36}*72
      54 vertex figures:
         36 of {2}*4
         18 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 4.
      5 facets:
         2 of {18}*36
         3 of {36}*72
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      4 facets:
         4 of {36}*72
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
      4 facets:
         4 of {36}*72
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 4.
      6 facets:
         4 of {18}*36
         2 of {36}*72
      36 vertex figures:
         36 of {4}*8

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