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}*576c
if this polytope has a name.
Group : SmallGroup(576,4970)
Rank : 3
Schlafli Type : {36,4}
Number of vertices, edges, etc : 72, 144, 8
Order of s0s1s2 : 18
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {36,4,2} of size 1152
Vertex Figure Of :
   {2,36,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {18,4}*288
   3-fold quotients : {12,4}*192c
   4-fold quotients : {9,4}*144, {18,4}*144b, {18,4}*144c
   6-fold quotients : {6,4}*96
   8-fold quotients : {9,4}*72, {18,2}*72
   12-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   16-fold quotients : {9,2}*36
   24-fold quotients : {3,4}*24, {6,2}*24
   48-fold quotients : {3,2}*12
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,4}*1152d, {36,8}*1152g, {36,8}*1152h, {72,4}*1152e, {72,4}*1152f
   3-fold covers : {108,4}*1728c, {36,12}*1728g, {36,12}*1728h
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      4 facets:
         4 of {36}*72
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
      4 facets:
         4 of {36}*72
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s1*s2*s1*s2> of order 2.
      4 facets:
         4 of {36}*72
      48 vertex figures:
         24 of {2}*4
         24 of {4}*8

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

Twisty Puzzle