Polytope of Type {4,36}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,36}*576a
if this polytope has a name.
Group : SmallGroup(576,303)
Rank : 3
Schlafli Type : {4,36}
Number of vertices, edges, etc : 8, 144, 72
Order of s0s1s2 : 36
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,36,2} of size 1152
Vertex Figure Of :
   {2,4,36} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,36}*288a
   3-fold quotients : {4,12}*192a
   4-fold quotients : {2,36}*144, {4,18}*144a
   6-fold quotients : {4,12}*96a
   8-fold quotients : {2,18}*72
   9-fold quotients : {4,4}*64
   12-fold quotients : {2,12}*48, {4,6}*48a
   16-fold quotients : {2,9}*36
   18-fold quotients : {4,4}*32
   24-fold quotients : {2,6}*24
   36-fold quotients : {2,4}*16, {4,2}*16
   48-fold quotients : {2,3}*12
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,36}*1152a, {4,72}*1152a, {4,36}*1152a, {4,72}*1152b, {8,36}*1152b
   3-fold covers : {4,108}*1728a, {12,36}*1728a, {12,36}*1728b
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1> of order 2.
      54 facets:
         36 of {2}*4
         18 of {4}*8
      4 vertex figures:
         4 of {36}*72
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      36 facets:
         36 of {4}*8
      6 vertex figures:
         4 of {18}*36
         2 of {36}*72

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