Polytope of Type {18,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,4}*576b
if this polytope has a name.
Group : SmallGroup(576,4970)
Rank : 3
Schlafli Type : {18,4}
Number of vertices, edges, etc : 72, 144, 16
Order of s0s1s2 : 36
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {18,4,2} of size 1152
Vertex Figure Of :
   {2,18,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {18,4}*288
   3-fold quotients : {6,4}*192b
   4-fold quotients : {18,4}*144a, {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 : {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
   16-fold quotients : {9,2}*36
   24-fold quotients : {3,4}*24, {6,2}*24
   36-fold quotients : {2,4}*16
   48-fold quotients : {3,2}*12
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,4}*1152d, {18,8}*1152f, {18,8}*1152g, {36,4}*1152e, {18,4}*1152b
   3-fold covers : {54,4}*1728b, {18,12}*1728c, {18,12}*1728d
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 2.
      8 facets:
         8 of {18}*36
      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*s1*s0*s1> of order 2.
      12 facets:
         8 of {9}*18
         4 of {18}*36
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      8 facets:
         8 of {18}*36
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s1*s2*s1*s2> of order 2.
      8 facets:
         8 of {18}*36
      48 vertex figures:
         24 of {2}*4
         24 of {4}*8
   P/N, where N=<s1*s2*s1*s2, 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 {9}*18
         2 of {18}*36
      24 vertex figures:
         12 of {2}*4
         12 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1> of order 4.
      4 facets:
         4 of {18}*36
      30 vertex figures:
         6 of {4}*8
         24 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0> of order 4.
      4 facets:
         4 of {18}*36
      18 vertex figures:
         18 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)( 74, 75)( 77, 81)( 78, 83)( 79, 82)( 80, 84)( 85,105)( 86,107)( 87,106)( 88,108)( 89,101)( 90,103)( 91,102)( 92,104)( 93, 97)( 94, 99)( 95, 98)( 96,100)(110,111)(113,117)(114,119)(115,118)(116,120)(121,141)(122,143)(123,142)(124,144)(125,137)(126,139)(127,138)(128,140)(129,133)(130,135)(131,134)(132,136);;
s1 := (  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 21)(  6, 22)(  7, 24)(  8, 23)(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 25, 33)( 26, 34)( 27, 36)( 28, 35)( 31, 32)( 37, 49)( 38, 50)( 39, 52)( 40, 51)( 41, 57)( 42, 58)( 43, 60)( 44, 59)( 45, 53)( 46, 54)( 47, 56)( 48, 55)( 61, 69)( 62, 70)( 63, 72)( 64, 71)( 67, 68)( 73,121)( 74,122)( 75,124)( 76,123)( 77,129)( 78,130)( 79,132)( 80,131)( 81,125)( 82,126)( 83,128)( 84,127)( 85,109)( 86,110)( 87,112)( 88,111)( 89,117)( 90,118)( 91,120)( 92,119)( 93,113)( 94,114)( 95,116)( 96,115)( 97,141)( 98,142)( 99,144)(100,143)(101,137)(102,138)(103,140)(104,139)(105,133)(106,134)(107,136)(108,135);;
s2 := (  1, 76)(  2, 75)(  3, 74)(  4, 73)(  5, 80)(  6, 79)(  7, 78)(  8, 77)(  9, 84)( 10, 83)( 11, 82)( 12, 81)( 13, 88)( 14, 87)( 15, 86)( 16, 85)( 17, 92)( 18, 91)( 19, 90)( 20, 89)( 21, 96)( 22, 95)( 23, 94)( 24, 93)( 25,100)( 26, 99)( 27, 98)( 28, 97)( 29,104)( 30,103)( 31,102)( 32,101)( 33,108)( 34,107)( 35,106)( 36,105)( 37,112)( 38,111)( 39,110)( 40,109)( 41,116)( 42,115)( 43,114)( 44,113)( 45,120)( 46,119)( 47,118)( 48,117)( 49,124)( 50,123)( 51,122)( 52,121)( 53,128)( 54,127)( 55,126)( 56,125)( 57,132)( 58,131)( 59,130)( 60,129)( 61,136)( 62,135)( 63,134)( 64,133)( 65,140)( 66,139)( 67,138)( 68,137)( 69,144)( 70,143)( 71,142)( 72,141);;
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*s1*s0*s1*s0*s1*s0*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)( 74, 75)( 77, 81)( 78, 83)( 79, 82)( 80, 84)( 85,105)( 86,107)( 87,106)( 88,108)( 89,101)( 90,103)( 91,102)( 92,104)( 93, 97)( 94, 99)( 95, 98)( 96,100)(110,111)(113,117)(114,119)(115,118)(116,120)(121,141)(122,143)(123,142)(124,144)(125,137)(126,139)(127,138)(128,140)(129,133)(130,135)(131,134)(132,136);
s1 := Sym(144)!(  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 21)(  6, 22)(  7, 24)(  8, 23)(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 25, 33)( 26, 34)( 27, 36)( 28, 35)( 31, 32)( 37, 49)( 38, 50)( 39, 52)( 40, 51)( 41, 57)( 42, 58)( 43, 60)( 44, 59)( 45, 53)( 46, 54)( 47, 56)( 48, 55)( 61, 69)( 62, 70)( 63, 72)( 64, 71)( 67, 68)( 73,121)( 74,122)( 75,124)( 76,123)( 77,129)( 78,130)( 79,132)( 80,131)( 81,125)( 82,126)( 83,128)( 84,127)( 85,109)( 86,110)( 87,112)( 88,111)( 89,117)( 90,118)( 91,120)( 92,119)( 93,113)( 94,114)( 95,116)( 96,115)( 97,141)( 98,142)( 99,144)(100,143)(101,137)(102,138)(103,140)(104,139)(105,133)(106,134)(107,136)(108,135);
s2 := Sym(144)!(  1, 76)(  2, 75)(  3, 74)(  4, 73)(  5, 80)(  6, 79)(  7, 78)(  8, 77)(  9, 84)( 10, 83)( 11, 82)( 12, 81)( 13, 88)( 14, 87)( 15, 86)( 16, 85)( 17, 92)( 18, 91)( 19, 90)( 20, 89)( 21, 96)( 22, 95)( 23, 94)( 24, 93)( 25,100)( 26, 99)( 27, 98)( 28, 97)( 29,104)( 30,103)( 31,102)( 32,101)( 33,108)( 34,107)( 35,106)( 36,105)( 37,112)( 38,111)( 39,110)( 40,109)( 41,116)( 42,115)( 43,114)( 44,113)( 45,120)( 46,119)( 47,118)( 48,117)( 49,124)( 50,123)( 51,122)( 52,121)( 53,128)( 54,127)( 55,126)( 56,125)( 57,132)( 58,131)( 59,130)( 60,129)( 61,136)( 62,135)( 63,134)( 64,133)( 65,140)( 66,139)( 67,138)( 68,137)( 69,144)( 70,143)( 71,142)( 72,141);
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*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
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