Polytope of Type {4,8}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,8}*1152b
if this polytope has a name.
Group : SmallGroup(1152,32552)
Rank : 3
Schlafli Type : {4,8}
Number of vertices, edges, etc : 72, 288, 144
Order of s0s1s2 : 12
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,4}*576
   4-fold quotients : {4,4}*288
   8-fold quotients : {4,4}*144
   9-fold quotients : {4,8}*128b
   16-fold quotients : {4,4}*72
   18-fold quotients : {4,4}*64
   36-fold quotients : {4,4}*32
   72-fold quotients : {2,4}*16, {4,2}*16
   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=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 2.
      72 facets:
         72 of {4}*8
      36 vertex figures:
         36 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      72 facets:
         72 of {4}*8
      36 vertex figures:
         36 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      48 facets:
         48 of {4}*8
      24 vertex figures:
         24 of {8}*16
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 3.
      48 facets:
         48 of {4}*8
      24 vertex figures:
         24 of {8}*16
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      36 facets:
         36 of {4}*8
      18 vertex figures:
         18 of {8}*16
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      36 facets:
         36 of {4}*8
      18 vertex figures:
         18 of {8}*16
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 4.
      36 facets:
         36 of {4}*8
      18 vertex figures:
         18 of {8}*16
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0> of order 4.
      36 facets:
         36 of {4}*8
      18 vertex figures:
         18 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 4.
      36 facets:
         36 of {4}*8
      18 vertex figures:
         18 of {8}*16
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      36 facets:
         36 of {4}*8
      18 vertex figures:
         18 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 6.
      24 facets:
         24 of {4}*8
      12 vertex figures:
         12 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 6.
      24 facets:
         24 of {4}*8
      12 vertex figures:
         12 of {8}*16
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
      24 facets:
         24 of {4}*8
      12 vertex figures:
         12 of {8}*16
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
      24 facets:
         24 of {4}*8
      12 vertex figures:
         12 of {8}*16
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 6.
      24 facets:
         24 of {4}*8
      12 vertex figures:
         12 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 12.
      12 facets:
         12 of {4}*8
      6 vertex figures:
         6 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 12.
      12 facets:
         12 of {4}*8
      6 vertex figures:
         6 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 12.
      12 facets:
         12 of {4}*8
      6 vertex figures:
         6 of {8}*16
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 12.
      12 facets:
         12 of {4}*8
      6 vertex figures:
         6 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 12.
      12 facets:
         12 of {4}*8
      6 vertex figures:
         6 of {8}*16
   P/N, where N=<s0*s2*s1*s0*s2*s1*s2*s1*s0*s1> of order 12.
      12 facets:
         12 of {4}*8
      6 vertex figures:
         6 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 12.
      12 facets:
         12 of {4}*8
      6 vertex figures:
         6 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 12.
      12 facets:
         12 of {4}*8
      6 vertex figures:
         6 of {8}*16

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

Twisty Puzzle