Polytope of Type {24,3}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,3}*1728
Also Known As : {24,3}6. if this polytope has another name.
Group : SmallGroup(1728,12317)
Rank : 3
Schlafli Type : {24,3}
Number of vertices, edges, etc : 288, 432, 36
Order of s0s1s2 : 6
Order of s0s1s2s1 : 24
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) :
   3-fold quotients : {24,3}*576
   4-fold quotients : {12,3}*432
   9-fold quotients : {8,3}*192
   12-fold quotients : {12,3}*144
   16-fold quotients : {6,3}*108
   36-fold quotients : {4,3}*48
   48-fold quotients : {6,3}*36
   72-fold quotients : {4,3}*24
   144-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   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> of order 2.
      24 facets:
         12 of {12}*24
         12 of {24}*48
      144 vertex figures:
         144 of {3}*6
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 2.
      18 facets:
         18 of {24}*48
      144 vertex figures:
         144 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      20 facets:
         12 of {8}*16
         8 of {24}*48
      96 vertex figures:
         96 of {3}*6
   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*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 4.
      12 facets:
         6 of {12}*24
         6 of {24}*48
      72 vertex figures:
         72 of {3}*6
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*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> of order 4.
      12 facets:
         6 of {12}*24
         6 of {24}*48
      72 vertex figures:
         72 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 4.
      18 facets:
         12 of {6}*12
         6 of {24}*48
      72 vertex figures:
         72 of {3}*6
   P/N, where N=<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*s1*s0*s1*s0*s1*s2*s1> of order 6.
      10 facets:
         6 of {8}*16
         4 of {24}*48
      48 vertex figures:
         48 of {3}*6
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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> of order 6.
      12 facets:
         4 of {12}*24
         2 of {24}*48
         6 of {8}*16
      48 vertex figures:
         48 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      16 facets:
         12 of {4}*8
         4 of {24}*48
      48 vertex figures:
         48 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 12.
      8 facets:
         4 of {6}*12
         1 of {24}*48
         3 of {8}*16
      24 vertex figures:
         24 of {3}*6
   P/N, where N=<s0*s1*s0*s1> of order 12.
      14 facets:
         12 of {2}*4
         2 of {24}*48
      24 vertex figures:
         24 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*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*s2*s1> of order 12.
      8 facets:
         6 of {4}*8
         2 of {24}*48
      24 vertex figures:
         24 of {3}*6
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1> of order 12.
      6 facets:
         2 of {12}*24
         1 of {24}*48
         3 of {8}*16
      24 vertex figures:
         24 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2> of order 12.
      8 facets:
         6 of {4}*8
         2 of {24}*48
      24 vertex figures:
         24 of {3}*6
   P/N, where N=<s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 12.
      6 facets:
         2 of {12}*24
         1 of {24}*48
         3 of {8}*16
      24 vertex figures:
         24 of {3}*6

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

Twisty Puzzle