Polytope of Type {3,24}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,24}*1728
Also Known As : {3,24}6. if this polytope has another name.
Group : SmallGroup(1728,12317)
Rank : 3
Schlafli Type : {3,24}
Number of vertices, edges, etc : 36, 432, 288
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 : {3,24}*576
   4-fold quotients : {3,12}*432
   9-fold quotients : {3,8}*192
   12-fold quotients : {3,12}*144
   16-fold quotients : {3,6}*108
   36-fold quotients : {3,4}*48
   48-fold quotients : {3,6}*36
   72-fold quotients : {3,4}*24
   144-fold quotients : {3,2}*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*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 2.
      144 facets:
         144 of {3}*6
      24 vertex figures:
         12 of {24}*48
         12 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1> of order 2.
      144 facets:
         144 of {3}*6
      18 vertex figures:
         18 of {24}*48
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 3.
      96 facets:
         96 of {3}*6
      20 vertex figures:
         8 of {24}*48
         12 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      72 facets:
         72 of {3}*6
      12 vertex figures:
         6 of {24}*48
         6 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1, s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      72 facets:
         72 of {3}*6
      12 vertex figures:
         6 of {24}*48
         6 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 4.
      72 facets:
         72 of {3}*6
      18 vertex figures:
         6 of {24}*48
         12 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1, s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1> of order 6.
      48 facets:
         48 of {3}*6
      10 vertex figures:
         4 of {24}*48
         6 of {8}*16
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2> of order 6.
      48 facets:
         48 of {3}*6
      12 vertex figures:
         6 of {8}*16
         2 of {24}*48
         4 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 6.
      48 facets:
         48 of {3}*6
      16 vertex figures:
         4 of {24}*48
         12 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 12.
      24 facets:
         24 of {3}*6
      8 vertex figures:
         3 of {8}*16
         1 of {24}*48
         4 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1> of order 12.
      24 facets:
         24 of {3}*6
      14 vertex figures:
         2 of {24}*48
         12 of {2}*4
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1, s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 12.
      24 facets:
         24 of {3}*6
      8 vertex figures:
         2 of {24}*48
         6 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2> of order 12.
      24 facets:
         24 of {3}*6
      6 vertex figures:
         3 of {8}*16
         1 of {24}*48
         2 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1, s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1> of order 12.
      24 facets:
         24 of {3}*6
      8 vertex figures:
         2 of {24}*48
         6 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2> of order 12.
      24 facets:
         24 of {3}*6
      6 vertex figures:
         3 of {8}*16
         1 of {24}*48
         2 of {12}*24

Permutation Representation (GAP) : 
s0 := (  3,  4)(  7,  8)(  9, 15)( 10, 16)( 11, 14)( 12, 13)( 19, 20)( 23, 24)( 25, 31)( 26, 32)( 27, 30)( 28, 29)( 35, 36)( 39, 40)( 41, 47)( 42, 48)( 43, 46)( 44, 45)( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,101)( 54,102)( 55,104)( 56,103)( 57,111)( 58,112)( 59,110)( 60,109)( 61,108)( 62,107)( 63,105)( 64,106)( 65,113)( 66,114)( 67,116)( 68,115)( 69,117)( 70,118)( 71,120)( 72,119)( 73,127)( 74,128)( 75,126)( 76,125)( 77,124)( 78,123)( 79,121)( 80,122)( 81,129)( 82,130)( 83,132)( 84,131)( 85,133)( 86,134)( 87,136)( 88,135)( 89,143)( 90,144)( 91,142)( 92,141)( 93,140)( 94,139)( 95,137)( 96,138);;
s1 := (  1, 65)(  2, 67)(  3, 66)(  4, 68)(  5, 76)(  6, 74)(  7, 75)(  8, 73)(  9, 72)( 10, 70)( 11, 71)( 12, 69)( 13, 77)( 14, 79)( 15, 78)( 16, 80)( 17, 81)( 18, 83)( 19, 82)( 20, 84)( 21, 92)( 22, 90)( 23, 91)( 24, 89)( 25, 88)( 26, 86)( 27, 87)( 28, 85)( 29, 93)( 30, 95)( 31, 94)( 32, 96)( 33, 49)( 34, 51)( 35, 50)( 36, 52)( 37, 60)( 38, 58)( 39, 59)( 40, 57)( 41, 56)( 42, 54)( 43, 55)( 44, 53)( 45, 61)( 46, 63)( 47, 62)( 48, 64)( 98, 99)(101,108)(102,106)(103,107)(104,105)(110,111)(114,115)(117,124)(118,122)(119,123)(120,121)(126,127)(130,131)(133,140)(134,138)(135,139)(136,137)(142,143);;
s2 := (  1,  5)(  2,  6)(  3,  7)(  4,  8)(  9, 15)( 10, 16)( 11, 13)( 12, 14)( 17, 37)( 18, 38)( 19, 39)( 20, 40)( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 25, 47)( 26, 48)( 27, 45)( 28, 46)( 29, 43)( 30, 44)( 31, 41)( 32, 42)( 49,101)( 50,102)( 51,103)( 52,104)( 53, 97)( 54, 98)( 55, 99)( 56,100)( 57,111)( 58,112)( 59,109)( 60,110)( 61,107)( 62,108)( 63,105)( 64,106)( 65,133)( 66,134)( 67,135)( 68,136)( 69,129)( 70,130)( 71,131)( 72,132)( 73,143)( 74,144)( 75,141)( 76,142)( 77,139)( 78,140)( 79,137)( 80,138)( 81,117)( 82,118)( 83,119)( 84,120)( 85,113)( 86,114)( 87,115)( 88,116)( 89,127)( 90,128)( 91,125)( 92,126)( 93,123)( 94,124)( 95,121)( 96,122);;
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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
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)!(  3,  4)(  7,  8)(  9, 15)( 10, 16)( 11, 14)( 12, 13)( 19, 20)( 23, 24)( 25, 31)( 26, 32)( 27, 30)( 28, 29)( 35, 36)( 39, 40)( 41, 47)( 42, 48)( 43, 46)( 44, 45)( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,101)( 54,102)( 55,104)( 56,103)( 57,111)( 58,112)( 59,110)( 60,109)( 61,108)( 62,107)( 63,105)( 64,106)( 65,113)( 66,114)( 67,116)( 68,115)( 69,117)( 70,118)( 71,120)( 72,119)( 73,127)( 74,128)( 75,126)( 76,125)( 77,124)( 78,123)( 79,121)( 80,122)( 81,129)( 82,130)( 83,132)( 84,131)( 85,133)( 86,134)( 87,136)( 88,135)( 89,143)( 90,144)( 91,142)( 92,141)( 93,140)( 94,139)( 95,137)( 96,138);
s1 := Sym(144)!(  1, 65)(  2, 67)(  3, 66)(  4, 68)(  5, 76)(  6, 74)(  7, 75)(  8, 73)(  9, 72)( 10, 70)( 11, 71)( 12, 69)( 13, 77)( 14, 79)( 15, 78)( 16, 80)( 17, 81)( 18, 83)( 19, 82)( 20, 84)( 21, 92)( 22, 90)( 23, 91)( 24, 89)( 25, 88)( 26, 86)( 27, 87)( 28, 85)( 29, 93)( 30, 95)( 31, 94)( 32, 96)( 33, 49)( 34, 51)( 35, 50)( 36, 52)( 37, 60)( 38, 58)( 39, 59)( 40, 57)( 41, 56)( 42, 54)( 43, 55)( 44, 53)( 45, 61)( 46, 63)( 47, 62)( 48, 64)( 98, 99)(101,108)(102,106)(103,107)(104,105)(110,111)(114,115)(117,124)(118,122)(119,123)(120,121)(126,127)(130,131)(133,140)(134,138)(135,139)(136,137)(142,143);
s2 := Sym(144)!(  1,  5)(  2,  6)(  3,  7)(  4,  8)(  9, 15)( 10, 16)( 11, 13)( 12, 14)( 17, 37)( 18, 38)( 19, 39)( 20, 40)( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 25, 47)( 26, 48)( 27, 45)( 28, 46)( 29, 43)( 30, 44)( 31, 41)( 32, 42)( 49,101)( 50,102)( 51,103)( 52,104)( 53, 97)( 54, 98)( 55, 99)( 56,100)( 57,111)( 58,112)( 59,109)( 60,110)( 61,107)( 62,108)( 63,105)( 64,106)( 65,133)( 66,134)( 67,135)( 68,136)( 69,129)( 70,130)( 71,131)( 72,132)( 73,143)( 74,144)( 75,141)( 76,142)( 77,139)( 78,140)( 79,137)( 80,138)( 81,117)( 82,118)( 83,119)( 84,120)( 85,113)( 86,114)( 87,115)( 88,116)( 89,127)( 90,128)( 91,125)( 92,126)( 93,123)( 94,124)( 95,121)( 96,122);
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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
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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