Polytope of Type {6,24}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,24}*1728a
Also Known As : {6,24}3. if this polytope has another name.
Group : SmallGroup(1728,12317)
Rank : 3
Schlafli Type : {6,24}
Number of vertices, edges, etc : 36, 432, 144
Order of s0s1s2 : 3
Order of s0s1s2s1 : 24
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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 : {6,24}*576a
   4-fold quotients : {6,12}*432d
   9-fold quotients : {6,8}*192a
   12-fold quotients : {6,12}*144d
   16-fold quotients : {6,6}*108
   36-fold quotients : {6,4}*48b
   72-fold quotients : {3,4}*24
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*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2> of order 2.
      72 facets:
         72 of {6}*12
      24 vertex figures:
         12 of {24}*48
         12 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2> of order 3.
      48 facets:
         48 of {6}*12
      20 vertex figures:
         8 of {24}*48
         12 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 4.
      36 facets:
         36 of {6}*12
      12 vertex figures:
         6 of {24}*48
         6 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      36 facets:
         36 of {6}*12
      18 vertex figures:
         6 of {24}*48
         12 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 6.
      24 facets:
         24 of {6}*12
      12 vertex figures:
         2 of {24}*48
         6 of {8}*16
         4 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 6.
      24 facets:
         24 of {6}*12
      16 vertex figures:
         4 of {24}*48
         12 of {4}*8

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