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Polytope of Type {12,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,3}*576
Also Known As : {{12,6|2},{6,3}4}. if this polytope has another name.
Group : SmallGroup(576,8313)
Rank : 4
Schlafli Type : {12,6,3}
Number of vertices, edges, etc : 12, 48, 12, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {12,6,3,2} of size 1152
Vertex Figure Of :
   {2,12,6,3} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6,3}*288
   3-fold quotients : {4,6,3}*192
   6-fold quotients : {2,6,3}*96
   12-fold quotients : {2,3,3}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,12,3}*1152, {24,6,3}*1152, {12,6,6}*1152a
   3-fold covers : {36,6,3}*1728, {12,6,3}*1728
Permutation Representation (GAP) :
s0 := (  5,  9)(  6, 10)(  7, 11)(  8, 12)( 17, 21)( 18, 22)( 19, 23)( 20, 24)
( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 69)( 66, 70)( 67, 71)( 68, 72)
( 73,109)( 74,110)( 75,111)( 76,112)( 77,117)( 78,118)( 79,119)( 80,120)
( 81,113)( 82,114)( 83,115)( 84,116)( 85,121)( 86,122)( 87,123)( 88,124)
( 89,129)( 90,130)( 91,131)( 92,132)( 93,125)( 94,126)( 95,127)( 96,128)
( 97,133)( 98,134)( 99,135)(100,136)(101,141)(102,142)(103,143)(104,144)
(105,137)(106,138)(107,139)(108,140);;
s1 := (  1, 77)(  2, 79)(  3, 78)(  4, 80)(  5, 73)(  6, 75)(  7, 74)(  8, 76)
(  9, 81)( 10, 83)( 11, 82)( 12, 84)( 13,101)( 14,103)( 15,102)( 16,104)
( 17, 97)( 18, 99)( 19, 98)( 20,100)( 21,105)( 22,107)( 23,106)( 24,108)
( 25, 89)( 26, 91)( 27, 90)( 28, 92)( 29, 85)( 30, 87)( 31, 86)( 32, 88)
( 33, 93)( 34, 95)( 35, 94)( 36, 96)( 37,113)( 38,115)( 39,114)( 40,116)
( 41,109)( 42,111)( 43,110)( 44,112)( 45,117)( 46,119)( 47,118)( 48,120)
( 49,137)( 50,139)( 51,138)( 52,140)( 53,133)( 54,135)( 55,134)( 56,136)
( 57,141)( 58,143)( 59,142)( 60,144)( 61,125)( 62,127)( 63,126)( 64,128)
( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)( 71,130)( 72,132);;
s2 := (  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 17)(  6, 18)(  7, 20)(  8, 19)
(  9, 21)( 10, 22)( 11, 24)( 12, 23)( 27, 28)( 31, 32)( 35, 36)( 37, 49)
( 38, 50)( 39, 52)( 40, 51)( 41, 53)( 42, 54)( 43, 56)( 44, 55)( 45, 57)
( 46, 58)( 47, 60)( 48, 59)( 63, 64)( 67, 68)( 71, 72)( 73, 85)( 74, 86)
( 75, 88)( 76, 87)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 81, 93)( 82, 94)
( 83, 96)( 84, 95)( 99,100)(103,104)(107,108)(109,121)(110,122)(111,124)
(112,123)(113,125)(114,126)(115,128)(116,127)(117,129)(118,130)(119,132)
(120,131)(135,136)(139,140)(143,144);;
s3 := (  1,  4)(  5,  8)(  9, 12)( 13, 28)( 14, 26)( 15, 27)( 16, 25)( 17, 32)
( 18, 30)( 19, 31)( 20, 29)( 21, 36)( 22, 34)( 23, 35)( 24, 33)( 37, 40)
( 41, 44)( 45, 48)( 49, 64)( 50, 62)( 51, 63)( 52, 61)( 53, 68)( 54, 66)
( 55, 67)( 56, 65)( 57, 72)( 58, 70)( 59, 71)( 60, 69)( 73, 76)( 77, 80)
( 81, 84)( 85,100)( 86, 98)( 87, 99)( 88, 97)( 89,104)( 90,102)( 91,103)
( 92,101)( 93,108)( 94,106)( 95,107)( 96,105)(109,112)(113,116)(117,120)
(121,136)(122,134)(123,135)(124,133)(125,140)(126,138)(127,139)(128,137)
(129,144)(130,142)(131,143)(132,141);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(144)!(  5,  9)(  6, 10)(  7, 11)(  8, 12)( 17, 21)( 18, 22)( 19, 23)
( 20, 24)( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 41, 45)( 42, 46)( 43, 47)
( 44, 48)( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 69)( 66, 70)( 67, 71)
( 68, 72)( 73,109)( 74,110)( 75,111)( 76,112)( 77,117)( 78,118)( 79,119)
( 80,120)( 81,113)( 82,114)( 83,115)( 84,116)( 85,121)( 86,122)( 87,123)
( 88,124)( 89,129)( 90,130)( 91,131)( 92,132)( 93,125)( 94,126)( 95,127)
( 96,128)( 97,133)( 98,134)( 99,135)(100,136)(101,141)(102,142)(103,143)
(104,144)(105,137)(106,138)(107,139)(108,140);
s1 := Sym(144)!(  1, 77)(  2, 79)(  3, 78)(  4, 80)(  5, 73)(  6, 75)(  7, 74)
(  8, 76)(  9, 81)( 10, 83)( 11, 82)( 12, 84)( 13,101)( 14,103)( 15,102)
( 16,104)( 17, 97)( 18, 99)( 19, 98)( 20,100)( 21,105)( 22,107)( 23,106)
( 24,108)( 25, 89)( 26, 91)( 27, 90)( 28, 92)( 29, 85)( 30, 87)( 31, 86)
( 32, 88)( 33, 93)( 34, 95)( 35, 94)( 36, 96)( 37,113)( 38,115)( 39,114)
( 40,116)( 41,109)( 42,111)( 43,110)( 44,112)( 45,117)( 46,119)( 47,118)
( 48,120)( 49,137)( 50,139)( 51,138)( 52,140)( 53,133)( 54,135)( 55,134)
( 56,136)( 57,141)( 58,143)( 59,142)( 60,144)( 61,125)( 62,127)( 63,126)
( 64,128)( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)( 71,130)
( 72,132);
s2 := Sym(144)!(  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 17)(  6, 18)(  7, 20)
(  8, 19)(  9, 21)( 10, 22)( 11, 24)( 12, 23)( 27, 28)( 31, 32)( 35, 36)
( 37, 49)( 38, 50)( 39, 52)( 40, 51)( 41, 53)( 42, 54)( 43, 56)( 44, 55)
( 45, 57)( 46, 58)( 47, 60)( 48, 59)( 63, 64)( 67, 68)( 71, 72)( 73, 85)
( 74, 86)( 75, 88)( 76, 87)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 81, 93)
( 82, 94)( 83, 96)( 84, 95)( 99,100)(103,104)(107,108)(109,121)(110,122)
(111,124)(112,123)(113,125)(114,126)(115,128)(116,127)(117,129)(118,130)
(119,132)(120,131)(135,136)(139,140)(143,144);
s3 := Sym(144)!(  1,  4)(  5,  8)(  9, 12)( 13, 28)( 14, 26)( 15, 27)( 16, 25)
( 17, 32)( 18, 30)( 19, 31)( 20, 29)( 21, 36)( 22, 34)( 23, 35)( 24, 33)
( 37, 40)( 41, 44)( 45, 48)( 49, 64)( 50, 62)( 51, 63)( 52, 61)( 53, 68)
( 54, 66)( 55, 67)( 56, 65)( 57, 72)( 58, 70)( 59, 71)( 60, 69)( 73, 76)
( 77, 80)( 81, 84)( 85,100)( 86, 98)( 87, 99)( 88, 97)( 89,104)( 90,102)
( 91,103)( 92,101)( 93,108)( 94,106)( 95,107)( 96,105)(109,112)(113,116)
(117,120)(121,136)(122,134)(123,135)(124,133)(125,140)(126,138)(127,139)
(128,137)(129,144)(130,142)(131,143)(132,141);
poly := sub<Sym(144)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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 >; 
 
References : None.
to this polytope