Polytope of Type {12,4,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4,3}*576
Also Known As : {{12,4|2},{4,3}}. if this polytope has another name.
Group : SmallGroup(576,8313)
Rank : 4
Schlafli Type : {12,4,3}
Number of vertices, edges, etc : 12, 48, 12, 6
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {12,4,3,2} of size 1152
Vertex Figure Of :
   {2,12,4,3} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,4,3}*288
   3-fold quotients : {4,4,3}*192b
   4-fold quotients : {12,2,3}*144
   6-fold quotients : {2,4,3}*96
   8-fold quotients : {6,2,3}*72
   12-fold quotients : {4,2,3}*48, {2,4,3}*48
   16-fold quotients : {3,2,3}*36
   24-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,8,3}*1152, {24,4,3}*1152, {12,4,6}*1152c
   3-fold covers : {36,4,3}*1728, {12,4,9}*1728, {12,12,3}*1728a, {12,12,3}*1728b
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, 79)(  2, 80)(  3, 77)(  4, 78)(  5, 75)(  6, 76)(  7, 73)(  8, 74)
(  9, 83)( 10, 84)( 11, 81)( 12, 82)( 13, 91)( 14, 92)( 15, 89)( 16, 90)
( 17, 87)( 18, 88)( 19, 85)( 20, 86)( 21, 95)( 22, 96)( 23, 93)( 24, 94)
( 25,103)( 26,104)( 27,101)( 28,102)( 29, 99)( 30,100)( 31, 97)( 32, 98)
( 33,107)( 34,108)( 35,105)( 36,106)( 37,115)( 38,116)( 39,113)( 40,114)
( 41,111)( 42,112)( 43,109)( 44,110)( 45,119)( 46,120)( 47,117)( 48,118)
( 49,127)( 50,128)( 51,125)( 52,126)( 53,123)( 54,124)( 55,121)( 56,122)
( 57,131)( 58,132)( 59,129)( 60,130)( 61,139)( 62,140)( 63,137)( 64,138)
( 65,135)( 66,136)( 67,133)( 68,134)( 69,143)( 70,144)( 71,141)( 72,142);;
s2 := (  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)( 17, 29)
( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 38, 39)
( 42, 43)( 46, 47)( 49, 61)( 50, 63)( 51, 62)( 52, 64)( 53, 65)( 54, 67)
( 55, 66)( 56, 68)( 57, 69)( 58, 71)( 59, 70)( 60, 72)( 74, 75)( 78, 79)
( 82, 83)( 85, 97)( 86, 99)( 87, 98)( 88,100)( 89,101)( 90,103)( 91,102)
( 92,104)( 93,105)( 94,107)( 95,106)( 96,108)(110,111)(114,115)(118,119)
(121,133)(122,135)(123,134)(124,136)(125,137)(126,139)(127,138)(128,140)
(129,141)(130,143)(131,142)(132,144);;
s3 := (  1, 25)(  2, 28)(  3, 27)(  4, 26)(  5, 29)(  6, 32)(  7, 31)(  8, 30)
(  9, 33)( 10, 36)( 11, 35)( 12, 34)( 14, 16)( 18, 20)( 22, 24)( 37, 61)
( 38, 64)( 39, 63)( 40, 62)( 41, 65)( 42, 68)( 43, 67)( 44, 66)( 45, 69)
( 46, 72)( 47, 71)( 48, 70)( 50, 52)( 54, 56)( 58, 60)( 73, 97)( 74,100)
( 75, 99)( 76, 98)( 77,101)( 78,104)( 79,103)( 80,102)( 81,105)( 82,108)
( 83,107)( 84,106)( 86, 88)( 90, 92)( 94, 96)(109,133)(110,136)(111,135)
(112,134)(113,137)(114,140)(115,139)(116,138)(117,141)(118,144)(119,143)
(120,142)(122,124)(126,128)(130,132);;
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*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, 79)(  2, 80)(  3, 77)(  4, 78)(  5, 75)(  6, 76)(  7, 73)
(  8, 74)(  9, 83)( 10, 84)( 11, 81)( 12, 82)( 13, 91)( 14, 92)( 15, 89)
( 16, 90)( 17, 87)( 18, 88)( 19, 85)( 20, 86)( 21, 95)( 22, 96)( 23, 93)
( 24, 94)( 25,103)( 26,104)( 27,101)( 28,102)( 29, 99)( 30,100)( 31, 97)
( 32, 98)( 33,107)( 34,108)( 35,105)( 36,106)( 37,115)( 38,116)( 39,113)
( 40,114)( 41,111)( 42,112)( 43,109)( 44,110)( 45,119)( 46,120)( 47,117)
( 48,118)( 49,127)( 50,128)( 51,125)( 52,126)( 53,123)( 54,124)( 55,121)
( 56,122)( 57,131)( 58,132)( 59,129)( 60,130)( 61,139)( 62,140)( 63,137)
( 64,138)( 65,135)( 66,136)( 67,133)( 68,134)( 69,143)( 70,144)( 71,141)
( 72,142);
s2 := Sym(144)!(  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)
( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)
( 38, 39)( 42, 43)( 46, 47)( 49, 61)( 50, 63)( 51, 62)( 52, 64)( 53, 65)
( 54, 67)( 55, 66)( 56, 68)( 57, 69)( 58, 71)( 59, 70)( 60, 72)( 74, 75)
( 78, 79)( 82, 83)( 85, 97)( 86, 99)( 87, 98)( 88,100)( 89,101)( 90,103)
( 91,102)( 92,104)( 93,105)( 94,107)( 95,106)( 96,108)(110,111)(114,115)
(118,119)(121,133)(122,135)(123,134)(124,136)(125,137)(126,139)(127,138)
(128,140)(129,141)(130,143)(131,142)(132,144);
s3 := Sym(144)!(  1, 25)(  2, 28)(  3, 27)(  4, 26)(  5, 29)(  6, 32)(  7, 31)
(  8, 30)(  9, 33)( 10, 36)( 11, 35)( 12, 34)( 14, 16)( 18, 20)( 22, 24)
( 37, 61)( 38, 64)( 39, 63)( 40, 62)( 41, 65)( 42, 68)( 43, 67)( 44, 66)
( 45, 69)( 46, 72)( 47, 71)( 48, 70)( 50, 52)( 54, 56)( 58, 60)( 73, 97)
( 74,100)( 75, 99)( 76, 98)( 77,101)( 78,104)( 79,103)( 80,102)( 81,105)
( 82,108)( 83,107)( 84,106)( 86, 88)( 90, 92)( 94, 96)(109,133)(110,136)
(111,135)(112,134)(113,137)(114,140)(115,139)(116,138)(117,141)(118,144)
(119,143)(120,142)(122,124)(126,128)(130,132);
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*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