Polytope of Type {12,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,4}*576a
Also Known As : {{12,6|2},{6,4|2}}. if this polytope has another name.
Group : SmallGroup(576,6159)
Rank : 4
Schlafli Type : {12,6,4}
Number of vertices, edges, etc : 12, 36, 12, 4
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {12,6,4,2} of size 1152
Vertex Figure Of :
   {2,12,6,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,6,2}*288a, {6,6,4}*288a
   3-fold quotients : {12,2,4}*192, {4,6,4}*192a
   4-fold quotients : {6,6,2}*144a
   6-fold quotients : {12,2,2}*96, {2,6,4}*96a, {4,6,2}*96a, {6,2,4}*96
   9-fold quotients : {4,2,4}*64
   12-fold quotients : {3,2,4}*48, {2,6,2}*48, {6,2,2}*48
   18-fold quotients : {2,2,4}*32, {4,2,2}*32
   24-fold quotients : {2,3,2}*24, {3,2,2}*24
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,12,4}*1152b, {12,6,8}*1152b, {24,6,4}*1152b
   3-fold covers : {36,6,4}*1728a, {12,18,4}*1728a, {12,6,4}*1728a, {12,6,12}*1728b, {12,6,12}*1728d, {12,6,4}*1728h
Permutation Representation (GAP) :

s0 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 37, 55)( 38, 56)( 39, 57)( 40, 61)
( 41, 62)( 42, 63)( 43, 58)( 44, 59)( 45, 60)( 46, 64)( 47, 65)( 48, 66)
( 49, 70)( 50, 71)( 51, 72)( 52, 67)( 53, 68)( 54, 69)( 76, 79)( 77, 80)
( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)( 96, 99)(103,106)
(104,107)(105,108)(109,127)(110,128)(111,129)(112,133)(113,134)(114,135)
(115,130)(116,131)(117,132)(118,136)(119,137)(120,138)(121,142)(122,143)
(123,144)(124,139)(125,140)(126,141);;
s1 := (  1, 40)(  2, 42)(  3, 41)(  4, 37)(  5, 39)(  6, 38)(  7, 43)(  8, 45)
(  9, 44)( 10, 49)( 11, 51)( 12, 50)( 13, 46)( 14, 48)( 15, 47)( 16, 52)
( 17, 54)( 18, 53)( 19, 58)( 20, 60)( 21, 59)( 22, 55)( 23, 57)( 24, 56)
( 25, 61)( 26, 63)( 27, 62)( 28, 67)( 29, 69)( 30, 68)( 31, 64)( 32, 66)
( 33, 65)( 34, 70)( 35, 72)( 36, 71)( 73,121)( 74,123)( 75,122)( 76,118)
( 77,120)( 78,119)( 79,124)( 80,126)( 81,125)( 82,112)( 83,114)( 84,113)
( 85,109)( 86,111)( 87,110)( 88,115)( 89,117)( 90,116)( 91,139)( 92,141)
( 93,140)( 94,136)( 95,138)( 96,137)( 97,142)( 98,144)( 99,143)(100,130)
(101,132)(102,131)(103,127)(104,129)(105,128)(106,133)(107,135)(108,134);;
s2 := (  1,  2)(  4,  5)(  7,  8)( 10, 11)( 13, 14)( 16, 17)( 19, 20)( 22, 23)
( 25, 26)( 28, 29)( 31, 32)( 34, 35)( 37, 38)( 40, 41)( 43, 44)( 46, 47)
( 49, 50)( 52, 53)( 55, 56)( 58, 59)( 61, 62)( 64, 65)( 67, 68)( 70, 71)
( 73, 83)( 74, 82)( 75, 84)( 76, 86)( 77, 85)( 78, 87)( 79, 89)( 80, 88)
( 81, 90)( 91,101)( 92,100)( 93,102)( 94,104)( 95,103)( 96,105)( 97,107)
( 98,106)( 99,108)(109,119)(110,118)(111,120)(112,122)(113,121)(114,123)
(115,125)(116,124)(117,126)(127,137)(128,136)(129,138)(130,140)(131,139)
(132,141)(133,143)(134,142)(135,144);;
s3 := (  1, 73)(  2, 74)(  3, 75)(  4, 76)(  5, 77)(  6, 78)(  7, 79)(  8, 80)
(  9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)( 16, 88)
( 17, 89)( 18, 90)( 19, 91)( 20, 92)( 21, 93)( 22, 94)( 23, 95)( 24, 96)
( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,101)( 30,102)( 31,103)( 32,104)
( 33,105)( 34,106)( 35,107)( 36,108)( 37,118)( 38,119)( 39,120)( 40,121)
( 41,122)( 42,123)( 43,124)( 44,125)( 45,126)( 46,109)( 47,110)( 48,111)
( 49,112)( 50,113)( 51,114)( 52,115)( 53,116)( 54,117)( 55,136)( 56,137)
( 57,138)( 58,139)( 59,140)( 60,141)( 61,142)( 62,143)( 63,144)( 64,127)
( 65,128)( 66,129)( 67,130)( 68,131)( 69,132)( 70,133)( 71,134)( 72,135);;
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, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
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)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 37, 55)( 38, 56)( 39, 57)
( 40, 61)( 41, 62)( 42, 63)( 43, 58)( 44, 59)( 45, 60)( 46, 64)( 47, 65)
( 48, 66)( 49, 70)( 50, 71)( 51, 72)( 52, 67)( 53, 68)( 54, 69)( 76, 79)
( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)( 96, 99)
(103,106)(104,107)(105,108)(109,127)(110,128)(111,129)(112,133)(113,134)
(114,135)(115,130)(116,131)(117,132)(118,136)(119,137)(120,138)(121,142)
(122,143)(123,144)(124,139)(125,140)(126,141);
s1 := Sym(144)!(  1, 40)(  2, 42)(  3, 41)(  4, 37)(  5, 39)(  6, 38)(  7, 43)
(  8, 45)(  9, 44)( 10, 49)( 11, 51)( 12, 50)( 13, 46)( 14, 48)( 15, 47)
( 16, 52)( 17, 54)( 18, 53)( 19, 58)( 20, 60)( 21, 59)( 22, 55)( 23, 57)
( 24, 56)( 25, 61)( 26, 63)( 27, 62)( 28, 67)( 29, 69)( 30, 68)( 31, 64)
( 32, 66)( 33, 65)( 34, 70)( 35, 72)( 36, 71)( 73,121)( 74,123)( 75,122)
( 76,118)( 77,120)( 78,119)( 79,124)( 80,126)( 81,125)( 82,112)( 83,114)
( 84,113)( 85,109)( 86,111)( 87,110)( 88,115)( 89,117)( 90,116)( 91,139)
( 92,141)( 93,140)( 94,136)( 95,138)( 96,137)( 97,142)( 98,144)( 99,143)
(100,130)(101,132)(102,131)(103,127)(104,129)(105,128)(106,133)(107,135)
(108,134);
s2 := Sym(144)!(  1,  2)(  4,  5)(  7,  8)( 10, 11)( 13, 14)( 16, 17)( 19, 20)
( 22, 23)( 25, 26)( 28, 29)( 31, 32)( 34, 35)( 37, 38)( 40, 41)( 43, 44)
( 46, 47)( 49, 50)( 52, 53)( 55, 56)( 58, 59)( 61, 62)( 64, 65)( 67, 68)
( 70, 71)( 73, 83)( 74, 82)( 75, 84)( 76, 86)( 77, 85)( 78, 87)( 79, 89)
( 80, 88)( 81, 90)( 91,101)( 92,100)( 93,102)( 94,104)( 95,103)( 96,105)
( 97,107)( 98,106)( 99,108)(109,119)(110,118)(111,120)(112,122)(113,121)
(114,123)(115,125)(116,124)(117,126)(127,137)(128,136)(129,138)(130,140)
(131,139)(132,141)(133,143)(134,142)(135,144);
s3 := Sym(144)!(  1, 73)(  2, 74)(  3, 75)(  4, 76)(  5, 77)(  6, 78)(  7, 79)
(  8, 80)(  9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)
( 16, 88)( 17, 89)( 18, 90)( 19, 91)( 20, 92)( 21, 93)( 22, 94)( 23, 95)
( 24, 96)( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,101)( 30,102)( 31,103)
( 32,104)( 33,105)( 34,106)( 35,107)( 36,108)( 37,118)( 38,119)( 39,120)
( 40,121)( 41,122)( 42,123)( 43,124)( 44,125)( 45,126)( 46,109)( 47,110)
( 48,111)( 49,112)( 50,113)( 51,114)( 52,115)( 53,116)( 54,117)( 55,136)
( 56,137)( 57,138)( 58,139)( 59,140)( 60,141)( 61,142)( 62,143)( 63,144)
( 64,127)( 65,128)( 66,129)( 67,130)( 68,131)( 69,132)( 70,133)( 71,134)
( 72,135);
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, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, 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