Polytope of Type {12,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,4}*576b
if this polytope has a name.
Group : SmallGroup(576,6956)
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}*288b, {6,6,4}*288c
   3-fold quotients : {12,2,4}*192
   4-fold quotients : {3,6,4}*144, {6,6,2}*144c
   6-fold quotients : {12,2,2}*96, {6,2,4}*96
   8-fold quotients : {3,6,2}*72
   9-fold quotients : {4,2,4}*64
   12-fold quotients : {3,2,4}*48, {6,2,2}*48
   18-fold quotients : {2,2,4}*32, {4,2,2}*32
   24-fold quotients : {3,2,2}*24
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,12,4}*1152c, {12,6,8}*1152c, {24,6,4}*1152c
   3-fold covers : {36,6,4}*1728b, {12,6,4}*1728b, {12,6,12}*1728f, {12,6,4}*1728h
Permutation Representation (GAP) :

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