Polytope of Type {4,6,12}

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

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

Permutation Representation (Magma) :

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

References : None.
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