Polytope of Type {3,6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,12}*576
Also Known As : {{3,6}₄,{6,12|2}}. if this polytope has another name.
Group : SmallGroup(576,8313)
Rank : 4
Schlafli Type : {3,6,12}
Number of vertices, edges, etc : 4, 12, 48, 12
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,12,2} of size 1152
Vertex Figure Of :
   {2,3,6,12} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,6}*288
   3-fold quotients : {3,6,4}*192
   6-fold quotients : {3,6,2}*96
   12-fold quotients : {3,3,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,12,12}*1152, {3,6,24}*1152, {6,6,12}*1152a
   3-fold covers : {3,6,36}*1728, {3,6,12}*1728
Permutation Representation (GAP) :

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