Polytope of Type {12,3,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,3,4}*1728
Also Known As : {{12,3}₆,{3,4}₃}. if this polytope has another name.
Group : SmallGroup(1728,46119)
Rank : 4
Schlafli Type : {12,3,4}
Number of vertices, edges, etc : 72, 108, 36, 4
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {12,3,4}*576
   4-fold quotients : {6,3,4}*432
   9-fold quotients : {4,3,4}*192b
   12-fold quotients : {6,3,4}*144
   18-fold quotients : {4,3,4}*96
   36-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope