Polytope of Type {12,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,4}*1728i
if this polytope has a name.
Group : SmallGroup(1728,46119)
Rank : 4
Schlafli Type : {12,6,4}
Number of vertices, edges, etc : 36, 108, 36, 4
Order of s₀s₁s₂s₃ : 3
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 : {12,6,4}*576f
   4-fold quotients : {6,6,4}*432
   9-fold quotients : {4,6,4}*192f
   18-fold quotients : {4,3,4}*96
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)( 17, 33)( 18, 36)
( 19, 35)( 20, 34)( 21, 45)( 22, 48)( 23, 47)( 24, 46)( 25, 41)( 26, 44)
( 27, 43)( 28, 42)( 29, 37)( 30, 40)( 31, 39)( 32, 38)( 49, 65)( 50, 68)
( 51, 67)( 52, 66)( 53, 77)( 54, 80)( 55, 79)( 56, 78)( 57, 73)( 58, 76)
( 59, 75)( 60, 74)( 61, 69)( 62, 72)( 63, 71)( 64, 70)( 82, 84)( 85, 93)
( 86, 96)( 87, 95)( 88, 94)( 90, 92)( 97,129)( 98,132)( 99,131)(100,130)
(101,141)(102,144)(103,143)(104,142)(105,137)(106,140)(107,139)(108,138)
(109,133)(110,136)(111,135)(112,134)(114,116)(117,125)(118,128)(119,127)
(120,126)(122,124);;
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, s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s0*s1*s2*s0*s1*s2, s3*s2*s1*s3*s2*s3*s2*s1*s2, 
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)!(  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)( 17, 33)
( 18, 36)( 19, 35)( 20, 34)( 21, 45)( 22, 48)( 23, 47)( 24, 46)( 25, 41)
( 26, 44)( 27, 43)( 28, 42)( 29, 37)( 30, 40)( 31, 39)( 32, 38)( 49, 65)
( 50, 68)( 51, 67)( 52, 66)( 53, 77)( 54, 80)( 55, 79)( 56, 78)( 57, 73)
( 58, 76)( 59, 75)( 60, 74)( 61, 69)( 62, 72)( 63, 71)( 64, 70)( 82, 84)
( 85, 93)( 86, 96)( 87, 95)( 88, 94)( 90, 92)( 97,129)( 98,132)( 99,131)
(100,130)(101,141)(102,144)(103,143)(104,142)(105,137)(106,140)(107,139)
(108,138)(109,133)(110,136)(111,135)(112,134)(114,116)(117,125)(118,128)
(119,127)(120,126)(122,124);
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, 
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s0*s1*s2*s0*s1*s2, 
s3*s2*s1*s3*s2*s3*s2*s1*s2, 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