Polytope of Type {6,3,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3,4}*1728
Also Known As : {{6,3}₁₂,{3,4}₃}. if this polytope has another name.
Group : SmallGroup(1728,46099)
Rank : 4
Schlafli Type : {6,3,4}
Number of vertices, edges, etc : 72, 108, 72, 4
Order of s₀s₁s₂s₃ : 12
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 : {6,3,4}*576
   4-fold quotients : {6,3,4}*432
   12-fold quotients : {6,3,4}*144
   36-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope