Polytope of Type {9,4,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,4,3}*1728
if this polytope has a name.
Group : SmallGroup(1728,46100)
Rank : 4
Schlafli Type : {9,4,3}
Number of vertices, edges, etc : 36, 144, 48, 12
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 4
Special Properties :
   Orientable
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 : {3,4,3}*576
   16-fold quotients : {9,2,3}*108
   48-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope