Polytope of Type {3,4,4,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4,4,9}*1728
if this polytope has a name.
Group : SmallGroup(1728,46100)
Rank : 5
Schlafli Type : {3,4,4,9}
Number of vertices, edges, etc : 3, 12, 16, 36, 9
Order of s₀s₁s₂s₃s₄ : 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 4
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 : {3,4,4,3}*576
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  5)(  3, 13)(  4,  9)(  7, 14)(  8, 10)( 11, 16)( 18, 21)( 19, 29)
( 20, 25)( 23, 30)( 24, 26)( 27, 32)( 34, 37)( 35, 45)( 36, 41)( 39, 46)
( 40, 42)( 43, 48)( 50, 53)( 51, 61)( 52, 57)( 55, 62)( 56, 58)( 59, 64)
( 66, 69)( 67, 77)( 68, 73)( 71, 78)( 72, 74)( 75, 80)( 82, 85)( 83, 93)
( 84, 89)( 87, 94)( 88, 90)( 91, 96)( 98,101)( 99,109)(100,105)(103,110)
(104,106)(107,112)(114,117)(115,125)(116,121)(119,126)(120,122)(123,128)
(130,133)(131,141)(132,137)(135,142)(136,138)(139,144);;
s1 := (  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);;
s2 := (  1,  6)(  2,  5)(  3,  8)(  4,  7)(  9, 14)( 10, 13)( 11, 16)( 12, 15)
( 17, 22)( 18, 21)( 19, 24)( 20, 23)( 25, 30)( 26, 29)( 27, 32)( 28, 31)
( 33, 38)( 34, 37)( 35, 40)( 36, 39)( 41, 46)( 42, 45)( 43, 48)( 44, 47)
( 49, 54)( 50, 53)( 51, 56)( 52, 55)( 57, 62)( 58, 61)( 59, 64)( 60, 63)
( 65, 70)( 66, 69)( 67, 72)( 68, 71)( 73, 78)( 74, 77)( 75, 80)( 76, 79)
( 81, 86)( 82, 85)( 83, 88)( 84, 87)( 89, 94)( 90, 93)( 91, 96)( 92, 95)
( 97,102)( 98,101)( 99,104)(100,103)(105,110)(106,109)(107,112)(108,111)
(113,118)(114,117)(115,120)(116,119)(121,126)(122,125)(123,128)(124,127)
(129,134)(130,133)(131,136)(132,135)(137,142)(138,141)(139,144)(140,143);;
s3 := (  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);;
s4 := (  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);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s2*s1*s0*s2*s1*s0*s2*s1, s2*s3*s4*s3*s2*s3*s4*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(144)!(  2,  5)(  3, 13)(  4,  9)(  7, 14)(  8, 10)( 11, 16)( 18, 21)
( 19, 29)( 20, 25)( 23, 30)( 24, 26)( 27, 32)( 34, 37)( 35, 45)( 36, 41)
( 39, 46)( 40, 42)( 43, 48)( 50, 53)( 51, 61)( 52, 57)( 55, 62)( 56, 58)
( 59, 64)( 66, 69)( 67, 77)( 68, 73)( 71, 78)( 72, 74)( 75, 80)( 82, 85)
( 83, 93)( 84, 89)( 87, 94)( 88, 90)( 91, 96)( 98,101)( 99,109)(100,105)
(103,110)(104,106)(107,112)(114,117)(115,125)(116,121)(119,126)(120,122)
(123,128)(130,133)(131,141)(132,137)(135,142)(136,138)(139,144);
s1 := 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);
s2 := Sym(144)!(  1,  6)(  2,  5)(  3,  8)(  4,  7)(  9, 14)( 10, 13)( 11, 16)
( 12, 15)( 17, 22)( 18, 21)( 19, 24)( 20, 23)( 25, 30)( 26, 29)( 27, 32)
( 28, 31)( 33, 38)( 34, 37)( 35, 40)( 36, 39)( 41, 46)( 42, 45)( 43, 48)
( 44, 47)( 49, 54)( 50, 53)( 51, 56)( 52, 55)( 57, 62)( 58, 61)( 59, 64)
( 60, 63)( 65, 70)( 66, 69)( 67, 72)( 68, 71)( 73, 78)( 74, 77)( 75, 80)
( 76, 79)( 81, 86)( 82, 85)( 83, 88)( 84, 87)( 89, 94)( 90, 93)( 91, 96)
( 92, 95)( 97,102)( 98,101)( 99,104)(100,103)(105,110)(106,109)(107,112)
(108,111)(113,118)(114,117)(115,120)(116,119)(121,126)(122,125)(123,128)
(124,127)(129,134)(130,133)(131,136)(132,135)(137,142)(138,141)(139,144)
(140,143);
s3 := 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);
s4 := 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);
poly := sub<Sym(144)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s0*s2*s1*s0*s2*s1*s0*s2*s1, 
s2*s3*s4*s3*s2*s3*s4*s2*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

References : None.
to this polytope