Polytope of Type {4,6,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,36}*1728d
if this polytope has a name.
Group : SmallGroup(1728,46117)
Rank : 4
Schlafli Type : {4,6,36}
Number of vertices, edges, etc : 4, 12, 108, 36
Order of s₀s₁s₂s₃ : 9
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 : {4,6,12}*576f
   9-fold quotients : {4,6,4}*192e
   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, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)( 24, 32)
( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)
( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)( 56, 64)
( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)( 72, 80)
( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)( 88, 96)
( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)(104,112)
(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)
(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144);;
s1 := (  3,  4)(  7,  8)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 19, 20)( 23, 24)
( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 35, 36)( 39, 40)( 41, 45)( 42, 46)
( 43, 48)( 44, 47)( 51, 52)( 55, 56)( 57, 61)( 58, 62)( 59, 64)( 60, 63)
( 67, 68)( 71, 72)( 73, 77)( 74, 78)( 75, 80)( 76, 79)( 83, 84)( 87, 88)
( 89, 93)( 90, 94)( 91, 96)( 92, 95)( 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)( 49,129)( 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)( 65,113)( 66,116)
( 67,115)( 68,114)( 69,125)( 70,128)( 71,127)( 72,126)( 73,121)( 74,124)
( 75,123)( 76,122)( 77,117)( 78,120)( 79,119)( 80,118)( 81, 97)( 82,100)
( 83, 99)( 84, 98)( 85,109)( 86,112)( 87,111)( 88,110)( 89,105)( 90,108)
( 91,107)( 92,106)( 93,101)( 94,104)( 95,103)( 96,102);;
s3 := (  1, 50)(  2, 49)(  3, 52)(  4, 51)(  5, 54)(  6, 53)(  7, 56)(  8, 55)
(  9, 58)( 10, 57)( 11, 60)( 12, 59)( 13, 62)( 14, 61)( 15, 64)( 16, 63)
( 17, 82)( 18, 81)( 19, 84)( 20, 83)( 21, 86)( 22, 85)( 23, 88)( 24, 87)
( 25, 90)( 26, 89)( 27, 92)( 28, 91)( 29, 94)( 30, 93)( 31, 96)( 32, 95)
( 33, 66)( 34, 65)( 35, 68)( 36, 67)( 37, 70)( 38, 69)( 39, 72)( 40, 71)
( 41, 74)( 42, 73)( 43, 76)( 44, 75)( 45, 78)( 46, 77)( 47, 80)( 48, 79)
( 97,130)( 98,129)( 99,132)(100,131)(101,134)(102,133)(103,136)(104,135)
(105,138)(106,137)(107,140)(108,139)(109,142)(110,141)(111,144)(112,143)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128);;
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3 ];;
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, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)
( 24, 32)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)
( 40, 48)( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)
( 56, 64)( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)
( 72, 80)( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)
( 88, 96)( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)
(104,112)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)
(120,128)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)
(136,144);
s1 := Sym(144)!(  3,  4)(  7,  8)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 19, 20)
( 23, 24)( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 35, 36)( 39, 40)( 41, 45)
( 42, 46)( 43, 48)( 44, 47)( 51, 52)( 55, 56)( 57, 61)( 58, 62)( 59, 64)
( 60, 63)( 67, 68)( 71, 72)( 73, 77)( 74, 78)( 75, 80)( 76, 79)( 83, 84)
( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)( 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)( 49,129)
( 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)( 65,113)
( 66,116)( 67,115)( 68,114)( 69,125)( 70,128)( 71,127)( 72,126)( 73,121)
( 74,124)( 75,123)( 76,122)( 77,117)( 78,120)( 79,119)( 80,118)( 81, 97)
( 82,100)( 83, 99)( 84, 98)( 85,109)( 86,112)( 87,111)( 88,110)( 89,105)
( 90,108)( 91,107)( 92,106)( 93,101)( 94,104)( 95,103)( 96,102);
s3 := Sym(144)!(  1, 50)(  2, 49)(  3, 52)(  4, 51)(  5, 54)(  6, 53)(  7, 56)
(  8, 55)(  9, 58)( 10, 57)( 11, 60)( 12, 59)( 13, 62)( 14, 61)( 15, 64)
( 16, 63)( 17, 82)( 18, 81)( 19, 84)( 20, 83)( 21, 86)( 22, 85)( 23, 88)
( 24, 87)( 25, 90)( 26, 89)( 27, 92)( 28, 91)( 29, 94)( 30, 93)( 31, 96)
( 32, 95)( 33, 66)( 34, 65)( 35, 68)( 36, 67)( 37, 70)( 38, 69)( 39, 72)
( 40, 71)( 41, 74)( 42, 73)( 43, 76)( 44, 75)( 45, 78)( 46, 77)( 47, 80)
( 48, 79)( 97,130)( 98,129)( 99,132)(100,131)(101,134)(102,133)(103,136)
(104,135)(105,138)(106,137)(107,140)(108,139)(109,142)(110,141)(111,144)
(112,143)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128);
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, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3 >;

References : None.
to this polytope