Polytope of Type {36,6,4}

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

References : None.
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