Polytope of Type {3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6}*1728
Also Known As : {3,6}_(12,0), {3,6}₂₄. if this polytope has another name.
Group : SmallGroup(1728,12317)
Rank : 3
Schlafli Type : {3,6}
Number of vertices, edges, etc : 144, 432, 288
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,6}*576
   4-fold quotients : {3,6}*432
   9-fold quotients : {3,6}*192
   12-fold quotients : {3,6}*144
   16-fold quotients : {3,6}*108
   36-fold quotients : {3,6}*48
   48-fold quotients : {3,6}*36
   72-fold quotients : {3,3}*24
   144-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

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

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

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

References : None.
to this polytope