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Polytope of Type {3,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,12}*1728
if this polytope has a name.
Group : SmallGroup(1728,46099)
Rank : 3
Schlafli Type : {3,12}
Number of vertices, edges, etc : 72, 432, 288
Order of s0s1s2 : 12
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   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 : {3,12}*576
   4-fold quotients : {3,6}*432, {3,12}*432
   12-fold quotients : {3,6}*144, {3,12}*144
   16-fold quotients : {3,6}*108
   36-fold quotients : {3,4}*48, {3,6}*48
   48-fold quotients : {3,6}*36
   72-fold quotients : {3,3}*24, {3,4}*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, 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)( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,101)( 54,102)
( 55,104)( 56,103)( 57,109)( 58,110)( 59,112)( 60,111)( 61,105)( 62,106)
( 63,108)( 64,107)( 65,113)( 66,114)( 67,116)( 68,115)( 69,117)( 70,118)
( 71,120)( 72,119)( 73,125)( 74,126)( 75,128)( 76,127)( 77,121)( 78,122)
( 79,124)( 80,123)( 81,129)( 82,130)( 83,132)( 84,131)( 85,133)( 86,134)
( 87,136)( 88,135)( 89,141)( 90,142)( 91,144)( 92,143)( 93,137)( 94,138)
( 95,140)( 96,139);;
s1 := (  1, 65)(  2, 67)(  3, 66)(  4, 68)(  5, 77)(  6, 79)(  7, 78)(  8, 80)
(  9, 73)( 10, 75)( 11, 74)( 12, 76)( 13, 69)( 14, 71)( 15, 70)( 16, 72)
( 17, 81)( 18, 83)( 19, 82)( 20, 84)( 21, 93)( 22, 95)( 23, 94)( 24, 96)
( 25, 89)( 26, 91)( 27, 90)( 28, 92)( 29, 85)( 30, 87)( 31, 86)( 32, 88)
( 33, 49)( 34, 51)( 35, 50)( 36, 52)( 37, 61)( 38, 63)( 39, 62)( 40, 64)
( 41, 57)( 42, 59)( 43, 58)( 44, 60)( 45, 53)( 46, 55)( 47, 54)( 48, 56)
( 98, 99)(101,109)(102,111)(103,110)(104,112)(106,107)(114,115)(117,125)
(118,127)(119,126)(120,128)(122,123)(130,131)(133,141)(134,143)(135,142)
(136,144)(138,139);;
s2 := (  1,  6)(  2,  5)(  3,  7)(  4,  8)(  9, 14)( 10, 13)( 11, 15)( 12, 16)
( 17, 38)( 18, 37)( 19, 39)( 20, 40)( 21, 34)( 22, 33)( 23, 35)( 24, 36)
( 25, 46)( 26, 45)( 27, 47)( 28, 48)( 29, 42)( 30, 41)( 31, 43)( 32, 44)
( 49, 54)( 50, 53)( 51, 55)( 52, 56)( 57, 62)( 58, 61)( 59, 63)( 60, 64)
( 65, 86)( 66, 85)( 67, 87)( 68, 88)( 69, 82)( 70, 81)( 71, 83)( 72, 84)
( 73, 94)( 74, 93)( 75, 95)( 76, 96)( 77, 90)( 78, 89)( 79, 91)( 80, 92)
( 97,102)( 98,101)( 99,103)(100,104)(105,110)(106,109)(107,111)(108,112)
(113,134)(114,133)(115,135)(116,136)(117,130)(118,129)(119,131)(120,132)
(121,142)(122,141)(123,143)(124,144)(125,138)(126,137)(127,139)(128,140);;
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*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, 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)( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,101)
( 54,102)( 55,104)( 56,103)( 57,109)( 58,110)( 59,112)( 60,111)( 61,105)
( 62,106)( 63,108)( 64,107)( 65,113)( 66,114)( 67,116)( 68,115)( 69,117)
( 70,118)( 71,120)( 72,119)( 73,125)( 74,126)( 75,128)( 76,127)( 77,121)
( 78,122)( 79,124)( 80,123)( 81,129)( 82,130)( 83,132)( 84,131)( 85,133)
( 86,134)( 87,136)( 88,135)( 89,141)( 90,142)( 91,144)( 92,143)( 93,137)
( 94,138)( 95,140)( 96,139);
s1 := Sym(144)!(  1, 65)(  2, 67)(  3, 66)(  4, 68)(  5, 77)(  6, 79)(  7, 78)
(  8, 80)(  9, 73)( 10, 75)( 11, 74)( 12, 76)( 13, 69)( 14, 71)( 15, 70)
( 16, 72)( 17, 81)( 18, 83)( 19, 82)( 20, 84)( 21, 93)( 22, 95)( 23, 94)
( 24, 96)( 25, 89)( 26, 91)( 27, 90)( 28, 92)( 29, 85)( 30, 87)( 31, 86)
( 32, 88)( 33, 49)( 34, 51)( 35, 50)( 36, 52)( 37, 61)( 38, 63)( 39, 62)
( 40, 64)( 41, 57)( 42, 59)( 43, 58)( 44, 60)( 45, 53)( 46, 55)( 47, 54)
( 48, 56)( 98, 99)(101,109)(102,111)(103,110)(104,112)(106,107)(114,115)
(117,125)(118,127)(119,126)(120,128)(122,123)(130,131)(133,141)(134,143)
(135,142)(136,144)(138,139);
s2 := Sym(144)!(  1,  6)(  2,  5)(  3,  7)(  4,  8)(  9, 14)( 10, 13)( 11, 15)
( 12, 16)( 17, 38)( 18, 37)( 19, 39)( 20, 40)( 21, 34)( 22, 33)( 23, 35)
( 24, 36)( 25, 46)( 26, 45)( 27, 47)( 28, 48)( 29, 42)( 30, 41)( 31, 43)
( 32, 44)( 49, 54)( 50, 53)( 51, 55)( 52, 56)( 57, 62)( 58, 61)( 59, 63)
( 60, 64)( 65, 86)( 66, 85)( 67, 87)( 68, 88)( 69, 82)( 70, 81)( 71, 83)
( 72, 84)( 73, 94)( 74, 93)( 75, 95)( 76, 96)( 77, 90)( 78, 89)( 79, 91)
( 80, 92)( 97,102)( 98,101)( 99,103)(100,104)(105,110)(106,109)(107,111)
(108,112)(113,134)(114,133)(115,135)(116,136)(117,130)(118,129)(119,131)
(120,132)(121,142)(122,141)(123,143)(124,144)(125,138)(126,137)(127,139)
(128,140);
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*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.
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