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Polytope of Type {6,156}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,156}*1872d
if this polytope has a name.
Group : SmallGroup(1872,1037)
Rank : 3
Schlafli Type : {6,156}
Number of vertices, edges, etc : 6, 468, 156
Order of s0s1s2 : 39
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {6,52}*624b
   13-fold quotients : {6,12}*144d
   39-fold quotients : {6,4}*48b
   78-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)
( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)( 63, 64)
( 67, 68)( 71, 72)( 75, 76)( 79, 80)( 83, 84)( 87, 88)( 91, 92)( 95, 96)
( 99,100)(103,104)(107,108)(111,112)(115,116)(119,120)(123,124)(127,128)
(131,132)(135,136)(139,140)(143,144)(147,148)(151,152)(155,156);;
s1 := (  2,  4)(  5, 49)(  6, 52)(  7, 51)(  8, 50)(  9, 45)( 10, 48)( 11, 47)
( 12, 46)( 13, 41)( 14, 44)( 15, 43)( 16, 42)( 17, 37)( 18, 40)( 19, 39)
( 20, 38)( 21, 33)( 22, 36)( 23, 35)( 24, 34)( 25, 29)( 26, 32)( 27, 31)
( 28, 30)( 53,105)( 54,108)( 55,107)( 56,106)( 57,153)( 58,156)( 59,155)
( 60,154)( 61,149)( 62,152)( 63,151)( 64,150)( 65,145)( 66,148)( 67,147)
( 68,146)( 69,141)( 70,144)( 71,143)( 72,142)( 73,137)( 74,140)( 75,139)
( 76,138)( 77,133)( 78,136)( 79,135)( 80,134)( 81,129)( 82,132)( 83,131)
( 84,130)( 85,125)( 86,128)( 87,127)( 88,126)( 89,121)( 90,124)( 91,123)
( 92,122)( 93,117)( 94,120)( 95,119)( 96,118)( 97,113)( 98,116)( 99,115)
(100,114)(101,109)(102,112)(103,111)(104,110);;
s2 := (  1, 58)(  2, 57)(  3, 60)(  4, 59)(  5, 54)(  6, 53)(  7, 56)(  8, 55)
(  9,102)( 10,101)( 11,104)( 12,103)( 13, 98)( 14, 97)( 15,100)( 16, 99)
( 17, 94)( 18, 93)( 19, 96)( 20, 95)( 21, 90)( 22, 89)( 23, 92)( 24, 91)
( 25, 86)( 26, 85)( 27, 88)( 28, 87)( 29, 82)( 30, 81)( 31, 84)( 32, 83)
( 33, 78)( 34, 77)( 35, 80)( 36, 79)( 37, 74)( 38, 73)( 39, 76)( 40, 75)
( 41, 70)( 42, 69)( 43, 72)( 44, 71)( 45, 66)( 46, 65)( 47, 68)( 48, 67)
( 49, 62)( 50, 61)( 51, 64)( 52, 63)(105,110)(106,109)(107,112)(108,111)
(113,154)(114,153)(115,156)(116,155)(117,150)(118,149)(119,152)(120,151)
(121,146)(122,145)(123,148)(124,147)(125,142)(126,141)(127,144)(128,143)
(129,138)(130,137)(131,140)(132,139)(133,134)(135,136);;
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*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(156)!(  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)
( 31, 32)( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)
( 63, 64)( 67, 68)( 71, 72)( 75, 76)( 79, 80)( 83, 84)( 87, 88)( 91, 92)
( 95, 96)( 99,100)(103,104)(107,108)(111,112)(115,116)(119,120)(123,124)
(127,128)(131,132)(135,136)(139,140)(143,144)(147,148)(151,152)(155,156);
s1 := Sym(156)!(  2,  4)(  5, 49)(  6, 52)(  7, 51)(  8, 50)(  9, 45)( 10, 48)
( 11, 47)( 12, 46)( 13, 41)( 14, 44)( 15, 43)( 16, 42)( 17, 37)( 18, 40)
( 19, 39)( 20, 38)( 21, 33)( 22, 36)( 23, 35)( 24, 34)( 25, 29)( 26, 32)
( 27, 31)( 28, 30)( 53,105)( 54,108)( 55,107)( 56,106)( 57,153)( 58,156)
( 59,155)( 60,154)( 61,149)( 62,152)( 63,151)( 64,150)( 65,145)( 66,148)
( 67,147)( 68,146)( 69,141)( 70,144)( 71,143)( 72,142)( 73,137)( 74,140)
( 75,139)( 76,138)( 77,133)( 78,136)( 79,135)( 80,134)( 81,129)( 82,132)
( 83,131)( 84,130)( 85,125)( 86,128)( 87,127)( 88,126)( 89,121)( 90,124)
( 91,123)( 92,122)( 93,117)( 94,120)( 95,119)( 96,118)( 97,113)( 98,116)
( 99,115)(100,114)(101,109)(102,112)(103,111)(104,110);
s2 := Sym(156)!(  1, 58)(  2, 57)(  3, 60)(  4, 59)(  5, 54)(  6, 53)(  7, 56)
(  8, 55)(  9,102)( 10,101)( 11,104)( 12,103)( 13, 98)( 14, 97)( 15,100)
( 16, 99)( 17, 94)( 18, 93)( 19, 96)( 20, 95)( 21, 90)( 22, 89)( 23, 92)
( 24, 91)( 25, 86)( 26, 85)( 27, 88)( 28, 87)( 29, 82)( 30, 81)( 31, 84)
( 32, 83)( 33, 78)( 34, 77)( 35, 80)( 36, 79)( 37, 74)( 38, 73)( 39, 76)
( 40, 75)( 41, 70)( 42, 69)( 43, 72)( 44, 71)( 45, 66)( 46, 65)( 47, 68)
( 48, 67)( 49, 62)( 50, 61)( 51, 64)( 52, 63)(105,110)(106,109)(107,112)
(108,111)(113,154)(114,153)(115,156)(116,155)(117,150)(118,149)(119,152)
(120,151)(121,146)(122,145)(123,148)(124,147)(125,142)(126,141)(127,144)
(128,143)(129,138)(130,137)(131,140)(132,139)(133,134)(135,136);
poly := sub<Sym(156)|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*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
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