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Polytope of Type {6,58}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,58}*696
Also Known As : {6,58|2}. if this polytope has another name.
Group : SmallGroup(696,39)
Rank : 3
Schlafli Type : {6,58}
Number of vertices, edges, etc : 6, 174, 58
Order of s0s1s2 : 174
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,58,2} of size 1392
Vertex Figure Of :
   {2,6,58} of size 1392
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,58}*232
   6-fold quotients : {2,29}*116
   29-fold quotients : {6,2}*24
   58-fold quotients : {3,2}*12
   87-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,58}*1392, {6,116}*1392a
Permutation Representation (GAP) :
s0 := ( 30, 59)( 31, 60)( 32, 61)( 33, 62)( 34, 63)( 35, 64)( 36, 65)( 37, 66)
( 38, 67)( 39, 68)( 40, 69)( 41, 70)( 42, 71)( 43, 72)( 44, 73)( 45, 74)
( 46, 75)( 47, 76)( 48, 77)( 49, 78)( 50, 79)( 51, 80)( 52, 81)( 53, 82)
( 54, 83)( 55, 84)( 56, 85)( 57, 86)( 58, 87)(117,146)(118,147)(119,148)
(120,149)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)
(128,157)(129,158)(130,159)(131,160)(132,161)(133,162)(134,163)(135,164)
(136,165)(137,166)(138,167)(139,168)(140,169)(141,170)(142,171)(143,172)
(144,173)(145,174);;
s1 := (  1, 30)(  2, 58)(  3, 57)(  4, 56)(  5, 55)(  6, 54)(  7, 53)(  8, 52)
(  9, 51)( 10, 50)( 11, 49)( 12, 48)( 13, 47)( 14, 46)( 15, 45)( 16, 44)
( 17, 43)( 18, 42)( 19, 41)( 20, 40)( 21, 39)( 22, 38)( 23, 37)( 24, 36)
( 25, 35)( 26, 34)( 27, 33)( 28, 32)( 29, 31)( 60, 87)( 61, 86)( 62, 85)
( 63, 84)( 64, 83)( 65, 82)( 66, 81)( 67, 80)( 68, 79)( 69, 78)( 70, 77)
( 71, 76)( 72, 75)( 73, 74)( 88,117)( 89,145)( 90,144)( 91,143)( 92,142)
( 93,141)( 94,140)( 95,139)( 96,138)( 97,137)( 98,136)( 99,135)(100,134)
(101,133)(102,132)(103,131)(104,130)(105,129)(106,128)(107,127)(108,126)
(109,125)(110,124)(111,123)(112,122)(113,121)(114,120)(115,119)(116,118)
(147,174)(148,173)(149,172)(150,171)(151,170)(152,169)(153,168)(154,167)
(155,166)(156,165)(157,164)(158,163)(159,162)(160,161);;
s2 := (  1, 89)(  2, 88)(  3,116)(  4,115)(  5,114)(  6,113)(  7,112)(  8,111)
(  9,110)( 10,109)( 11,108)( 12,107)( 13,106)( 14,105)( 15,104)( 16,103)
( 17,102)( 18,101)( 19,100)( 20, 99)( 21, 98)( 22, 97)( 23, 96)( 24, 95)
( 25, 94)( 26, 93)( 27, 92)( 28, 91)( 29, 90)( 30,118)( 31,117)( 32,145)
( 33,144)( 34,143)( 35,142)( 36,141)( 37,140)( 38,139)( 39,138)( 40,137)
( 41,136)( 42,135)( 43,134)( 44,133)( 45,132)( 46,131)( 47,130)( 48,129)
( 49,128)( 50,127)( 51,126)( 52,125)( 53,124)( 54,123)( 55,122)( 56,121)
( 57,120)( 58,119)( 59,147)( 60,146)( 61,174)( 62,173)( 63,172)( 64,171)
( 65,170)( 66,169)( 67,168)( 68,167)( 69,166)( 70,165)( 71,164)( 72,163)
( 73,162)( 74,161)( 75,160)( 76,159)( 77,158)( 78,157)( 79,156)( 80,155)
( 81,154)( 82,153)( 83,152)( 84,151)( 85,150)( 86,149)( 87,148);;
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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*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*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*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(174)!( 30, 59)( 31, 60)( 32, 61)( 33, 62)( 34, 63)( 35, 64)( 36, 65)
( 37, 66)( 38, 67)( 39, 68)( 40, 69)( 41, 70)( 42, 71)( 43, 72)( 44, 73)
( 45, 74)( 46, 75)( 47, 76)( 48, 77)( 49, 78)( 50, 79)( 51, 80)( 52, 81)
( 53, 82)( 54, 83)( 55, 84)( 56, 85)( 57, 86)( 58, 87)(117,146)(118,147)
(119,148)(120,149)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)
(127,156)(128,157)(129,158)(130,159)(131,160)(132,161)(133,162)(134,163)
(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(141,170)(142,171)
(143,172)(144,173)(145,174);
s1 := Sym(174)!(  1, 30)(  2, 58)(  3, 57)(  4, 56)(  5, 55)(  6, 54)(  7, 53)
(  8, 52)(  9, 51)( 10, 50)( 11, 49)( 12, 48)( 13, 47)( 14, 46)( 15, 45)
( 16, 44)( 17, 43)( 18, 42)( 19, 41)( 20, 40)( 21, 39)( 22, 38)( 23, 37)
( 24, 36)( 25, 35)( 26, 34)( 27, 33)( 28, 32)( 29, 31)( 60, 87)( 61, 86)
( 62, 85)( 63, 84)( 64, 83)( 65, 82)( 66, 81)( 67, 80)( 68, 79)( 69, 78)
( 70, 77)( 71, 76)( 72, 75)( 73, 74)( 88,117)( 89,145)( 90,144)( 91,143)
( 92,142)( 93,141)( 94,140)( 95,139)( 96,138)( 97,137)( 98,136)( 99,135)
(100,134)(101,133)(102,132)(103,131)(104,130)(105,129)(106,128)(107,127)
(108,126)(109,125)(110,124)(111,123)(112,122)(113,121)(114,120)(115,119)
(116,118)(147,174)(148,173)(149,172)(150,171)(151,170)(152,169)(153,168)
(154,167)(155,166)(156,165)(157,164)(158,163)(159,162)(160,161);
s2 := Sym(174)!(  1, 89)(  2, 88)(  3,116)(  4,115)(  5,114)(  6,113)(  7,112)
(  8,111)(  9,110)( 10,109)( 11,108)( 12,107)( 13,106)( 14,105)( 15,104)
( 16,103)( 17,102)( 18,101)( 19,100)( 20, 99)( 21, 98)( 22, 97)( 23, 96)
( 24, 95)( 25, 94)( 26, 93)( 27, 92)( 28, 91)( 29, 90)( 30,118)( 31,117)
( 32,145)( 33,144)( 34,143)( 35,142)( 36,141)( 37,140)( 38,139)( 39,138)
( 40,137)( 41,136)( 42,135)( 43,134)( 44,133)( 45,132)( 46,131)( 47,130)
( 48,129)( 49,128)( 50,127)( 51,126)( 52,125)( 53,124)( 54,123)( 55,122)
( 56,121)( 57,120)( 58,119)( 59,147)( 60,146)( 61,174)( 62,173)( 63,172)
( 64,171)( 65,170)( 66,169)( 67,168)( 68,167)( 69,166)( 70,165)( 71,164)
( 72,163)( 73,162)( 74,161)( 75,160)( 76,159)( 77,158)( 78,157)( 79,156)
( 80,155)( 81,154)( 82,153)( 83,152)( 84,151)( 85,150)( 86,149)( 87,148);
poly := sub<Sym(174)|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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*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*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*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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