Polytope of Type {58,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {58,6}*696
Also Known As : {58,6|2}. if this polytope has another name.
Group : SmallGroup(696,39)
Rank : 3
Schlafli Type : {58,6}
Number of vertices, edges, etc : 58, 174, 6
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 :
   {58,6,2} of size 1392
Vertex Figure Of :
   {2,58,6} of size 1392
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {58,2}*232
   6-fold quotients : {29,2}*116
   29-fold quotients : {2,6}*24
   58-fold quotients : {2,3}*12
   87-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {58,12}*1392, {116,6}*1392a
Permutation Representation (GAP) :

s0 := (  2, 29)(  3, 28)(  4, 27)(  5, 26)(  6, 25)(  7, 24)(  8, 23)(  9, 22)
( 10, 21)( 11, 20)( 12, 19)( 13, 18)( 14, 17)( 15, 16)( 31, 58)( 32, 57)
( 33, 56)( 34, 55)( 35, 54)( 36, 53)( 37, 52)( 38, 51)( 39, 50)( 40, 49)
( 41, 48)( 42, 47)( 43, 46)( 44, 45)( 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)( 89,116)( 90,115)( 91,114)( 92,113)( 93,112)( 94,111)
( 95,110)( 96,109)( 97,108)( 98,107)( 99,106)(100,105)(101,104)(102,103)
(118,145)(119,144)(120,143)(121,142)(122,141)(123,140)(124,139)(125,138)
(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)(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);;
s1 := (  1,  2)(  3, 29)(  4, 28)(  5, 27)(  6, 26)(  7, 25)(  8, 24)(  9, 23)
( 10, 22)( 11, 21)( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 30, 60)( 31, 59)
( 32, 87)( 33, 86)( 34, 85)( 35, 84)( 36, 83)( 37, 82)( 38, 81)( 39, 80)
( 40, 79)( 41, 78)( 42, 77)( 43, 76)( 44, 75)( 45, 74)( 46, 73)( 47, 72)
( 48, 71)( 49, 70)( 50, 69)( 51, 68)( 52, 67)( 53, 66)( 54, 65)( 55, 64)
( 56, 63)( 57, 62)( 58, 61)( 88, 89)( 90,116)( 91,115)( 92,114)( 93,113)
( 94,112)( 95,111)( 96,110)( 97,109)( 98,108)( 99,107)(100,106)(101,105)
(102,104)(117,147)(118,146)(119,174)(120,173)(121,172)(122,171)(123,170)
(124,169)(125,168)(126,167)(127,166)(128,165)(129,164)(130,163)(131,162)
(132,161)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)(139,154)
(140,153)(141,152)(142,151)(143,150)(144,149)(145,148);;
s2 := (  1,117)(  2,118)(  3,119)(  4,120)(  5,121)(  6,122)(  7,123)(  8,124)
(  9,125)( 10,126)( 11,127)( 12,128)( 13,129)( 14,130)( 15,131)( 16,132)
( 17,133)( 18,134)( 19,135)( 20,136)( 21,137)( 22,138)( 23,139)( 24,140)
( 25,141)( 26,142)( 27,143)( 28,144)( 29,145)( 30, 88)( 31, 89)( 32, 90)
( 33, 91)( 34, 92)( 35, 93)( 36, 94)( 37, 95)( 38, 96)( 39, 97)( 40, 98)
( 41, 99)( 42,100)( 43,101)( 44,102)( 45,103)( 46,104)( 47,105)( 48,106)
( 49,107)( 50,108)( 51,109)( 52,110)( 53,111)( 54,112)( 55,113)( 56,114)
( 57,115)( 58,116)( 59,146)( 60,147)( 61,148)( 62,149)( 63,150)( 64,151)
( 65,152)( 66,153)( 67,154)( 68,155)( 69,156)( 70,157)( 71,158)( 72,159)
( 73,160)( 74,161)( 75,162)( 76,163)( 77,164)( 78,165)( 79,166)( 80,167)
( 81,168)( 82,169)( 83,170)( 84,171)( 85,172)( 86,173)( 87,174);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(174)!(  2, 29)(  3, 28)(  4, 27)(  5, 26)(  6, 25)(  7, 24)(  8, 23)
(  9, 22)( 10, 21)( 11, 20)( 12, 19)( 13, 18)( 14, 17)( 15, 16)( 31, 58)
( 32, 57)( 33, 56)( 34, 55)( 35, 54)( 36, 53)( 37, 52)( 38, 51)( 39, 50)
( 40, 49)( 41, 48)( 42, 47)( 43, 46)( 44, 45)( 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)( 89,116)( 90,115)( 91,114)( 92,113)( 93,112)
( 94,111)( 95,110)( 96,109)( 97,108)( 98,107)( 99,106)(100,105)(101,104)
(102,103)(118,145)(119,144)(120,143)(121,142)(122,141)(123,140)(124,139)
(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)(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);
s1 := Sym(174)!(  1,  2)(  3, 29)(  4, 28)(  5, 27)(  6, 26)(  7, 25)(  8, 24)
(  9, 23)( 10, 22)( 11, 21)( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 30, 60)
( 31, 59)( 32, 87)( 33, 86)( 34, 85)( 35, 84)( 36, 83)( 37, 82)( 38, 81)
( 39, 80)( 40, 79)( 41, 78)( 42, 77)( 43, 76)( 44, 75)( 45, 74)( 46, 73)
( 47, 72)( 48, 71)( 49, 70)( 50, 69)( 51, 68)( 52, 67)( 53, 66)( 54, 65)
( 55, 64)( 56, 63)( 57, 62)( 58, 61)( 88, 89)( 90,116)( 91,115)( 92,114)
( 93,113)( 94,112)( 95,111)( 96,110)( 97,109)( 98,108)( 99,107)(100,106)
(101,105)(102,104)(117,147)(118,146)(119,174)(120,173)(121,172)(122,171)
(123,170)(124,169)(125,168)(126,167)(127,166)(128,165)(129,164)(130,163)
(131,162)(132,161)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)
(139,154)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148);
s2 := Sym(174)!(  1,117)(  2,118)(  3,119)(  4,120)(  5,121)(  6,122)(  7,123)
(  8,124)(  9,125)( 10,126)( 11,127)( 12,128)( 13,129)( 14,130)( 15,131)
( 16,132)( 17,133)( 18,134)( 19,135)( 20,136)( 21,137)( 22,138)( 23,139)
( 24,140)( 25,141)( 26,142)( 27,143)( 28,144)( 29,145)( 30, 88)( 31, 89)
( 32, 90)( 33, 91)( 34, 92)( 35, 93)( 36, 94)( 37, 95)( 38, 96)( 39, 97)
( 40, 98)( 41, 99)( 42,100)( 43,101)( 44,102)( 45,103)( 46,104)( 47,105)
( 48,106)( 49,107)( 50,108)( 51,109)( 52,110)( 53,111)( 54,112)( 55,113)
( 56,114)( 57,115)( 58,116)( 59,146)( 60,147)( 61,148)( 62,149)( 63,150)
( 64,151)( 65,152)( 66,153)( 67,154)( 68,155)( 69,156)( 70,157)( 71,158)
( 72,159)( 73,160)( 74,161)( 75,162)( 76,163)( 77,164)( 78,165)( 79,166)
( 80,167)( 81,168)( 82,169)( 83,170)( 84,171)( 85,172)( 86,173)( 87,174);
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
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