Polytope of Type {2,10,34}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,10,34}*1360
if this polytope has a name.
Group : SmallGroup(1360,241)
Rank : 4
Schlafli Type : {2,10,34}
Number of vertices, edges, etc : 2, 10, 170, 34
Order of s₀s₁s₂s₃ : 170
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   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) :
   5-fold quotients : {2,2,34}*272
   10-fold quotients : {2,2,17}*136
   17-fold quotients : {2,10,2}*80
   34-fold quotients : {2,5,2}*40
   85-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 20, 71)( 21, 72)( 22, 73)( 23, 74)( 24, 75)( 25, 76)( 26, 77)( 27, 78)
( 28, 79)( 29, 80)( 30, 81)( 31, 82)( 32, 83)( 33, 84)( 34, 85)( 35, 86)
( 36, 87)( 37, 54)( 38, 55)( 39, 56)( 40, 57)( 41, 58)( 42, 59)( 43, 60)
( 44, 61)( 45, 62)( 46, 63)( 47, 64)( 48, 65)( 49, 66)( 50, 67)( 51, 68)
( 52, 69)( 53, 70)(105,156)(106,157)(107,158)(108,159)(109,160)(110,161)
(111,162)(112,163)(113,164)(114,165)(115,166)(116,167)(117,168)(118,169)
(119,170)(120,171)(121,172)(122,139)(123,140)(124,141)(125,142)(126,143)
(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)(134,151)
(135,152)(136,153)(137,154)(138,155);;
s2 := (  3, 20)(  4, 36)(  5, 35)(  6, 34)(  7, 33)(  8, 32)(  9, 31)( 10, 30)
( 11, 29)( 12, 28)( 13, 27)( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)
( 19, 21)( 37, 71)( 38, 87)( 39, 86)( 40, 85)( 41, 84)( 42, 83)( 43, 82)
( 44, 81)( 45, 80)( 46, 79)( 47, 78)( 48, 77)( 49, 76)( 50, 75)( 51, 74)
( 52, 73)( 53, 72)( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)
( 61, 64)( 62, 63)( 88,105)( 89,121)( 90,120)( 91,119)( 92,118)( 93,117)
( 94,116)( 95,115)( 96,114)( 97,113)( 98,112)( 99,111)(100,110)(101,109)
(102,108)(103,107)(104,106)(122,156)(123,172)(124,171)(125,170)(126,169)
(127,168)(128,167)(129,166)(130,165)(131,164)(132,163)(133,162)(134,161)
(135,160)(136,159)(137,158)(138,157)(140,155)(141,154)(142,153)(143,152)
(144,151)(145,150)(146,149)(147,148);;
s3 := (  3, 89)(  4, 88)(  5,104)(  6,103)(  7,102)(  8,101)(  9,100)( 10, 99)
( 11, 98)( 12, 97)( 13, 96)( 14, 95)( 15, 94)( 16, 93)( 17, 92)( 18, 91)
( 19, 90)( 20,106)( 21,105)( 22,121)( 23,120)( 24,119)( 25,118)( 26,117)
( 27,116)( 28,115)( 29,114)( 30,113)( 31,112)( 32,111)( 33,110)( 34,109)
( 35,108)( 36,107)( 37,123)( 38,122)( 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)( 54,140)( 55,139)( 56,155)( 57,154)( 58,153)
( 59,152)( 60,151)( 61,150)( 62,149)( 63,148)( 64,147)( 65,146)( 66,145)
( 67,144)( 68,143)( 69,142)( 70,141)( 71,157)( 72,156)( 73,172)( 74,171)
( 75,170)( 76,169)( 77,168)( 78,167)( 79,166)( 80,165)( 81,164)( 82,163)
( 83,162)( 84,161)( 85,160)( 86,159)( 87,158);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(172)!(1,2);
s1 := Sym(172)!( 20, 71)( 21, 72)( 22, 73)( 23, 74)( 24, 75)( 25, 76)( 26, 77)
( 27, 78)( 28, 79)( 29, 80)( 30, 81)( 31, 82)( 32, 83)( 33, 84)( 34, 85)
( 35, 86)( 36, 87)( 37, 54)( 38, 55)( 39, 56)( 40, 57)( 41, 58)( 42, 59)
( 43, 60)( 44, 61)( 45, 62)( 46, 63)( 47, 64)( 48, 65)( 49, 66)( 50, 67)
( 51, 68)( 52, 69)( 53, 70)(105,156)(106,157)(107,158)(108,159)(109,160)
(110,161)(111,162)(112,163)(113,164)(114,165)(115,166)(116,167)(117,168)
(118,169)(119,170)(120,171)(121,172)(122,139)(123,140)(124,141)(125,142)
(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)
(134,151)(135,152)(136,153)(137,154)(138,155);
s2 := Sym(172)!(  3, 20)(  4, 36)(  5, 35)(  6, 34)(  7, 33)(  8, 32)(  9, 31)
( 10, 30)( 11, 29)( 12, 28)( 13, 27)( 14, 26)( 15, 25)( 16, 24)( 17, 23)
( 18, 22)( 19, 21)( 37, 71)( 38, 87)( 39, 86)( 40, 85)( 41, 84)( 42, 83)
( 43, 82)( 44, 81)( 45, 80)( 46, 79)( 47, 78)( 48, 77)( 49, 76)( 50, 75)
( 51, 74)( 52, 73)( 53, 72)( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)
( 60, 65)( 61, 64)( 62, 63)( 88,105)( 89,121)( 90,120)( 91,119)( 92,118)
( 93,117)( 94,116)( 95,115)( 96,114)( 97,113)( 98,112)( 99,111)(100,110)
(101,109)(102,108)(103,107)(104,106)(122,156)(123,172)(124,171)(125,170)
(126,169)(127,168)(128,167)(129,166)(130,165)(131,164)(132,163)(133,162)
(134,161)(135,160)(136,159)(137,158)(138,157)(140,155)(141,154)(142,153)
(143,152)(144,151)(145,150)(146,149)(147,148);
s3 := Sym(172)!(  3, 89)(  4, 88)(  5,104)(  6,103)(  7,102)(  8,101)(  9,100)
( 10, 99)( 11, 98)( 12, 97)( 13, 96)( 14, 95)( 15, 94)( 16, 93)( 17, 92)
( 18, 91)( 19, 90)( 20,106)( 21,105)( 22,121)( 23,120)( 24,119)( 25,118)
( 26,117)( 27,116)( 28,115)( 29,114)( 30,113)( 31,112)( 32,111)( 33,110)
( 34,109)( 35,108)( 36,107)( 37,123)( 38,122)( 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)( 54,140)( 55,139)( 56,155)( 57,154)
( 58,153)( 59,152)( 60,151)( 61,150)( 62,149)( 63,148)( 64,147)( 65,146)
( 66,145)( 67,144)( 68,143)( 69,142)( 70,141)( 71,157)( 72,156)( 73,172)
( 74,171)( 75,170)( 76,169)( 77,168)( 78,167)( 79,166)( 80,165)( 81,164)
( 82,163)( 83,162)( 84,161)( 85,160)( 86,159)( 87,158);
poly := sub<Sym(172)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope