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Polytope of Type {70,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {70,6}*840
Also Known As : {70,6|2}. if this polytope has another name.
Group : SmallGroup(840,174)
Rank : 3
Schlafli Type : {70,6}
Number of vertices, edges, etc : 70, 210, 6
Order of s0s1s2 : 210
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {70,6,2} of size 1680
Vertex Figure Of :
   {2,70,6} of size 1680
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {70,2}*280
   5-fold quotients : {14,6}*168
   6-fold quotients : {35,2}*140
   7-fold quotients : {10,6}*120
   15-fold quotients : {14,2}*56
   21-fold quotients : {10,2}*40
   30-fold quotients : {7,2}*28
   35-fold quotients : {2,6}*24
   42-fold quotients : {5,2}*20
   70-fold quotients : {2,3}*12
   105-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {70,12}*1680, {140,6}*1680a
Permutation Representation (GAP) :
s0 := (  2,  7)(  3,  6)(  4,  5)(  8, 29)(  9, 35)( 10, 34)( 11, 33)( 12, 32)
( 13, 31)( 14, 30)( 15, 22)( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)
( 21, 23)( 37, 42)( 38, 41)( 39, 40)( 43, 64)( 44, 70)( 45, 69)( 46, 68)
( 47, 67)( 48, 66)( 49, 65)( 50, 57)( 51, 63)( 52, 62)( 53, 61)( 54, 60)
( 55, 59)( 56, 58)( 72, 77)( 73, 76)( 74, 75)( 78, 99)( 79,105)( 80,104)
( 81,103)( 82,102)( 83,101)( 84,100)( 85, 92)( 86, 98)( 87, 97)( 88, 96)
( 89, 95)( 90, 94)( 91, 93)(107,112)(108,111)(109,110)(113,134)(114,140)
(115,139)(116,138)(117,137)(118,136)(119,135)(120,127)(121,133)(122,132)
(123,131)(124,130)(125,129)(126,128)(142,147)(143,146)(144,145)(148,169)
(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,162)(156,168)
(157,167)(158,166)(159,165)(160,164)(161,163)(177,182)(178,181)(179,180)
(183,204)(184,210)(185,209)(186,208)(187,207)(188,206)(189,205)(190,197)
(191,203)(192,202)(193,201)(194,200)(195,199)(196,198);;
s1 := (  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)( 15, 30)
( 16, 29)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 23)( 24, 28)
( 25, 27)( 36, 79)( 37, 78)( 38, 84)( 39, 83)( 40, 82)( 41, 81)( 42, 80)
( 43, 72)( 44, 71)( 45, 77)( 46, 76)( 47, 75)( 48, 74)( 49, 73)( 50,100)
( 51, 99)( 52,105)( 53,104)( 54,103)( 55,102)( 56,101)( 57, 93)( 58, 92)
( 59, 98)( 60, 97)( 61, 96)( 62, 95)( 63, 94)( 64, 86)( 65, 85)( 66, 91)
( 67, 90)( 68, 89)( 69, 88)( 70, 87)(106,114)(107,113)(108,119)(109,118)
(110,117)(111,116)(112,115)(120,135)(121,134)(122,140)(123,139)(124,138)
(125,137)(126,136)(127,128)(129,133)(130,132)(141,184)(142,183)(143,189)
(144,188)(145,187)(146,186)(147,185)(148,177)(149,176)(150,182)(151,181)
(152,180)(153,179)(154,178)(155,205)(156,204)(157,210)(158,209)(159,208)
(160,207)(161,206)(162,198)(163,197)(164,203)(165,202)(166,201)(167,200)
(168,199)(169,191)(170,190)(171,196)(172,195)(173,194)(174,193)(175,192);;
s2 := (  1,141)(  2,142)(  3,143)(  4,144)(  5,145)(  6,146)(  7,147)(  8,148)
(  9,149)( 10,150)( 11,151)( 12,152)( 13,153)( 14,154)( 15,155)( 16,156)
( 17,157)( 18,158)( 19,159)( 20,160)( 21,161)( 22,162)( 23,163)( 24,164)
( 25,165)( 26,166)( 27,167)( 28,168)( 29,169)( 30,170)( 31,171)( 32,172)
( 33,173)( 34,174)( 35,175)( 36,106)( 37,107)( 38,108)( 39,109)( 40,110)
( 41,111)( 42,112)( 43,113)( 44,114)( 45,115)( 46,116)( 47,117)( 48,118)
( 49,119)( 50,120)( 51,121)( 52,122)( 53,123)( 54,124)( 55,125)( 56,126)
( 57,127)( 58,128)( 59,129)( 60,130)( 61,131)( 62,132)( 63,133)( 64,134)
( 65,135)( 66,136)( 67,137)( 68,138)( 69,139)( 70,140)( 71,176)( 72,177)
( 73,178)( 74,179)( 75,180)( 76,181)( 77,182)( 78,183)( 79,184)( 80,185)
( 81,186)( 82,187)( 83,188)( 84,189)( 85,190)( 86,191)( 87,192)( 88,193)
( 89,194)( 90,195)( 91,196)( 92,197)( 93,198)( 94,199)( 95,200)( 96,201)
( 97,202)( 98,203)( 99,204)(100,205)(101,206)(102,207)(103,208)(104,209)
(105,210);;
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*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(210)!(  2,  7)(  3,  6)(  4,  5)(  8, 29)(  9, 35)( 10, 34)( 11, 33)
( 12, 32)( 13, 31)( 14, 30)( 15, 22)( 16, 28)( 17, 27)( 18, 26)( 19, 25)
( 20, 24)( 21, 23)( 37, 42)( 38, 41)( 39, 40)( 43, 64)( 44, 70)( 45, 69)
( 46, 68)( 47, 67)( 48, 66)( 49, 65)( 50, 57)( 51, 63)( 52, 62)( 53, 61)
( 54, 60)( 55, 59)( 56, 58)( 72, 77)( 73, 76)( 74, 75)( 78, 99)( 79,105)
( 80,104)( 81,103)( 82,102)( 83,101)( 84,100)( 85, 92)( 86, 98)( 87, 97)
( 88, 96)( 89, 95)( 90, 94)( 91, 93)(107,112)(108,111)(109,110)(113,134)
(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,127)(121,133)
(122,132)(123,131)(124,130)(125,129)(126,128)(142,147)(143,146)(144,145)
(148,169)(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,162)
(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)(177,182)(178,181)
(179,180)(183,204)(184,210)(185,209)(186,208)(187,207)(188,206)(189,205)
(190,197)(191,203)(192,202)(193,201)(194,200)(195,199)(196,198);
s1 := Sym(210)!(  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)
( 15, 30)( 16, 29)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 23)
( 24, 28)( 25, 27)( 36, 79)( 37, 78)( 38, 84)( 39, 83)( 40, 82)( 41, 81)
( 42, 80)( 43, 72)( 44, 71)( 45, 77)( 46, 76)( 47, 75)( 48, 74)( 49, 73)
( 50,100)( 51, 99)( 52,105)( 53,104)( 54,103)( 55,102)( 56,101)( 57, 93)
( 58, 92)( 59, 98)( 60, 97)( 61, 96)( 62, 95)( 63, 94)( 64, 86)( 65, 85)
( 66, 91)( 67, 90)( 68, 89)( 69, 88)( 70, 87)(106,114)(107,113)(108,119)
(109,118)(110,117)(111,116)(112,115)(120,135)(121,134)(122,140)(123,139)
(124,138)(125,137)(126,136)(127,128)(129,133)(130,132)(141,184)(142,183)
(143,189)(144,188)(145,187)(146,186)(147,185)(148,177)(149,176)(150,182)
(151,181)(152,180)(153,179)(154,178)(155,205)(156,204)(157,210)(158,209)
(159,208)(160,207)(161,206)(162,198)(163,197)(164,203)(165,202)(166,201)
(167,200)(168,199)(169,191)(170,190)(171,196)(172,195)(173,194)(174,193)
(175,192);
s2 := Sym(210)!(  1,141)(  2,142)(  3,143)(  4,144)(  5,145)(  6,146)(  7,147)
(  8,148)(  9,149)( 10,150)( 11,151)( 12,152)( 13,153)( 14,154)( 15,155)
( 16,156)( 17,157)( 18,158)( 19,159)( 20,160)( 21,161)( 22,162)( 23,163)
( 24,164)( 25,165)( 26,166)( 27,167)( 28,168)( 29,169)( 30,170)( 31,171)
( 32,172)( 33,173)( 34,174)( 35,175)( 36,106)( 37,107)( 38,108)( 39,109)
( 40,110)( 41,111)( 42,112)( 43,113)( 44,114)( 45,115)( 46,116)( 47,117)
( 48,118)( 49,119)( 50,120)( 51,121)( 52,122)( 53,123)( 54,124)( 55,125)
( 56,126)( 57,127)( 58,128)( 59,129)( 60,130)( 61,131)( 62,132)( 63,133)
( 64,134)( 65,135)( 66,136)( 67,137)( 68,138)( 69,139)( 70,140)( 71,176)
( 72,177)( 73,178)( 74,179)( 75,180)( 76,181)( 77,182)( 78,183)( 79,184)
( 80,185)( 81,186)( 82,187)( 83,188)( 84,189)( 85,190)( 86,191)( 87,192)
( 88,193)( 89,194)( 90,195)( 91,196)( 92,197)( 93,198)( 94,199)( 95,200)
( 96,201)( 97,202)( 98,203)( 99,204)(100,205)(101,206)(102,207)(103,208)
(104,209)(105,210);
poly := sub<Sym(210)|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*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.
to this polytope