Polytope of Type {2,210}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,210}*840
if this polytope has a name.
Group : SmallGroup(840,185)
Rank : 3
Schlafli Type : {2,210}
Number of vertices, edges, etc : 2, 210, 210
Order of s₀s₁s₂ : 210
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,210,2} of size 1680
Vertex Figure Of :
   {2,2,210} of size 1680
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,105}*420
   3-fold quotients : {2,70}*280
   5-fold quotients : {2,42}*168
   6-fold quotients : {2,35}*140
   7-fold quotients : {2,30}*120
   10-fold quotients : {2,21}*84
   14-fold quotients : {2,15}*60
   15-fold quotients : {2,14}*56
   21-fold quotients : {2,10}*40
   30-fold quotients : {2,7}*28
   35-fold quotients : {2,6}*24
   42-fold quotients : {2,5}*20
   70-fold quotients : {2,3}*12
   105-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,420}*1680, {4,210}*1680a
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  9)(  5,  8)(  6,  7)( 10, 31)( 11, 37)( 12, 36)( 13, 35)( 14, 34)
( 15, 33)( 16, 32)( 17, 24)( 18, 30)( 19, 29)( 20, 28)( 21, 27)( 22, 26)
( 23, 25)( 38, 73)( 39, 79)( 40, 78)( 41, 77)( 42, 76)( 43, 75)( 44, 74)
( 45,101)( 46,107)( 47,106)( 48,105)( 49,104)( 50,103)( 51,102)( 52, 94)
( 53,100)( 54, 99)( 55, 98)( 56, 97)( 57, 96)( 58, 95)( 59, 87)( 60, 93)
( 61, 92)( 62, 91)( 63, 90)( 64, 89)( 65, 88)( 66, 80)( 67, 86)( 68, 85)
( 69, 84)( 70, 83)( 71, 82)( 72, 81)(109,114)(110,113)(111,112)(115,136)
(116,142)(117,141)(118,140)(119,139)(120,138)(121,137)(122,129)(123,135)
(124,134)(125,133)(126,132)(127,131)(128,130)(143,178)(144,184)(145,183)
(146,182)(147,181)(148,180)(149,179)(150,206)(151,212)(152,211)(153,210)
(154,209)(155,208)(156,207)(157,199)(158,205)(159,204)(160,203)(161,202)
(162,201)(163,200)(164,192)(165,198)(166,197)(167,196)(168,195)(169,194)
(170,193)(171,185)(172,191)(173,190)(174,189)(175,188)(176,187)(177,186);;
s2 := (  3,151)(  4,150)(  5,156)(  6,155)(  7,154)(  8,153)(  9,152)( 10,144)
( 11,143)( 12,149)( 13,148)( 14,147)( 15,146)( 16,145)( 17,172)( 18,171)
( 19,177)( 20,176)( 21,175)( 22,174)( 23,173)( 24,165)( 25,164)( 26,170)
( 27,169)( 28,168)( 29,167)( 30,166)( 31,158)( 32,157)( 33,163)( 34,162)
( 35,161)( 36,160)( 37,159)( 38,116)( 39,115)( 40,121)( 41,120)( 42,119)
( 43,118)( 44,117)( 45,109)( 46,108)( 47,114)( 48,113)( 49,112)( 50,111)
( 51,110)( 52,137)( 53,136)( 54,142)( 55,141)( 56,140)( 57,139)( 58,138)
( 59,130)( 60,129)( 61,135)( 62,134)( 63,133)( 64,132)( 65,131)( 66,123)
( 67,122)( 68,128)( 69,127)( 70,126)( 71,125)( 72,124)( 73,186)( 74,185)
( 75,191)( 76,190)( 77,189)( 78,188)( 79,187)( 80,179)( 81,178)( 82,184)
( 83,183)( 84,182)( 85,181)( 86,180)( 87,207)( 88,206)( 89,212)( 90,211)
( 91,210)( 92,209)( 93,208)( 94,200)( 95,199)( 96,205)( 97,204)( 98,203)
( 99,202)(100,201)(101,193)(102,192)(103,198)(104,197)(105,196)(106,195)
(107,194);;
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*s1*s0*s1, s0*s2*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*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*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*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*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(212)!(1,2);
s1 := Sym(212)!(  4,  9)(  5,  8)(  6,  7)( 10, 31)( 11, 37)( 12, 36)( 13, 35)
( 14, 34)( 15, 33)( 16, 32)( 17, 24)( 18, 30)( 19, 29)( 20, 28)( 21, 27)
( 22, 26)( 23, 25)( 38, 73)( 39, 79)( 40, 78)( 41, 77)( 42, 76)( 43, 75)
( 44, 74)( 45,101)( 46,107)( 47,106)( 48,105)( 49,104)( 50,103)( 51,102)
( 52, 94)( 53,100)( 54, 99)( 55, 98)( 56, 97)( 57, 96)( 58, 95)( 59, 87)
( 60, 93)( 61, 92)( 62, 91)( 63, 90)( 64, 89)( 65, 88)( 66, 80)( 67, 86)
( 68, 85)( 69, 84)( 70, 83)( 71, 82)( 72, 81)(109,114)(110,113)(111,112)
(115,136)(116,142)(117,141)(118,140)(119,139)(120,138)(121,137)(122,129)
(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(143,178)(144,184)
(145,183)(146,182)(147,181)(148,180)(149,179)(150,206)(151,212)(152,211)
(153,210)(154,209)(155,208)(156,207)(157,199)(158,205)(159,204)(160,203)
(161,202)(162,201)(163,200)(164,192)(165,198)(166,197)(167,196)(168,195)
(169,194)(170,193)(171,185)(172,191)(173,190)(174,189)(175,188)(176,187)
(177,186);
s2 := Sym(212)!(  3,151)(  4,150)(  5,156)(  6,155)(  7,154)(  8,153)(  9,152)
( 10,144)( 11,143)( 12,149)( 13,148)( 14,147)( 15,146)( 16,145)( 17,172)
( 18,171)( 19,177)( 20,176)( 21,175)( 22,174)( 23,173)( 24,165)( 25,164)
( 26,170)( 27,169)( 28,168)( 29,167)( 30,166)( 31,158)( 32,157)( 33,163)
( 34,162)( 35,161)( 36,160)( 37,159)( 38,116)( 39,115)( 40,121)( 41,120)
( 42,119)( 43,118)( 44,117)( 45,109)( 46,108)( 47,114)( 48,113)( 49,112)
( 50,111)( 51,110)( 52,137)( 53,136)( 54,142)( 55,141)( 56,140)( 57,139)
( 58,138)( 59,130)( 60,129)( 61,135)( 62,134)( 63,133)( 64,132)( 65,131)
( 66,123)( 67,122)( 68,128)( 69,127)( 70,126)( 71,125)( 72,124)( 73,186)
( 74,185)( 75,191)( 76,190)( 77,189)( 78,188)( 79,187)( 80,179)( 81,178)
( 82,184)( 83,183)( 84,182)( 85,181)( 86,180)( 87,207)( 88,206)( 89,212)
( 90,211)( 91,210)( 92,209)( 93,208)( 94,200)( 95,199)( 96,205)( 97,204)
( 98,203)( 99,202)(100,201)(101,193)(102,192)(103,198)(104,197)(105,196)
(106,195)(107,194);
poly := sub<Sym(212)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

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

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