Polytope of Type {210}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {210}*420
Also Known As : 210-gon, {210}. if this polytope has another name.
Group : SmallGroup(420,40)
Rank : 2
Schlafli Type : {210}
Number of vertices, edges, etc : 210, 210
Order of s0s1 : 210
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{210,2} of size 840
{210,4} of size 1680
{210,4} of size 1680
{210,4} of size 1680
Vertex Figure Of :
{2,210} of size 840
{4,210} of size 1680
{4,210} of size 1680
{4,210} of size 1680
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {105}*210
3-fold quotients : {70}*140
5-fold quotients : {42}*84
6-fold quotients : {35}*70
7-fold quotients : {30}*60
10-fold quotients : {21}*42
14-fold quotients : {15}*30
15-fold quotients : {14}*28
21-fold quotients : {10}*20
30-fold quotients : {7}*14
35-fold quotients : {6}*12
42-fold quotients : {5}*10
70-fold quotients : {3}*6
105-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {420}*840
3-fold covers : {630}*1260
4-fold covers : {840}*1680
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)( 36, 71)( 37, 77)( 38, 76)( 39, 75)( 40, 74)( 41, 73)( 42, 72)
( 43, 99)( 44,105)( 45,104)( 46,103)( 47,102)( 48,101)( 49,100)( 50, 92)
( 51, 98)( 52, 97)( 53, 96)( 54, 95)( 55, 94)( 56, 93)( 57, 85)( 58, 91)
( 59, 90)( 60, 89)( 61, 88)( 62, 87)( 63, 86)( 64, 78)( 65, 84)( 66, 83)
( 67, 82)( 68, 81)( 69, 80)( 70, 79)(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)(141,176)(142,182)(143,181)
(144,180)(145,179)(146,178)(147,177)(148,204)(149,210)(150,209)(151,208)
(152,207)(153,206)(154,205)(155,197)(156,203)(157,202)(158,201)(159,200)
(160,199)(161,198)(162,190)(163,196)(164,195)(165,194)(166,193)(167,192)
(168,191)(169,183)(170,189)(171,188)(172,187)(173,186)(174,185)(175,184);;
s1 := ( 1,149)( 2,148)( 3,154)( 4,153)( 5,152)( 6,151)( 7,150)( 8,142)
( 9,141)( 10,147)( 11,146)( 12,145)( 13,144)( 14,143)( 15,170)( 16,169)
( 17,175)( 18,174)( 19,173)( 20,172)( 21,171)( 22,163)( 23,162)( 24,168)
( 25,167)( 26,166)( 27,165)( 28,164)( 29,156)( 30,155)( 31,161)( 32,160)
( 33,159)( 34,158)( 35,157)( 36,114)( 37,113)( 38,119)( 39,118)( 40,117)
( 41,116)( 42,115)( 43,107)( 44,106)( 45,112)( 46,111)( 47,110)( 48,109)
( 49,108)( 50,135)( 51,134)( 52,140)( 53,139)( 54,138)( 55,137)( 56,136)
( 57,128)( 58,127)( 59,133)( 60,132)( 61,131)( 62,130)( 63,129)( 64,121)
( 65,120)( 66,126)( 67,125)( 68,124)( 69,123)( 70,122)( 71,184)( 72,183)
( 73,189)( 74,188)( 75,187)( 76,186)( 77,185)( 78,177)( 79,176)( 80,182)
( 81,181)( 82,180)( 83,179)( 84,178)( 85,205)( 86,204)( 87,210)( 88,209)
( 89,208)( 90,207)( 91,206)( 92,198)( 93,197)( 94,203)( 95,202)( 96,201)
( 97,200)( 98,199)( 99,191)(100,190)(101,196)(102,195)(103,194)(104,193)
(105,192);;
poly := Group([s0,s1]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1");;
s0 := F.1;; s1 := F.2;;
rels := [ s0*s0, s1*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*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*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*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)( 36, 71)( 37, 77)( 38, 76)( 39, 75)( 40, 74)( 41, 73)
( 42, 72)( 43, 99)( 44,105)( 45,104)( 46,103)( 47,102)( 48,101)( 49,100)
( 50, 92)( 51, 98)( 52, 97)( 53, 96)( 54, 95)( 55, 94)( 56, 93)( 57, 85)
( 58, 91)( 59, 90)( 60, 89)( 61, 88)( 62, 87)( 63, 86)( 64, 78)( 65, 84)
( 66, 83)( 67, 82)( 68, 81)( 69, 80)( 70, 79)(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)(141,176)(142,182)
(143,181)(144,180)(145,179)(146,178)(147,177)(148,204)(149,210)(150,209)
(151,208)(152,207)(153,206)(154,205)(155,197)(156,203)(157,202)(158,201)
(159,200)(160,199)(161,198)(162,190)(163,196)(164,195)(165,194)(166,193)
(167,192)(168,191)(169,183)(170,189)(171,188)(172,187)(173,186)(174,185)
(175,184);
s1 := Sym(210)!( 1,149)( 2,148)( 3,154)( 4,153)( 5,152)( 6,151)( 7,150)
( 8,142)( 9,141)( 10,147)( 11,146)( 12,145)( 13,144)( 14,143)( 15,170)
( 16,169)( 17,175)( 18,174)( 19,173)( 20,172)( 21,171)( 22,163)( 23,162)
( 24,168)( 25,167)( 26,166)( 27,165)( 28,164)( 29,156)( 30,155)( 31,161)
( 32,160)( 33,159)( 34,158)( 35,157)( 36,114)( 37,113)( 38,119)( 39,118)
( 40,117)( 41,116)( 42,115)( 43,107)( 44,106)( 45,112)( 46,111)( 47,110)
( 48,109)( 49,108)( 50,135)( 51,134)( 52,140)( 53,139)( 54,138)( 55,137)
( 56,136)( 57,128)( 58,127)( 59,133)( 60,132)( 61,131)( 62,130)( 63,129)
( 64,121)( 65,120)( 66,126)( 67,125)( 68,124)( 69,123)( 70,122)( 71,184)
( 72,183)( 73,189)( 74,188)( 75,187)( 76,186)( 77,185)( 78,177)( 79,176)
( 80,182)( 81,181)( 82,180)( 83,179)( 84,178)( 85,205)( 86,204)( 87,210)
( 88,209)( 89,208)( 90,207)( 91,206)( 92,198)( 93,197)( 94,203)( 95,202)
( 96,201)( 97,200)( 98,199)( 99,191)(100,190)(101,196)(102,195)(103,194)
(104,193)(105,192);
poly := sub<Sym(210)|s0,s1>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1> := Group< s0,s1 | s0*s0, s1*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*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*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*s0*s1 >;
References : None.
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