Part of the Atlas of Small Regular Polytopes

Polytope of Type {210}

Atlas Canonical Name {210}*420

Overview

Group
SmallGroup(420,40)
Rank
2
Schläfli Type
{210}
Vertices, edges, …
210, 210
Order of s0s1
210
Also known as
210-gon, {210}. if this polytope has another name.

Special Properties

  • Universal
  • Spherical
  • Locally Spherical
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

2-fold

3-fold

5-fold

6-fold

7-fold

10-fold

14-fold

15-fold

21-fold

30-fold

35-fold

42-fold

70-fold

105-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

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.

to this polytope.