Polytope of Type {10,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,42}*840
Also Known As : {10,42|2}. if this polytope has another name.
Group : SmallGroup(840,173)
Rank : 3
Schlafli Type : {10,42}
Number of vertices, edges, etc : 10, 210, 42
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 :
   {10,42,2} of size 1680
Vertex Figure Of :
   {2,10,42} of size 1680
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {10,14}*280
   5-fold quotients : {2,42}*168
   7-fold quotients : {10,6}*120
   10-fold quotients : {2,21}*84
   15-fold quotients : {2,14}*56
   21-fold quotients : {10,2}*40
   30-fold quotients : {2,7}*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 : {20,42}*1680a, {10,84}*1680
Permutation Representation (GAP) :

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

References : None.
to this polytope