Polytope of Type {42,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,10}*840
Also Known As : {42,10|2}. if this polytope has another name.
Group : SmallGroup(840,173)
Rank : 3
Schlafli Type : {42,10}
Number of vertices, edges, etc : 42, 210, 10
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 :
   {42,10,2} of size 1680
Vertex Figure Of :
   {2,42,10} of size 1680
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {14,10}*280
   5-fold quotients : {42,2}*168
   7-fold quotients : {6,10}*120
   10-fold quotients : {21,2}*84
   15-fold quotients : {14,2}*56
   21-fold quotients : {2,10}*40
   30-fold quotients : {7,2}*28
   35-fold quotients : {6,2}*24
   42-fold quotients : {2,5}*20
   70-fold quotients : {3,2}*12
   105-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {42,20}*1680a, {84,10}*1680
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope