Polytope of Type {2,42,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,42,10}*1680
if this polytope has a name.
Group : SmallGroup(1680,990)
Rank : 4
Schlafli Type : {2,42,10}
Number of vertices, edges, etc : 2, 42, 210, 10
Order of s₀s₁s₂s₃ : 210
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,14,10}*560
   5-fold quotients : {2,42,2}*336
   7-fold quotients : {2,6,10}*240
   10-fold quotients : {2,21,2}*168
   15-fold quotients : {2,14,2}*112
   21-fold quotients : {2,2,10}*80
   30-fold quotients : {2,7,2}*56
   35-fold quotients : {2,6,2}*48
   42-fold quotients : {2,2,5}*40
   70-fold quotients : {2,3,2}*24
   105-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  9)(  5,  8)(  6,  7)( 11, 16)( 12, 15)( 13, 14)( 18, 23)( 19, 22)
( 20, 21)( 25, 30)( 26, 29)( 27, 28)( 32, 37)( 33, 36)( 34, 35)( 38, 73)
( 39, 79)( 40, 78)( 41, 77)( 42, 76)( 43, 75)( 44, 74)( 45, 80)( 46, 86)
( 47, 85)( 48, 84)( 49, 83)( 50, 82)( 51, 81)( 52, 87)( 53, 93)( 54, 92)
( 55, 91)( 56, 90)( 57, 89)( 58, 88)( 59, 94)( 60,100)( 61, 99)( 62, 98)
( 63, 97)( 64, 96)( 65, 95)( 66,101)( 67,107)( 68,106)( 69,105)( 70,104)
( 71,103)( 72,102)(109,114)(110,113)(111,112)(116,121)(117,120)(118,119)
(123,128)(124,127)(125,126)(130,135)(131,134)(132,133)(137,142)(138,141)
(139,140)(143,178)(144,184)(145,183)(146,182)(147,181)(148,180)(149,179)
(150,185)(151,191)(152,190)(153,189)(154,188)(155,187)(156,186)(157,192)
(158,198)(159,197)(160,196)(161,195)(162,194)(163,193)(164,199)(165,205)
(166,204)(167,203)(168,202)(169,201)(170,200)(171,206)(172,212)(173,211)
(174,210)(175,209)(176,208)(177,207);;
s2 := (  3, 39)(  4, 38)(  5, 44)(  6, 43)(  7, 42)(  8, 41)(  9, 40)( 10, 67)
( 11, 66)( 12, 72)( 13, 71)( 14, 70)( 15, 69)( 16, 68)( 17, 60)( 18, 59)
( 19, 65)( 20, 64)( 21, 63)( 22, 62)( 23, 61)( 24, 53)( 25, 52)( 26, 58)
( 27, 57)( 28, 56)( 29, 55)( 30, 54)( 31, 46)( 32, 45)( 33, 51)( 34, 50)
( 35, 49)( 36, 48)( 37, 47)( 73, 74)( 75, 79)( 76, 78)( 80,102)( 81,101)
( 82,107)( 83,106)( 84,105)( 85,104)( 86,103)( 87, 95)( 88, 94)( 89,100)
( 90, 99)( 91, 98)( 92, 97)( 93, 96)(108,144)(109,143)(110,149)(111,148)
(112,147)(113,146)(114,145)(115,172)(116,171)(117,177)(118,176)(119,175)
(120,174)(121,173)(122,165)(123,164)(124,170)(125,169)(126,168)(127,167)
(128,166)(129,158)(130,157)(131,163)(132,162)(133,161)(134,160)(135,159)
(136,151)(137,150)(138,156)(139,155)(140,154)(141,153)(142,152)(178,179)
(180,184)(181,183)(185,207)(186,206)(187,212)(188,211)(189,210)(190,209)
(191,208)(192,200)(193,199)(194,205)(195,204)(196,203)(197,202)(198,201);;
s3 := (  3,115)(  4,116)(  5,117)(  6,118)(  7,119)(  8,120)(  9,121)( 10,108)
( 11,109)( 12,110)( 13,111)( 14,112)( 15,113)( 16,114)( 17,136)( 18,137)
( 19,138)( 20,139)( 21,140)( 22,141)( 23,142)( 24,129)( 25,130)( 26,131)
( 27,132)( 28,133)( 29,134)( 30,135)( 31,122)( 32,123)( 33,124)( 34,125)
( 35,126)( 36,127)( 37,128)( 38,150)( 39,151)( 40,152)( 41,153)( 42,154)
( 43,155)( 44,156)( 45,143)( 46,144)( 47,145)( 48,146)( 49,147)( 50,148)
( 51,149)( 52,171)( 53,172)( 54,173)( 55,174)( 56,175)( 57,176)( 58,177)
( 59,164)( 60,165)( 61,166)( 62,167)( 63,168)( 64,169)( 65,170)( 66,157)
( 67,158)( 68,159)( 69,160)( 70,161)( 71,162)( 72,163)( 73,185)( 74,186)
( 75,187)( 76,188)( 77,189)( 78,190)( 79,191)( 80,178)( 81,179)( 82,180)
( 83,181)( 84,182)( 85,183)( 86,184)( 87,206)( 88,207)( 89,208)( 90,209)
( 91,210)( 92,211)( 93,212)( 94,199)( 95,200)( 96,201)( 97,202)( 98,203)
( 99,204)(100,205)(101,192)(102,193)(103,194)(104,195)(105,196)(106,197)
(107,198);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)( 11, 16)( 12, 15)( 13, 14)( 18, 23)
( 19, 22)( 20, 21)( 25, 30)( 26, 29)( 27, 28)( 32, 37)( 33, 36)( 34, 35)
( 38, 73)( 39, 79)( 40, 78)( 41, 77)( 42, 76)( 43, 75)( 44, 74)( 45, 80)
( 46, 86)( 47, 85)( 48, 84)( 49, 83)( 50, 82)( 51, 81)( 52, 87)( 53, 93)
( 54, 92)( 55, 91)( 56, 90)( 57, 89)( 58, 88)( 59, 94)( 60,100)( 61, 99)
( 62, 98)( 63, 97)( 64, 96)( 65, 95)( 66,101)( 67,107)( 68,106)( 69,105)
( 70,104)( 71,103)( 72,102)(109,114)(110,113)(111,112)(116,121)(117,120)
(118,119)(123,128)(124,127)(125,126)(130,135)(131,134)(132,133)(137,142)
(138,141)(139,140)(143,178)(144,184)(145,183)(146,182)(147,181)(148,180)
(149,179)(150,185)(151,191)(152,190)(153,189)(154,188)(155,187)(156,186)
(157,192)(158,198)(159,197)(160,196)(161,195)(162,194)(163,193)(164,199)
(165,205)(166,204)(167,203)(168,202)(169,201)(170,200)(171,206)(172,212)
(173,211)(174,210)(175,209)(176,208)(177,207);
s2 := Sym(212)!(  3, 39)(  4, 38)(  5, 44)(  6, 43)(  7, 42)(  8, 41)(  9, 40)
( 10, 67)( 11, 66)( 12, 72)( 13, 71)( 14, 70)( 15, 69)( 16, 68)( 17, 60)
( 18, 59)( 19, 65)( 20, 64)( 21, 63)( 22, 62)( 23, 61)( 24, 53)( 25, 52)
( 26, 58)( 27, 57)( 28, 56)( 29, 55)( 30, 54)( 31, 46)( 32, 45)( 33, 51)
( 34, 50)( 35, 49)( 36, 48)( 37, 47)( 73, 74)( 75, 79)( 76, 78)( 80,102)
( 81,101)( 82,107)( 83,106)( 84,105)( 85,104)( 86,103)( 87, 95)( 88, 94)
( 89,100)( 90, 99)( 91, 98)( 92, 97)( 93, 96)(108,144)(109,143)(110,149)
(111,148)(112,147)(113,146)(114,145)(115,172)(116,171)(117,177)(118,176)
(119,175)(120,174)(121,173)(122,165)(123,164)(124,170)(125,169)(126,168)
(127,167)(128,166)(129,158)(130,157)(131,163)(132,162)(133,161)(134,160)
(135,159)(136,151)(137,150)(138,156)(139,155)(140,154)(141,153)(142,152)
(178,179)(180,184)(181,183)(185,207)(186,206)(187,212)(188,211)(189,210)
(190,209)(191,208)(192,200)(193,199)(194,205)(195,204)(196,203)(197,202)
(198,201);
s3 := Sym(212)!(  3,115)(  4,116)(  5,117)(  6,118)(  7,119)(  8,120)(  9,121)
( 10,108)( 11,109)( 12,110)( 13,111)( 14,112)( 15,113)( 16,114)( 17,136)
( 18,137)( 19,138)( 20,139)( 21,140)( 22,141)( 23,142)( 24,129)( 25,130)
( 26,131)( 27,132)( 28,133)( 29,134)( 30,135)( 31,122)( 32,123)( 33,124)
( 34,125)( 35,126)( 36,127)( 37,128)( 38,150)( 39,151)( 40,152)( 41,153)
( 42,154)( 43,155)( 44,156)( 45,143)( 46,144)( 47,145)( 48,146)( 49,147)
( 50,148)( 51,149)( 52,171)( 53,172)( 54,173)( 55,174)( 56,175)( 57,176)
( 58,177)( 59,164)( 60,165)( 61,166)( 62,167)( 63,168)( 64,169)( 65,170)
( 66,157)( 67,158)( 68,159)( 69,160)( 70,161)( 71,162)( 72,163)( 73,185)
( 74,186)( 75,187)( 76,188)( 77,189)( 78,190)( 79,191)( 80,178)( 81,179)
( 82,180)( 83,181)( 84,182)( 85,183)( 86,184)( 87,206)( 88,207)( 89,208)
( 90,209)( 91,210)( 92,211)( 93,212)( 94,199)( 95,200)( 96,201)( 97,202)
( 98,203)( 99,204)(100,205)(101,192)(102,193)(103,194)(104,195)(105,196)
(106,197)(107,198);
poly := sub<Sym(212)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope