Polytope of Type {2,10,42}

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

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

Permutation Representation (Magma) :

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

to this polytope