Polytope of Type {2,70,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,70,6}*1680
if this polytope has a name.
Group : SmallGroup(1680,991)
Rank : 4
Schlafli Type : {2,70,6}
Number of vertices, edges, etc : 2, 70, 210, 6
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 : {2,70,2}*560
   5-fold quotients : {2,14,6}*336
   6-fold quotients : {2,35,2}*280
   7-fold quotients : {2,10,6}*240
   15-fold quotients : {2,14,2}*112
   21-fold quotients : {2,10,2}*80
   30-fold quotients : {2,7,2}*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 := (  4,  9)(  5,  8)(  6,  7)( 10, 31)( 11, 37)( 12, 36)( 13, 35)( 14, 34)
( 15, 33)( 16, 32)( 17, 24)( 18, 30)( 19, 29)( 20, 28)( 21, 27)( 22, 26)
( 23, 25)( 39, 44)( 40, 43)( 41, 42)( 45, 66)( 46, 72)( 47, 71)( 48, 70)
( 49, 69)( 50, 68)( 51, 67)( 52, 59)( 53, 65)( 54, 64)( 55, 63)( 56, 62)
( 57, 61)( 58, 60)( 74, 79)( 75, 78)( 76, 77)( 80,101)( 81,107)( 82,106)
( 83,105)( 84,104)( 85,103)( 86,102)( 87, 94)( 88,100)( 89, 99)( 90, 98)
( 91, 97)( 92, 96)( 93, 95)(109,114)(110,113)(111,112)(115,136)(116,142)
(117,141)(118,140)(119,139)(120,138)(121,137)(122,129)(123,135)(124,134)
(125,133)(126,132)(127,131)(128,130)(144,149)(145,148)(146,147)(150,171)
(151,177)(152,176)(153,175)(154,174)(155,173)(156,172)(157,164)(158,170)
(159,169)(160,168)(161,167)(162,166)(163,165)(179,184)(180,183)(181,182)
(185,206)(186,212)(187,211)(188,210)(189,209)(190,208)(191,207)(192,199)
(193,205)(194,204)(195,203)(196,202)(197,201)(198,200);;
s2 := (  3, 11)(  4, 10)(  5, 16)(  6, 15)(  7, 14)(  8, 13)(  9, 12)( 17, 32)
( 18, 31)( 19, 37)( 20, 36)( 21, 35)( 22, 34)( 23, 33)( 24, 25)( 26, 30)
( 27, 29)( 38, 81)( 39, 80)( 40, 86)( 41, 85)( 42, 84)( 43, 83)( 44, 82)
( 45, 74)( 46, 73)( 47, 79)( 48, 78)( 49, 77)( 50, 76)( 51, 75)( 52,102)
( 53,101)( 54,107)( 55,106)( 56,105)( 57,104)( 58,103)( 59, 95)( 60, 94)
( 61,100)( 62, 99)( 63, 98)( 64, 97)( 65, 96)( 66, 88)( 67, 87)( 68, 93)
( 69, 92)( 70, 91)( 71, 90)( 72, 89)(108,116)(109,115)(110,121)(111,120)
(112,119)(113,118)(114,117)(122,137)(123,136)(124,142)(125,141)(126,140)
(127,139)(128,138)(129,130)(131,135)(132,134)(143,186)(144,185)(145,191)
(146,190)(147,189)(148,188)(149,187)(150,179)(151,178)(152,184)(153,183)
(154,182)(155,181)(156,180)(157,207)(158,206)(159,212)(160,211)(161,210)
(162,209)(163,208)(164,200)(165,199)(166,205)(167,204)(168,203)(169,202)
(170,201)(171,193)(172,192)(173,198)(174,197)(175,196)(176,195)(177,194);;
s3 := (  3,143)(  4,144)(  5,145)(  6,146)(  7,147)(  8,148)(  9,149)( 10,150)
( 11,151)( 12,152)( 13,153)( 14,154)( 15,155)( 16,156)( 17,157)( 18,158)
( 19,159)( 20,160)( 21,161)( 22,162)( 23,163)( 24,164)( 25,165)( 26,166)
( 27,167)( 28,168)( 29,169)( 30,170)( 31,171)( 32,172)( 33,173)( 34,174)
( 35,175)( 36,176)( 37,177)( 38,108)( 39,109)( 40,110)( 41,111)( 42,112)
( 43,113)( 44,114)( 45,115)( 46,116)( 47,117)( 48,118)( 49,119)( 50,120)
( 51,121)( 52,122)( 53,123)( 54,124)( 55,125)( 56,126)( 57,127)( 58,128)
( 59,129)( 60,130)( 61,131)( 62,132)( 63,133)( 64,134)( 65,135)( 66,136)
( 67,137)( 68,138)( 69,139)( 70,140)( 71,141)( 72,142)( 73,178)( 74,179)
( 75,180)( 76,181)( 77,182)( 78,183)( 79,184)( 80,185)( 81,186)( 82,187)
( 83,188)( 84,189)( 85,190)( 86,191)( 87,192)( 88,193)( 89,194)( 90,195)
( 91,196)( 92,197)( 93,198)( 94,199)( 95,200)( 96,201)( 97,202)( 98,203)
( 99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)(106,211)
(107,212);;
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, 
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*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)( 10, 31)( 11, 37)( 12, 36)( 13, 35)
( 14, 34)( 15, 33)( 16, 32)( 17, 24)( 18, 30)( 19, 29)( 20, 28)( 21, 27)
( 22, 26)( 23, 25)( 39, 44)( 40, 43)( 41, 42)( 45, 66)( 46, 72)( 47, 71)
( 48, 70)( 49, 69)( 50, 68)( 51, 67)( 52, 59)( 53, 65)( 54, 64)( 55, 63)
( 56, 62)( 57, 61)( 58, 60)( 74, 79)( 75, 78)( 76, 77)( 80,101)( 81,107)
( 82,106)( 83,105)( 84,104)( 85,103)( 86,102)( 87, 94)( 88,100)( 89, 99)
( 90, 98)( 91, 97)( 92, 96)( 93, 95)(109,114)(110,113)(111,112)(115,136)
(116,142)(117,141)(118,140)(119,139)(120,138)(121,137)(122,129)(123,135)
(124,134)(125,133)(126,132)(127,131)(128,130)(144,149)(145,148)(146,147)
(150,171)(151,177)(152,176)(153,175)(154,174)(155,173)(156,172)(157,164)
(158,170)(159,169)(160,168)(161,167)(162,166)(163,165)(179,184)(180,183)
(181,182)(185,206)(186,212)(187,211)(188,210)(189,209)(190,208)(191,207)
(192,199)(193,205)(194,204)(195,203)(196,202)(197,201)(198,200);
s2 := Sym(212)!(  3, 11)(  4, 10)(  5, 16)(  6, 15)(  7, 14)(  8, 13)(  9, 12)
( 17, 32)( 18, 31)( 19, 37)( 20, 36)( 21, 35)( 22, 34)( 23, 33)( 24, 25)
( 26, 30)( 27, 29)( 38, 81)( 39, 80)( 40, 86)( 41, 85)( 42, 84)( 43, 83)
( 44, 82)( 45, 74)( 46, 73)( 47, 79)( 48, 78)( 49, 77)( 50, 76)( 51, 75)
( 52,102)( 53,101)( 54,107)( 55,106)( 56,105)( 57,104)( 58,103)( 59, 95)
( 60, 94)( 61,100)( 62, 99)( 63, 98)( 64, 97)( 65, 96)( 66, 88)( 67, 87)
( 68, 93)( 69, 92)( 70, 91)( 71, 90)( 72, 89)(108,116)(109,115)(110,121)
(111,120)(112,119)(113,118)(114,117)(122,137)(123,136)(124,142)(125,141)
(126,140)(127,139)(128,138)(129,130)(131,135)(132,134)(143,186)(144,185)
(145,191)(146,190)(147,189)(148,188)(149,187)(150,179)(151,178)(152,184)
(153,183)(154,182)(155,181)(156,180)(157,207)(158,206)(159,212)(160,211)
(161,210)(162,209)(163,208)(164,200)(165,199)(166,205)(167,204)(168,203)
(169,202)(170,201)(171,193)(172,192)(173,198)(174,197)(175,196)(176,195)
(177,194);
s3 := Sym(212)!(  3,143)(  4,144)(  5,145)(  6,146)(  7,147)(  8,148)(  9,149)
( 10,150)( 11,151)( 12,152)( 13,153)( 14,154)( 15,155)( 16,156)( 17,157)
( 18,158)( 19,159)( 20,160)( 21,161)( 22,162)( 23,163)( 24,164)( 25,165)
( 26,166)( 27,167)( 28,168)( 29,169)( 30,170)( 31,171)( 32,172)( 33,173)
( 34,174)( 35,175)( 36,176)( 37,177)( 38,108)( 39,109)( 40,110)( 41,111)
( 42,112)( 43,113)( 44,114)( 45,115)( 46,116)( 47,117)( 48,118)( 49,119)
( 50,120)( 51,121)( 52,122)( 53,123)( 54,124)( 55,125)( 56,126)( 57,127)
( 58,128)( 59,129)( 60,130)( 61,131)( 62,132)( 63,133)( 64,134)( 65,135)
( 66,136)( 67,137)( 68,138)( 69,139)( 70,140)( 71,141)( 72,142)( 73,178)
( 74,179)( 75,180)( 76,181)( 77,182)( 78,183)( 79,184)( 80,185)( 81,186)
( 82,187)( 83,188)( 84,189)( 85,190)( 86,191)( 87,192)( 88,193)( 89,194)
( 90,195)( 91,196)( 92,197)( 93,198)( 94,199)( 95,200)( 96,201)( 97,202)
( 98,203)( 99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)
(106,211)(107,212);
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, 
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*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