Polytope of Type {2,52,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,52,8}*1664b
if this polytope has a name.
Group : SmallGroup(1664,13835)
Rank : 4
Schlafli Type : {2,52,8}
Number of vertices, edges, etc : 2, 52, 208, 8
Order of s₀s₁s₂s₃ : 104
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) :
   2-fold quotients : {2,52,4}*832
   4-fold quotients : {2,52,2}*416, {2,26,4}*416
   8-fold quotients : {2,26,2}*208
   13-fold quotients : {2,4,8}*128b
   16-fold quotients : {2,13,2}*104
   26-fold quotients : {2,4,4}*64
   52-fold quotients : {2,2,4}*32, {2,4,2}*32
   104-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

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