Polytope of Type {8,52,2}

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

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