Polytope of Type {2,8,52}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,8,52}*1664a
if this polytope has a name.
Group : SmallGroup(1664,13687)
Rank : 4
Schlafli Type : {2,8,52}
Number of vertices, edges, etc : 2, 8, 208, 52
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 : {2,4,52}*832, {2,8,26}*832
   4-fold quotients : {2,2,52}*416, {2,4,26}*416
   8-fold quotients : {2,2,26}*208
   13-fold quotients : {2,8,4}*128a
   16-fold quotients : {2,2,13}*104
   26-fold quotients : {2,4,4}*64, {2,8,2}*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 := ( 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)( 81, 94)( 82, 95)( 83, 96)
( 84, 97)( 85, 98)( 86, 99)( 87,100)( 88,101)( 89,102)( 90,103)( 91,104)
( 92,105)( 93,106)(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)(159,185)(160,186)(161,187)(162,188)
(163,189)(164,190)(165,191)(166,192)(167,193)(168,194)(169,195)(170,196)
(171,197)(172,198)(173,199)(174,200)(175,201)(176,202)(177,203)(178,204)
(179,205)(180,206)(181,207)(182,208)(183,209)(184,210);;
s2 := (  3,107)(  4,119)(  5,118)(  6,117)(  7,116)(  8,115)(  9,114)( 10,113)
( 11,112)( 12,111)( 13,110)( 14,109)( 15,108)( 16,120)( 17,132)( 18,131)
( 19,130)( 20,129)( 21,128)( 22,127)( 23,126)( 24,125)( 25,124)( 26,123)
( 27,122)( 28,121)( 29,146)( 30,158)( 31,157)( 32,156)( 33,155)( 34,154)
( 35,153)( 36,152)( 37,151)( 38,150)( 39,149)( 40,148)( 41,147)( 42,133)
( 43,145)( 44,144)( 45,143)( 46,142)( 47,141)( 48,140)( 49,139)( 50,138)
( 51,137)( 52,136)( 53,135)( 54,134)( 55,159)( 56,171)( 57,170)( 58,169)
( 59,168)( 60,167)( 61,166)( 62,165)( 63,164)( 64,163)( 65,162)( 66,161)
( 67,160)( 68,172)( 69,184)( 70,183)( 71,182)( 72,181)( 73,180)( 74,179)
( 75,178)( 76,177)( 77,176)( 78,175)( 79,174)( 80,173)( 81,198)( 82,210)
( 83,209)( 84,208)( 85,207)( 86,206)( 87,205)( 88,204)( 89,203)( 90,202)
( 91,201)( 92,200)( 93,199)( 94,185)( 95,197)( 96,196)( 97,195)( 98,194)
( 99,193)(100,192)(101,191)(102,190)(103,189)(104,188)(105,187)(106,186);;
s3 := (  3,  4)(  5, 15)(  6, 14)(  7, 13)(  8, 12)(  9, 11)( 16, 17)( 18, 28)
( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 29, 30)( 31, 41)( 32, 40)( 33, 39)
( 34, 38)( 35, 37)( 42, 43)( 44, 54)( 45, 53)( 46, 52)( 47, 51)( 48, 50)
( 55, 56)( 57, 67)( 58, 66)( 59, 65)( 60, 64)( 61, 63)( 68, 69)( 70, 80)
( 71, 79)( 72, 78)( 73, 77)( 74, 76)( 81, 82)( 83, 93)( 84, 92)( 85, 91)
( 86, 90)( 87, 89)( 94, 95)( 96,106)( 97,105)( 98,104)( 99,103)(100,102)
(107,160)(108,159)(109,171)(110,170)(111,169)(112,168)(113,167)(114,166)
(115,165)(116,164)(117,163)(118,162)(119,161)(120,173)(121,172)(122,184)
(123,183)(124,182)(125,181)(126,180)(127,179)(128,178)(129,177)(130,176)
(131,175)(132,174)(133,186)(134,185)(135,197)(136,196)(137,195)(138,194)
(139,193)(140,192)(141,191)(142,190)(143,189)(144,188)(145,187)(146,199)
(147,198)(148,210)(149,209)(150,208)(151,207)(152,206)(153,205)(154,204)
(155,203)(156,202)(157,201)(158,200);;
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, 
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*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(210)!(1,2);
s1 := 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)( 81, 94)( 82, 95)
( 83, 96)( 84, 97)( 85, 98)( 86, 99)( 87,100)( 88,101)( 89,102)( 90,103)
( 91,104)( 92,105)( 93,106)(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)(159,185)(160,186)(161,187)
(162,188)(163,189)(164,190)(165,191)(166,192)(167,193)(168,194)(169,195)
(170,196)(171,197)(172,198)(173,199)(174,200)(175,201)(176,202)(177,203)
(178,204)(179,205)(180,206)(181,207)(182,208)(183,209)(184,210);
s2 := Sym(210)!(  3,107)(  4,119)(  5,118)(  6,117)(  7,116)(  8,115)(  9,114)
( 10,113)( 11,112)( 12,111)( 13,110)( 14,109)( 15,108)( 16,120)( 17,132)
( 18,131)( 19,130)( 20,129)( 21,128)( 22,127)( 23,126)( 24,125)( 25,124)
( 26,123)( 27,122)( 28,121)( 29,146)( 30,158)( 31,157)( 32,156)( 33,155)
( 34,154)( 35,153)( 36,152)( 37,151)( 38,150)( 39,149)( 40,148)( 41,147)
( 42,133)( 43,145)( 44,144)( 45,143)( 46,142)( 47,141)( 48,140)( 49,139)
( 50,138)( 51,137)( 52,136)( 53,135)( 54,134)( 55,159)( 56,171)( 57,170)
( 58,169)( 59,168)( 60,167)( 61,166)( 62,165)( 63,164)( 64,163)( 65,162)
( 66,161)( 67,160)( 68,172)( 69,184)( 70,183)( 71,182)( 72,181)( 73,180)
( 74,179)( 75,178)( 76,177)( 77,176)( 78,175)( 79,174)( 80,173)( 81,198)
( 82,210)( 83,209)( 84,208)( 85,207)( 86,206)( 87,205)( 88,204)( 89,203)
( 90,202)( 91,201)( 92,200)( 93,199)( 94,185)( 95,197)( 96,196)( 97,195)
( 98,194)( 99,193)(100,192)(101,191)(102,190)(103,189)(104,188)(105,187)
(106,186);
s3 := Sym(210)!(  3,  4)(  5, 15)(  6, 14)(  7, 13)(  8, 12)(  9, 11)( 16, 17)
( 18, 28)( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 29, 30)( 31, 41)( 32, 40)
( 33, 39)( 34, 38)( 35, 37)( 42, 43)( 44, 54)( 45, 53)( 46, 52)( 47, 51)
( 48, 50)( 55, 56)( 57, 67)( 58, 66)( 59, 65)( 60, 64)( 61, 63)( 68, 69)
( 70, 80)( 71, 79)( 72, 78)( 73, 77)( 74, 76)( 81, 82)( 83, 93)( 84, 92)
( 85, 91)( 86, 90)( 87, 89)( 94, 95)( 96,106)( 97,105)( 98,104)( 99,103)
(100,102)(107,160)(108,159)(109,171)(110,170)(111,169)(112,168)(113,167)
(114,166)(115,165)(116,164)(117,163)(118,162)(119,161)(120,173)(121,172)
(122,184)(123,183)(124,182)(125,181)(126,180)(127,179)(128,178)(129,177)
(130,176)(131,175)(132,174)(133,186)(134,185)(135,197)(136,196)(137,195)
(138,194)(139,193)(140,192)(141,191)(142,190)(143,189)(144,188)(145,187)
(146,199)(147,198)(148,210)(149,209)(150,208)(151,207)(152,206)(153,205)
(154,204)(155,203)(156,202)(157,201)(158,200);
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, s1*s2*s3*s2*s1*s2*s3*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope