Polytope of Type {2,4,52}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,52}*1664
if this polytope has a name.
Group : SmallGroup(1664,13895)
Rank : 4
Schlafli Type : {2,4,52}
Number of vertices, edges, etc : 2, 8, 208, 104
Order of s₀s₁s₂s₃ : 52
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
   4-fold quotients : {2,2,52}*416, {2,4,26}*416
   8-fold quotients : {2,2,26}*208
   13-fold quotients : {2,4,4}*128
   16-fold quotients : {2,2,13}*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 := ( 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)( 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)(159,198)(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);;
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)( 29,133)( 30,145)( 31,144)( 32,143)( 33,142)( 34,141)
( 35,140)( 36,139)( 37,138)( 38,137)( 39,136)( 40,135)( 41,134)( 42,146)
( 43,158)( 44,157)( 45,156)( 46,155)( 47,154)( 48,153)( 49,152)( 50,151)
( 51,150)( 52,149)( 53,148)( 54,147)( 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)( 81,185)( 82,197)
( 83,196)( 84,195)( 85,194)( 86,193)( 87,192)( 88,191)( 89,190)( 90,189)
( 91,188)( 92,187)( 93,186)( 94,198)( 95,210)( 96,209)( 97,208)( 98,207)
( 99,206)(100,205)(101,204)(102,203)(103,202)(104,201)(105,200)(106,199);;
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, 43)( 30, 42)( 31, 54)( 32, 53)
( 33, 52)( 34, 51)( 35, 50)( 36, 49)( 37, 48)( 38, 47)( 39, 46)( 40, 45)
( 41, 44)( 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, 95)( 82, 94)( 83,106)
( 84,105)( 85,104)( 86,103)( 87,102)( 88,101)( 89,100)( 90, 99)( 91, 98)
( 92, 97)( 93, 96)(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)(133,199)(134,198)(135,210)(136,209)
(137,208)(138,207)(139,206)(140,205)(141,204)(142,203)(143,202)(144,201)
(145,200)(146,186)(147,185)(148,197)(149,196)(150,195)(151,194)(152,193)
(153,192)(154,191)(155,190)(156,189)(157,188)(158,187);;
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*s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s2*s1*s2*s1*s3*s2*s3*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)!( 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)( 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)(159,198)(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);
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)( 29,133)( 30,145)( 31,144)( 32,143)( 33,142)
( 34,141)( 35,140)( 36,139)( 37,138)( 38,137)( 39,136)( 40,135)( 41,134)
( 42,146)( 43,158)( 44,157)( 45,156)( 46,155)( 47,154)( 48,153)( 49,152)
( 50,151)( 51,150)( 52,149)( 53,148)( 54,147)( 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)( 81,185)
( 82,197)( 83,196)( 84,195)( 85,194)( 86,193)( 87,192)( 88,191)( 89,190)
( 90,189)( 91,188)( 92,187)( 93,186)( 94,198)( 95,210)( 96,209)( 97,208)
( 98,207)( 99,206)(100,205)(101,204)(102,203)(103,202)(104,201)(105,200)
(106,199);
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, 43)( 30, 42)( 31, 54)
( 32, 53)( 33, 52)( 34, 51)( 35, 50)( 36, 49)( 37, 48)( 38, 47)( 39, 46)
( 40, 45)( 41, 44)( 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, 95)( 82, 94)
( 83,106)( 84,105)( 85,104)( 86,103)( 87,102)( 88,101)( 89,100)( 90, 99)
( 91, 98)( 92, 97)( 93, 96)(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)(133,199)(134,198)(135,210)
(136,209)(137,208)(138,207)(139,206)(140,205)(141,204)(142,203)(143,202)
(144,201)(145,200)(146,186)(147,185)(148,197)(149,196)(150,195)(151,194)
(152,193)(153,192)(154,191)(155,190)(156,189)(157,188)(158,187);
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*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s1*s2*s1*s3*s2*s3*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