Polytope of Type {4,104,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,104,2}*1664a
if this polytope has a name.
Group : SmallGroup(1664,13688)
Rank : 4
Schlafli Type : {4,104,2}
Number of vertices, edges, etc : 4, 208, 104, 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, {2,104,2}*832
   4-fold quotients : {2,52,2}*416, {4,26,2}*416
   8-fold quotients : {2,26,2}*208
   13-fold quotients : {4,8,2}*128a
   16-fold quotients : {2,13,2}*104
   26-fold quotients : {4,4,2}*64, {2,8,2}*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 := (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)(157,183)(158,184)(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);;
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)( 27,131)( 28,143)( 29,142)( 30,141)( 31,140)( 32,139)
( 33,138)( 34,137)( 35,136)( 36,135)( 37,134)( 38,133)( 39,132)( 40,144)
( 41,156)( 42,155)( 43,154)( 44,153)( 45,152)( 46,151)( 47,150)( 48,149)
( 49,148)( 50,147)( 51,146)( 52,145)( 53,170)( 54,182)( 55,181)( 56,180)
( 57,179)( 58,178)( 59,177)( 60,176)( 61,175)( 62,174)( 63,173)( 64,172)
( 65,171)( 66,157)( 67,169)( 68,168)( 69,167)( 70,166)( 71,165)( 72,164)
( 73,163)( 74,162)( 75,161)( 76,160)( 77,159)( 78,158)( 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, 28)( 29, 39)( 30, 38)( 31, 37)
( 32, 36)( 33, 35)( 40, 41)( 42, 52)( 43, 51)( 44, 50)( 45, 49)( 46, 48)
( 53, 67)( 54, 66)( 55, 78)( 56, 77)( 57, 76)( 58, 75)( 59, 74)( 60, 73)
( 61, 72)( 62, 71)( 63, 70)( 64, 69)( 65, 68)( 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)(131,184)(132,183)(133,195)(134,194)
(135,193)(136,192)(137,191)(138,190)(139,189)(140,188)(141,187)(142,186)
(143,185)(144,197)(145,196)(146,208)(147,207)(148,206)(149,205)(150,204)
(151,203)(152,202)(153,201)(154,200)(155,199)(156,198);;
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, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!(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)(157,183)(158,184)(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);
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)( 27,131)( 28,143)( 29,142)( 30,141)( 31,140)
( 32,139)( 33,138)( 34,137)( 35,136)( 36,135)( 37,134)( 38,133)( 39,132)
( 40,144)( 41,156)( 42,155)( 43,154)( 44,153)( 45,152)( 46,151)( 47,150)
( 48,149)( 49,148)( 50,147)( 51,146)( 52,145)( 53,170)( 54,182)( 55,181)
( 56,180)( 57,179)( 58,178)( 59,177)( 60,176)( 61,175)( 62,174)( 63,173)
( 64,172)( 65,171)( 66,157)( 67,169)( 68,168)( 69,167)( 70,166)( 71,165)
( 72,164)( 73,163)( 74,162)( 75,161)( 76,160)( 77,159)( 78,158)( 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, 28)( 29, 39)( 30, 38)
( 31, 37)( 32, 36)( 33, 35)( 40, 41)( 42, 52)( 43, 51)( 44, 50)( 45, 49)
( 46, 48)( 53, 67)( 54, 66)( 55, 78)( 56, 77)( 57, 76)( 58, 75)( 59, 74)
( 60, 73)( 61, 72)( 62, 71)( 63, 70)( 64, 69)( 65, 68)( 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)(131,184)(132,183)(133,195)
(134,194)(135,193)(136,192)(137,191)(138,190)(139,189)(140,188)(141,187)
(142,186)(143,185)(144,197)(145,196)(146,208)(147,207)(148,206)(149,205)
(150,204)(151,203)(152,202)(153,201)(154,200)(155,199)(156,198);
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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