Polytope of Type {104,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {104,4}*832b
if this polytope has a name.
Group : SmallGroup(832,374)
Rank : 3
Schlafli Type : {104,4}
Number of vertices, edges, etc : 104, 208, 4
Order of s₀s₁s₂ : 104
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {104,4,2} of size 1664
Vertex Figure Of :
   {2,104,4} of size 1664
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {52,4}*416
   4-fold quotients : {52,2}*208, {26,4}*208
   8-fold quotients : {26,2}*104
   13-fold quotients : {8,4}*64b
   16-fold quotients : {13,2}*52
   26-fold quotients : {4,4}*32
   52-fold quotients : {2,4}*16, {4,2}*16
   104-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {104,4}*1664a, {104,8}*1664a, {104,8}*1664d
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
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