Polytope of Type {104,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {104,4}*1664b
if this polytope has a name.
Group : SmallGroup(1664,6495)
Rank : 3
Schlafli Type : {104,4}
Number of vertices, edges, etc : 208, 416, 8
Order of s₀s₁s₂ : 52
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {52,4}*832
   4-fold quotients : {52,4}*416
   8-fold quotients : {52,2}*208, {26,4}*208
   13-fold quotients : {8,4}*128b
   16-fold quotients : {26,2}*104
   26-fold quotients : {4,4}*64
   32-fold quotients : {13,2}*52
   52-fold quotients : {4,4}*32
   104-fold quotients : {2,4}*16, {4,2}*16
   208-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  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)( 53,196)( 54,208)( 55,207)( 56,206)
( 57,205)( 58,204)( 59,203)( 60,202)( 61,201)( 62,200)( 63,199)( 64,198)
( 65,197)( 66,183)( 67,195)( 68,194)( 69,193)( 70,192)( 71,191)( 72,190)
( 73,189)( 74,188)( 75,187)( 76,186)( 77,185)( 78,184)( 79,170)( 80,182)
( 81,181)( 82,180)( 83,179)( 84,178)( 85,177)( 86,176)( 87,175)( 88,174)
( 89,173)( 90,172)( 91,171)( 92,157)( 93,169)( 94,168)( 95,167)( 96,166)
( 97,165)( 98,164)( 99,163)(100,162)(101,161)(102,160)(103,159)(104,158);;
s1 := (  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);;
s2 := ( 53, 92)( 54, 93)( 55, 94)( 56, 95)( 57, 96)( 58, 97)( 59, 98)( 60, 99)
( 61,100)( 62,101)( 63,102)( 64,103)( 65,104)( 66, 79)( 67, 80)( 68, 81)
( 69, 82)( 70, 83)( 71, 84)( 72, 85)( 73, 86)( 74, 87)( 75, 88)( 76, 89)
( 77, 90)( 78, 91)(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*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*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*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)!(  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)( 53,196)( 54,208)( 55,207)
( 56,206)( 57,205)( 58,204)( 59,203)( 60,202)( 61,201)( 62,200)( 63,199)
( 64,198)( 65,197)( 66,183)( 67,195)( 68,194)( 69,193)( 70,192)( 71,191)
( 72,190)( 73,189)( 74,188)( 75,187)( 76,186)( 77,185)( 78,184)( 79,170)
( 80,182)( 81,181)( 82,180)( 83,179)( 84,178)( 85,177)( 86,176)( 87,175)
( 88,174)( 89,173)( 90,172)( 91,171)( 92,157)( 93,169)( 94,168)( 95,167)
( 96,166)( 97,165)( 98,164)( 99,163)(100,162)(101,161)(102,160)(103,159)
(104,158);
s1 := Sym(208)!(  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);
s2 := Sym(208)!( 53, 92)( 54, 93)( 55, 94)( 56, 95)( 57, 96)( 58, 97)( 59, 98)
( 60, 99)( 61,100)( 62,101)( 63,102)( 64,103)( 65,104)( 66, 79)( 67, 80)
( 68, 81)( 69, 82)( 70, 83)( 71, 84)( 72, 85)( 73, 86)( 74, 87)( 75, 88)
( 76, 89)( 77, 90)( 78, 91)(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*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*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*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.
to this polytope