Polytope of Type {4,52}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,52}*1664
if this polytope has a name.
Group : SmallGroup(1664,6495)
Rank : 3
Schlafli Type : {4,52}
Number of vertices, edges, etc : 16, 416, 208
Order of s₀s₁s₂ : 104
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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}*832
   4-fold quotients : {4,52}*416
   8-fold quotients : {2,52}*208, {4,26}*208
   13-fold quotients : {4,4}*128
   16-fold quotients : {2,26}*104
   26-fold quotients : {4,4}*64
   32-fold quotients : {2,13}*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 := ( 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);;
s1 := (  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)( 54, 65)( 55, 64)( 56, 63)( 57, 62)( 58, 61)( 59, 60)( 67, 78)
( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 79, 92)( 80,104)( 81,103)
( 82,102)( 83,101)( 84,100)( 85, 99)( 86, 98)( 87, 97)( 88, 96)( 89, 95)
( 90, 94)( 91, 93)(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);;
s2 := (  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)( 53,158)( 54,157)( 55,169)( 56,168)
( 57,167)( 58,166)( 59,165)( 60,164)( 61,163)( 62,162)( 63,161)( 64,160)
( 65,159)( 66,171)( 67,170)( 68,182)( 69,181)( 70,180)( 71,179)( 72,178)
( 73,177)( 74,176)( 75,175)( 76,174)( 77,173)( 78,172)( 79,184)( 80,183)
( 81,195)( 82,194)( 83,193)( 84,192)( 85,191)( 86,190)( 87,189)( 88,188)
( 89,187)( 90,186)( 91,185)( 92,197)( 93,196)( 94,208)( 95,207)( 96,206)
( 97,205)( 98,204)( 99,203)(100,202)(101,201)(102,200)(103,199)(104,198);;
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope