Polytope of Type {4,26,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,26,4}*832
Also Known As : {{4,26|2},{26,4|2}}. if this polytope has another name.
Group : SmallGroup(832,1261)
Rank : 4
Schlafli Type : {4,26,4}
Number of vertices, edges, etc : 4, 52, 52, 4
Order of s₀s₁s₂s₃ : 52
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,26,4,2} of size 1664
Vertex Figure Of :
   {2,4,26,4} of size 1664
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,26,4}*416, {4,26,2}*416
   4-fold quotients : {2,26,2}*208
   8-fold quotients : {2,13,2}*104
   13-fold quotients : {4,2,4}*64
   26-fold quotients : {2,2,4}*32, {4,2,2}*32
   52-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,52,4}*1664, {4,26,8}*1664, {8,26,4}*1664
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)( 53,183)( 54,195)( 55,194)( 56,193)
( 57,192)( 58,191)( 59,190)( 60,189)( 61,188)( 62,187)( 63,186)( 64,185)
( 65,184)( 66,196)( 67,208)( 68,207)( 69,206)( 70,205)( 71,204)( 72,203)
( 73,202)( 74,201)( 75,200)( 76,199)( 77,198)( 78,197)( 79,157)( 80,169)
( 81,168)( 82,167)( 83,166)( 84,165)( 85,164)( 86,163)( 87,162)( 88,161)
( 89,160)( 90,159)( 91,158)( 92,170)( 93,182)( 94,181)( 95,180)( 96,179)
( 97,178)( 98,177)( 99,176)(100,175)(101,174)(102,173)(103,172)(104,171);;
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, 93)( 54, 92)( 55,104)( 56,103)( 57,102)( 58,101)( 59,100)( 60, 99)
( 61, 98)( 62, 97)( 63, 96)( 64, 95)( 65, 94)( 66, 80)( 67, 79)( 68, 91)
( 69, 90)( 70, 89)( 71, 88)( 72, 87)( 73, 86)( 74, 85)( 75, 84)( 76, 83)
( 77, 82)( 78, 81)(105,106)(107,117)(108,116)(109,115)(110,114)(111,113)
(118,119)(120,130)(121,129)(122,128)(123,127)(124,126)(131,132)(133,143)
(134,142)(135,141)(136,140)(137,139)(144,145)(146,156)(147,155)(148,154)
(149,153)(150,152)(157,197)(158,196)(159,208)(160,207)(161,206)(162,205)
(163,204)(164,203)(165,202)(166,201)(167,200)(168,199)(169,198)(170,184)
(171,183)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)(178,189)
(179,188)(180,187)(181,186)(182,185);;
s3 := (  1, 53)(  2, 54)(  3, 55)(  4, 56)(  5, 57)(  6, 58)(  7, 59)(  8, 60)
(  9, 61)( 10, 62)( 11, 63)( 12, 64)( 13, 65)( 14, 66)( 15, 67)( 16, 68)
( 17, 69)( 18, 70)( 19, 71)( 20, 72)( 21, 73)( 22, 74)( 23, 75)( 24, 76)
( 25, 77)( 26, 78)( 27, 79)( 28, 80)( 29, 81)( 30, 82)( 31, 83)( 32, 84)
( 33, 85)( 34, 86)( 35, 87)( 36, 88)( 37, 89)( 38, 90)( 39, 91)( 40, 92)
( 41, 93)( 42, 94)( 43, 95)( 44, 96)( 45, 97)( 46, 98)( 47, 99)( 48,100)
( 49,101)( 50,102)( 51,103)( 52,104)(105,183)(106,184)(107,185)(108,186)
(109,187)(110,188)(111,189)(112,190)(113,191)(114,192)(115,193)(116,194)
(117,195)(118,196)(119,197)(120,198)(121,199)(122,200)(123,201)(124,202)
(125,203)(126,204)(127,205)(128,206)(129,207)(130,208)(131,157)(132,158)
(133,159)(134,160)(135,161)(136,162)(137,163)(138,164)(139,165)(140,166)
(141,167)(142,168)(143,169)(144,170)(145,171)(146,172)(147,173)(148,174)
(149,175)(150,176)(151,177)(152,178)(153,179)(154,180)(155,181)(156,182);;
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!(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(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)( 53,183)( 54,195)( 55,194)
( 56,193)( 57,192)( 58,191)( 59,190)( 60,189)( 61,188)( 62,187)( 63,186)
( 64,185)( 65,184)( 66,196)( 67,208)( 68,207)( 69,206)( 70,205)( 71,204)
( 72,203)( 73,202)( 74,201)( 75,200)( 76,199)( 77,198)( 78,197)( 79,157)
( 80,169)( 81,168)( 82,167)( 83,166)( 84,165)( 85,164)( 86,163)( 87,162)
( 88,161)( 89,160)( 90,159)( 91,158)( 92,170)( 93,182)( 94,181)( 95,180)
( 96,179)( 97,178)( 98,177)( 99,176)(100,175)(101,174)(102,173)(103,172)
(104,171);
s2 := 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, 28)( 29, 39)( 30, 38)
( 31, 37)( 32, 36)( 33, 35)( 40, 41)( 42, 52)( 43, 51)( 44, 50)( 45, 49)
( 46, 48)( 53, 93)( 54, 92)( 55,104)( 56,103)( 57,102)( 58,101)( 59,100)
( 60, 99)( 61, 98)( 62, 97)( 63, 96)( 64, 95)( 65, 94)( 66, 80)( 67, 79)
( 68, 91)( 69, 90)( 70, 89)( 71, 88)( 72, 87)( 73, 86)( 74, 85)( 75, 84)
( 76, 83)( 77, 82)( 78, 81)(105,106)(107,117)(108,116)(109,115)(110,114)
(111,113)(118,119)(120,130)(121,129)(122,128)(123,127)(124,126)(131,132)
(133,143)(134,142)(135,141)(136,140)(137,139)(144,145)(146,156)(147,155)
(148,154)(149,153)(150,152)(157,197)(158,196)(159,208)(160,207)(161,206)
(162,205)(163,204)(164,203)(165,202)(166,201)(167,200)(168,199)(169,198)
(170,184)(171,183)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)
(178,189)(179,188)(180,187)(181,186)(182,185);
s3 := Sym(208)!(  1, 53)(  2, 54)(  3, 55)(  4, 56)(  5, 57)(  6, 58)(  7, 59)
(  8, 60)(  9, 61)( 10, 62)( 11, 63)( 12, 64)( 13, 65)( 14, 66)( 15, 67)
( 16, 68)( 17, 69)( 18, 70)( 19, 71)( 20, 72)( 21, 73)( 22, 74)( 23, 75)
( 24, 76)( 25, 77)( 26, 78)( 27, 79)( 28, 80)( 29, 81)( 30, 82)( 31, 83)
( 32, 84)( 33, 85)( 34, 86)( 35, 87)( 36, 88)( 37, 89)( 38, 90)( 39, 91)
( 40, 92)( 41, 93)( 42, 94)( 43, 95)( 44, 96)( 45, 97)( 46, 98)( 47, 99)
( 48,100)( 49,101)( 50,102)( 51,103)( 52,104)(105,183)(106,184)(107,185)
(108,186)(109,187)(110,188)(111,189)(112,190)(113,191)(114,192)(115,193)
(116,194)(117,195)(118,196)(119,197)(120,198)(121,199)(122,200)(123,201)
(124,202)(125,203)(126,204)(127,205)(128,206)(129,207)(130,208)(131,157)
(132,158)(133,159)(134,160)(135,161)(136,162)(137,163)(138,164)(139,165)
(140,166)(141,167)(142,168)(143,169)(144,170)(145,171)(146,172)(147,173)
(148,174)(149,175)(150,176)(151,177)(152,178)(153,179)(154,180)(155,181)
(156,182);
poly := sub<Sym(208)|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, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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