Polytope of Type {4,4,26}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,26}*1664
Also Known As : {{4,4}₄,{4,26|2}}. if this polytope has another name.
Group : SmallGroup(1664,13892)
Rank : 4
Schlafli Type : {4,4,26}
Number of vertices, edges, etc : 8, 16, 104, 26
Order of s₀s₁s₂s₃ : 52
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   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 : {4,4,26}*832
   4-fold quotients : {2,4,26}*416, {4,2,26}*416
   8-fold quotients : {4,2,13}*208, {2,2,26}*208
   13-fold quotients : {4,4,2}*128
   16-fold quotients : {2,2,13}*104
   26-fold quotients : {4,4,2}*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 := ( 27, 40)( 28, 41)( 29, 42)( 30, 43)( 31, 44)( 32, 45)( 33, 46)( 34, 47)
( 35, 48)( 36, 49)( 37, 50)( 38, 51)( 39, 52)( 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)(131,144)(132,145)(133,146)(134,147)(135,148)(136,149)
(137,150)(138,151)(139,152)(140,153)(141,154)(142,155)(143,156)(183,196)
(184,197)(185,198)(186,199)(187,200)(188,201)(189,202)(190,203)(191,204)
(192,205)(193,206)(194,207)(195,208);;
s1 := (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);;
s2 := (  1,157)(  2,169)(  3,168)(  4,167)(  5,166)(  6,165)(  7,164)(  8,163)
(  9,162)( 10,161)( 11,160)( 12,159)( 13,158)( 14,170)( 15,182)( 16,181)
( 17,180)( 18,179)( 19,178)( 20,177)( 21,176)( 22,175)( 23,174)( 24,173)
( 25,172)( 26,171)( 27,183)( 28,195)( 29,194)( 30,193)( 31,192)( 32,191)
( 33,190)( 34,189)( 35,188)( 36,187)( 37,186)( 38,185)( 39,184)( 40,196)
( 41,208)( 42,207)( 43,206)( 44,205)( 45,204)( 46,203)( 47,202)( 48,201)
( 49,200)( 50,199)( 51,198)( 52,197)( 53,105)( 54,117)( 55,116)( 56,115)
( 57,114)( 58,113)( 59,112)( 60,111)( 61,110)( 62,109)( 63,108)( 64,107)
( 65,106)( 66,118)( 67,130)( 68,129)( 69,128)( 70,127)( 71,126)( 72,125)
( 73,124)( 74,123)( 75,122)( 76,121)( 77,120)( 78,119)( 79,131)( 80,143)
( 81,142)( 82,141)( 83,140)( 84,139)( 85,138)( 86,137)( 87,136)( 88,135)
( 89,134)( 90,133)( 91,132)( 92,144)( 93,156)( 94,155)( 95,154)( 96,153)
( 97,152)( 98,151)( 99,150)(100,149)(101,148)(102,147)(103,146)(104,145);;
s3 := (  1, 54)(  2, 53)(  3, 65)(  4, 64)(  5, 63)(  6, 62)(  7, 61)(  8, 60)
(  9, 59)( 10, 58)( 11, 57)( 12, 56)( 13, 55)( 14, 67)( 15, 66)( 16, 78)
( 17, 77)( 18, 76)( 19, 75)( 20, 74)( 21, 73)( 22, 72)( 23, 71)( 24, 70)
( 25, 69)( 26, 68)( 27, 80)( 28, 79)( 29, 91)( 30, 90)( 31, 89)( 32, 88)
( 33, 87)( 34, 86)( 35, 85)( 36, 84)( 37, 83)( 38, 82)( 39, 81)( 40, 93)
( 41, 92)( 42,104)( 43,103)( 44,102)( 45,101)( 46,100)( 47, 99)( 48, 98)
( 49, 97)( 50, 96)( 51, 95)( 52, 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);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(208)!( 27, 40)( 28, 41)( 29, 42)( 30, 43)( 31, 44)( 32, 45)( 33, 46)
( 34, 47)( 35, 48)( 36, 49)( 37, 50)( 38, 51)( 39, 52)( 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)(131,144)(132,145)(133,146)(134,147)(135,148)
(136,149)(137,150)(138,151)(139,152)(140,153)(141,154)(142,155)(143,156)
(183,196)(184,197)(185,198)(186,199)(187,200)(188,201)(189,202)(190,203)
(191,204)(192,205)(193,206)(194,207)(195,208);
s1 := 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);
s2 := Sym(208)!(  1,157)(  2,169)(  3,168)(  4,167)(  5,166)(  6,165)(  7,164)
(  8,163)(  9,162)( 10,161)( 11,160)( 12,159)( 13,158)( 14,170)( 15,182)
( 16,181)( 17,180)( 18,179)( 19,178)( 20,177)( 21,176)( 22,175)( 23,174)
( 24,173)( 25,172)( 26,171)( 27,183)( 28,195)( 29,194)( 30,193)( 31,192)
( 32,191)( 33,190)( 34,189)( 35,188)( 36,187)( 37,186)( 38,185)( 39,184)
( 40,196)( 41,208)( 42,207)( 43,206)( 44,205)( 45,204)( 46,203)( 47,202)
( 48,201)( 49,200)( 50,199)( 51,198)( 52,197)( 53,105)( 54,117)( 55,116)
( 56,115)( 57,114)( 58,113)( 59,112)( 60,111)( 61,110)( 62,109)( 63,108)
( 64,107)( 65,106)( 66,118)( 67,130)( 68,129)( 69,128)( 70,127)( 71,126)
( 72,125)( 73,124)( 74,123)( 75,122)( 76,121)( 77,120)( 78,119)( 79,131)
( 80,143)( 81,142)( 82,141)( 83,140)( 84,139)( 85,138)( 86,137)( 87,136)
( 88,135)( 89,134)( 90,133)( 91,132)( 92,144)( 93,156)( 94,155)( 95,154)
( 96,153)( 97,152)( 98,151)( 99,150)(100,149)(101,148)(102,147)(103,146)
(104,145);
s3 := Sym(208)!(  1, 54)(  2, 53)(  3, 65)(  4, 64)(  5, 63)(  6, 62)(  7, 61)
(  8, 60)(  9, 59)( 10, 58)( 11, 57)( 12, 56)( 13, 55)( 14, 67)( 15, 66)
( 16, 78)( 17, 77)( 18, 76)( 19, 75)( 20, 74)( 21, 73)( 22, 72)( 23, 71)
( 24, 70)( 25, 69)( 26, 68)( 27, 80)( 28, 79)( 29, 91)( 30, 90)( 31, 89)
( 32, 88)( 33, 87)( 34, 86)( 35, 85)( 36, 84)( 37, 83)( 38, 82)( 39, 81)
( 40, 93)( 41, 92)( 42,104)( 43,103)( 44,102)( 45,101)( 46,100)( 47, 99)
( 48, 98)( 49, 97)( 50, 96)( 51, 95)( 52, 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);
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
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