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Polytope of Type {3,4,26}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4,26}*1248
Also Known As : {{3,4},{4,26|2}}. if this polytope has another name.
Group : SmallGroup(1248,1438)
Rank : 4
Schlafli Type : {3,4,26}
Number of vertices, edges, etc : 6, 12, 104, 26
Order of s0s1s2s3 : 78
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) :
   4-fold quotients : {3,2,26}*312
   8-fold quotients : {3,2,13}*156
   13-fold quotients : {3,4,2}*96
   26-fold quotients : {3,4,2}*48
   52-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)
( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 53,105)( 54,106)( 55,108)
( 56,107)( 57,109)( 58,110)( 59,112)( 60,111)( 61,113)( 62,114)( 63,116)
( 64,115)( 65,117)( 66,118)( 67,120)( 68,119)( 69,121)( 70,122)( 71,124)
( 72,123)( 73,125)( 74,126)( 75,128)( 76,127)( 77,129)( 78,130)( 79,132)
( 80,131)( 81,133)( 82,134)( 83,136)( 84,135)( 85,137)( 86,138)( 87,140)
( 88,139)( 89,141)( 90,142)( 91,144)( 92,143)( 93,145)( 94,146)( 95,148)
( 96,147)( 97,149)( 98,150)( 99,152)(100,151)(101,153)(102,154)(103,156)
(104,155)(159,160)(163,164)(167,168)(171,172)(175,176)(179,180)(183,184)
(187,188)(191,192)(195,196)(199,200)(203,204)(207,208)(209,261)(210,262)
(211,264)(212,263)(213,265)(214,266)(215,268)(216,267)(217,269)(218,270)
(219,272)(220,271)(221,273)(222,274)(223,276)(224,275)(225,277)(226,278)
(227,280)(228,279)(229,281)(230,282)(231,284)(232,283)(233,285)(234,286)
(235,288)(236,287)(237,289)(238,290)(239,292)(240,291)(241,293)(242,294)
(243,296)(244,295)(245,297)(246,298)(247,300)(248,299)(249,301)(250,302)
(251,304)(252,303)(253,305)(254,306)(255,308)(256,307)(257,309)(258,310)
(259,312)(260,311);;
s1 := (  1, 53)(  2, 56)(  3, 55)(  4, 54)(  5, 57)(  6, 60)(  7, 59)(  8, 58)
(  9, 61)( 10, 64)( 11, 63)( 12, 62)( 13, 65)( 14, 68)( 15, 67)( 16, 66)
( 17, 69)( 18, 72)( 19, 71)( 20, 70)( 21, 73)( 22, 76)( 23, 75)( 24, 74)
( 25, 77)( 26, 80)( 27, 79)( 28, 78)( 29, 81)( 30, 84)( 31, 83)( 32, 82)
( 33, 85)( 34, 88)( 35, 87)( 36, 86)( 37, 89)( 38, 92)( 39, 91)( 40, 90)
( 41, 93)( 42, 96)( 43, 95)( 44, 94)( 45, 97)( 46,100)( 47, 99)( 48, 98)
( 49,101)( 50,104)( 51,103)( 52,102)(106,108)(110,112)(114,116)(118,120)
(122,124)(126,128)(130,132)(134,136)(138,140)(142,144)(146,148)(150,152)
(154,156)(157,209)(158,212)(159,211)(160,210)(161,213)(162,216)(163,215)
(164,214)(165,217)(166,220)(167,219)(168,218)(169,221)(170,224)(171,223)
(172,222)(173,225)(174,228)(175,227)(176,226)(177,229)(178,232)(179,231)
(180,230)(181,233)(182,236)(183,235)(184,234)(185,237)(186,240)(187,239)
(188,238)(189,241)(190,244)(191,243)(192,242)(193,245)(194,248)(195,247)
(196,246)(197,249)(198,252)(199,251)(200,250)(201,253)(202,256)(203,255)
(204,254)(205,257)(206,260)(207,259)(208,258)(262,264)(266,268)(270,272)
(274,276)(278,280)(282,284)(286,288)(290,292)(294,296)(298,300)(302,304)
(306,308)(310,312);;
s2 := (  1,  2)(  3,  4)(  5, 50)(  6, 49)(  7, 52)(  8, 51)(  9, 46)( 10, 45)
( 11, 48)( 12, 47)( 13, 42)( 14, 41)( 15, 44)( 16, 43)( 17, 38)( 18, 37)
( 19, 40)( 20, 39)( 21, 34)( 22, 33)( 23, 36)( 24, 35)( 25, 30)( 26, 29)
( 27, 32)( 28, 31)( 53, 54)( 55, 56)( 57,102)( 58,101)( 59,104)( 60,103)
( 61, 98)( 62, 97)( 63,100)( 64, 99)( 65, 94)( 66, 93)( 67, 96)( 68, 95)
( 69, 90)( 70, 89)( 71, 92)( 72, 91)( 73, 86)( 74, 85)( 75, 88)( 76, 87)
( 77, 82)( 78, 81)( 79, 84)( 80, 83)(105,106)(107,108)(109,154)(110,153)
(111,156)(112,155)(113,150)(114,149)(115,152)(116,151)(117,146)(118,145)
(119,148)(120,147)(121,142)(122,141)(123,144)(124,143)(125,138)(126,137)
(127,140)(128,139)(129,134)(130,133)(131,136)(132,135)(157,158)(159,160)
(161,206)(162,205)(163,208)(164,207)(165,202)(166,201)(167,204)(168,203)
(169,198)(170,197)(171,200)(172,199)(173,194)(174,193)(175,196)(176,195)
(177,190)(178,189)(179,192)(180,191)(181,186)(182,185)(183,188)(184,187)
(209,210)(211,212)(213,258)(214,257)(215,260)(216,259)(217,254)(218,253)
(219,256)(220,255)(221,250)(222,249)(223,252)(224,251)(225,246)(226,245)
(227,248)(228,247)(229,242)(230,241)(231,244)(232,243)(233,238)(234,237)
(235,240)(236,239)(261,262)(263,264)(265,310)(266,309)(267,312)(268,311)
(269,306)(270,305)(271,308)(272,307)(273,302)(274,301)(275,304)(276,303)
(277,298)(278,297)(279,300)(280,299)(281,294)(282,293)(283,296)(284,295)
(285,290)(286,289)(287,292)(288,291);;
s3 := (  1,161)(  2,162)(  3,163)(  4,164)(  5,157)(  6,158)(  7,159)(  8,160)
(  9,205)( 10,206)( 11,207)( 12,208)( 13,201)( 14,202)( 15,203)( 16,204)
( 17,197)( 18,198)( 19,199)( 20,200)( 21,193)( 22,194)( 23,195)( 24,196)
( 25,189)( 26,190)( 27,191)( 28,192)( 29,185)( 30,186)( 31,187)( 32,188)
( 33,181)( 34,182)( 35,183)( 36,184)( 37,177)( 38,178)( 39,179)( 40,180)
( 41,173)( 42,174)( 43,175)( 44,176)( 45,169)( 46,170)( 47,171)( 48,172)
( 49,165)( 50,166)( 51,167)( 52,168)( 53,213)( 54,214)( 55,215)( 56,216)
( 57,209)( 58,210)( 59,211)( 60,212)( 61,257)( 62,258)( 63,259)( 64,260)
( 65,253)( 66,254)( 67,255)( 68,256)( 69,249)( 70,250)( 71,251)( 72,252)
( 73,245)( 74,246)( 75,247)( 76,248)( 77,241)( 78,242)( 79,243)( 80,244)
( 81,237)( 82,238)( 83,239)( 84,240)( 85,233)( 86,234)( 87,235)( 88,236)
( 89,229)( 90,230)( 91,231)( 92,232)( 93,225)( 94,226)( 95,227)( 96,228)
( 97,221)( 98,222)( 99,223)(100,224)(101,217)(102,218)(103,219)(104,220)
(105,265)(106,266)(107,267)(108,268)(109,261)(110,262)(111,263)(112,264)
(113,309)(114,310)(115,311)(116,312)(117,305)(118,306)(119,307)(120,308)
(121,301)(122,302)(123,303)(124,304)(125,297)(126,298)(127,299)(128,300)
(129,293)(130,294)(131,295)(132,296)(133,289)(134,290)(135,291)(136,292)
(137,285)(138,286)(139,287)(140,288)(141,281)(142,282)(143,283)(144,284)
(145,277)(146,278)(147,279)(148,280)(149,273)(150,274)(151,275)(152,276)
(153,269)(154,270)(155,271)(156,272);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
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(312)!(  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)
( 31, 32)( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 53,105)( 54,106)
( 55,108)( 56,107)( 57,109)( 58,110)( 59,112)( 60,111)( 61,113)( 62,114)
( 63,116)( 64,115)( 65,117)( 66,118)( 67,120)( 68,119)( 69,121)( 70,122)
( 71,124)( 72,123)( 73,125)( 74,126)( 75,128)( 76,127)( 77,129)( 78,130)
( 79,132)( 80,131)( 81,133)( 82,134)( 83,136)( 84,135)( 85,137)( 86,138)
( 87,140)( 88,139)( 89,141)( 90,142)( 91,144)( 92,143)( 93,145)( 94,146)
( 95,148)( 96,147)( 97,149)( 98,150)( 99,152)(100,151)(101,153)(102,154)
(103,156)(104,155)(159,160)(163,164)(167,168)(171,172)(175,176)(179,180)
(183,184)(187,188)(191,192)(195,196)(199,200)(203,204)(207,208)(209,261)
(210,262)(211,264)(212,263)(213,265)(214,266)(215,268)(216,267)(217,269)
(218,270)(219,272)(220,271)(221,273)(222,274)(223,276)(224,275)(225,277)
(226,278)(227,280)(228,279)(229,281)(230,282)(231,284)(232,283)(233,285)
(234,286)(235,288)(236,287)(237,289)(238,290)(239,292)(240,291)(241,293)
(242,294)(243,296)(244,295)(245,297)(246,298)(247,300)(248,299)(249,301)
(250,302)(251,304)(252,303)(253,305)(254,306)(255,308)(256,307)(257,309)
(258,310)(259,312)(260,311);
s1 := Sym(312)!(  1, 53)(  2, 56)(  3, 55)(  4, 54)(  5, 57)(  6, 60)(  7, 59)
(  8, 58)(  9, 61)( 10, 64)( 11, 63)( 12, 62)( 13, 65)( 14, 68)( 15, 67)
( 16, 66)( 17, 69)( 18, 72)( 19, 71)( 20, 70)( 21, 73)( 22, 76)( 23, 75)
( 24, 74)( 25, 77)( 26, 80)( 27, 79)( 28, 78)( 29, 81)( 30, 84)( 31, 83)
( 32, 82)( 33, 85)( 34, 88)( 35, 87)( 36, 86)( 37, 89)( 38, 92)( 39, 91)
( 40, 90)( 41, 93)( 42, 96)( 43, 95)( 44, 94)( 45, 97)( 46,100)( 47, 99)
( 48, 98)( 49,101)( 50,104)( 51,103)( 52,102)(106,108)(110,112)(114,116)
(118,120)(122,124)(126,128)(130,132)(134,136)(138,140)(142,144)(146,148)
(150,152)(154,156)(157,209)(158,212)(159,211)(160,210)(161,213)(162,216)
(163,215)(164,214)(165,217)(166,220)(167,219)(168,218)(169,221)(170,224)
(171,223)(172,222)(173,225)(174,228)(175,227)(176,226)(177,229)(178,232)
(179,231)(180,230)(181,233)(182,236)(183,235)(184,234)(185,237)(186,240)
(187,239)(188,238)(189,241)(190,244)(191,243)(192,242)(193,245)(194,248)
(195,247)(196,246)(197,249)(198,252)(199,251)(200,250)(201,253)(202,256)
(203,255)(204,254)(205,257)(206,260)(207,259)(208,258)(262,264)(266,268)
(270,272)(274,276)(278,280)(282,284)(286,288)(290,292)(294,296)(298,300)
(302,304)(306,308)(310,312);
s2 := Sym(312)!(  1,  2)(  3,  4)(  5, 50)(  6, 49)(  7, 52)(  8, 51)(  9, 46)
( 10, 45)( 11, 48)( 12, 47)( 13, 42)( 14, 41)( 15, 44)( 16, 43)( 17, 38)
( 18, 37)( 19, 40)( 20, 39)( 21, 34)( 22, 33)( 23, 36)( 24, 35)( 25, 30)
( 26, 29)( 27, 32)( 28, 31)( 53, 54)( 55, 56)( 57,102)( 58,101)( 59,104)
( 60,103)( 61, 98)( 62, 97)( 63,100)( 64, 99)( 65, 94)( 66, 93)( 67, 96)
( 68, 95)( 69, 90)( 70, 89)( 71, 92)( 72, 91)( 73, 86)( 74, 85)( 75, 88)
( 76, 87)( 77, 82)( 78, 81)( 79, 84)( 80, 83)(105,106)(107,108)(109,154)
(110,153)(111,156)(112,155)(113,150)(114,149)(115,152)(116,151)(117,146)
(118,145)(119,148)(120,147)(121,142)(122,141)(123,144)(124,143)(125,138)
(126,137)(127,140)(128,139)(129,134)(130,133)(131,136)(132,135)(157,158)
(159,160)(161,206)(162,205)(163,208)(164,207)(165,202)(166,201)(167,204)
(168,203)(169,198)(170,197)(171,200)(172,199)(173,194)(174,193)(175,196)
(176,195)(177,190)(178,189)(179,192)(180,191)(181,186)(182,185)(183,188)
(184,187)(209,210)(211,212)(213,258)(214,257)(215,260)(216,259)(217,254)
(218,253)(219,256)(220,255)(221,250)(222,249)(223,252)(224,251)(225,246)
(226,245)(227,248)(228,247)(229,242)(230,241)(231,244)(232,243)(233,238)
(234,237)(235,240)(236,239)(261,262)(263,264)(265,310)(266,309)(267,312)
(268,311)(269,306)(270,305)(271,308)(272,307)(273,302)(274,301)(275,304)
(276,303)(277,298)(278,297)(279,300)(280,299)(281,294)(282,293)(283,296)
(284,295)(285,290)(286,289)(287,292)(288,291);
s3 := Sym(312)!(  1,161)(  2,162)(  3,163)(  4,164)(  5,157)(  6,158)(  7,159)
(  8,160)(  9,205)( 10,206)( 11,207)( 12,208)( 13,201)( 14,202)( 15,203)
( 16,204)( 17,197)( 18,198)( 19,199)( 20,200)( 21,193)( 22,194)( 23,195)
( 24,196)( 25,189)( 26,190)( 27,191)( 28,192)( 29,185)( 30,186)( 31,187)
( 32,188)( 33,181)( 34,182)( 35,183)( 36,184)( 37,177)( 38,178)( 39,179)
( 40,180)( 41,173)( 42,174)( 43,175)( 44,176)( 45,169)( 46,170)( 47,171)
( 48,172)( 49,165)( 50,166)( 51,167)( 52,168)( 53,213)( 54,214)( 55,215)
( 56,216)( 57,209)( 58,210)( 59,211)( 60,212)( 61,257)( 62,258)( 63,259)
( 64,260)( 65,253)( 66,254)( 67,255)( 68,256)( 69,249)( 70,250)( 71,251)
( 72,252)( 73,245)( 74,246)( 75,247)( 76,248)( 77,241)( 78,242)( 79,243)
( 80,244)( 81,237)( 82,238)( 83,239)( 84,240)( 85,233)( 86,234)( 87,235)
( 88,236)( 89,229)( 90,230)( 91,231)( 92,232)( 93,225)( 94,226)( 95,227)
( 96,228)( 97,221)( 98,222)( 99,223)(100,224)(101,217)(102,218)(103,219)
(104,220)(105,265)(106,266)(107,267)(108,268)(109,261)(110,262)(111,263)
(112,264)(113,309)(114,310)(115,311)(116,312)(117,305)(118,306)(119,307)
(120,308)(121,301)(122,302)(123,303)(124,304)(125,297)(126,298)(127,299)
(128,300)(129,293)(130,294)(131,295)(132,296)(133,289)(134,290)(135,291)
(136,292)(137,285)(138,286)(139,287)(140,288)(141,281)(142,282)(143,283)
(144,284)(145,277)(146,278)(147,279)(148,280)(149,273)(150,274)(151,275)
(152,276)(153,269)(154,270)(155,271)(156,272);
poly := sub<Sym(312)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, 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.
to this polytope