Polytope of Type {4,6,26}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,26}*1248b
if this polytope has a name.
Group : SmallGroup(1248,1438)
Rank : 4
Schlafli Type : {4,6,26}
Number of vertices, edges, etc : 4, 12, 78, 26
Order of s₀s₁s₂s₃ : 78
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-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) :
   13-fold quotients : {4,6,2}*96c
   26-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312);;
s1 := (  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);;
s2 := (  1, 53)(  2, 56)(  3, 55)(  4, 54)(  5,101)(  6,104)(  7,103)(  8,102)
(  9, 97)( 10,100)( 11, 99)( 12, 98)( 13, 93)( 14, 96)( 15, 95)( 16, 94)
( 17, 89)( 18, 92)( 19, 91)( 20, 90)( 21, 85)( 22, 88)( 23, 87)( 24, 86)
( 25, 81)( 26, 84)( 27, 83)( 28, 82)( 29, 77)( 30, 80)( 31, 79)( 32, 78)
( 33, 73)( 34, 76)( 35, 75)( 36, 74)( 37, 69)( 38, 72)( 39, 71)( 40, 70)
( 41, 65)( 42, 68)( 43, 67)( 44, 66)( 45, 61)( 46, 64)( 47, 63)( 48, 62)
( 49, 57)( 50, 60)( 51, 59)( 52, 58)(106,108)(109,153)(110,156)(111,155)
(112,154)(113,149)(114,152)(115,151)(116,150)(117,145)(118,148)(119,147)
(120,146)(121,141)(122,144)(123,143)(124,142)(125,137)(126,140)(127,139)
(128,138)(129,133)(130,136)(131,135)(132,134)(157,209)(158,212)(159,211)
(160,210)(161,257)(162,260)(163,259)(164,258)(165,253)(166,256)(167,255)
(168,254)(169,249)(170,252)(171,251)(172,250)(173,245)(174,248)(175,247)
(176,246)(177,241)(178,244)(179,243)(180,242)(181,237)(182,240)(183,239)
(184,238)(185,233)(186,236)(187,235)(188,234)(189,229)(190,232)(191,231)
(192,230)(193,225)(194,228)(195,227)(196,226)(197,221)(198,224)(199,223)
(200,222)(201,217)(202,220)(203,219)(204,218)(205,213)(206,216)(207,215)
(208,214)(262,264)(265,309)(266,312)(267,311)(268,310)(269,305)(270,308)
(271,307)(272,306)(273,301)(274,304)(275,303)(276,302)(277,297)(278,300)
(279,299)(280,298)(281,293)(282,296)(283,295)(284,294)(285,289)(286,292)
(287,291)(288,290);;
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*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312);
s1 := 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);
s2 := Sym(312)!(  1, 53)(  2, 56)(  3, 55)(  4, 54)(  5,101)(  6,104)(  7,103)
(  8,102)(  9, 97)( 10,100)( 11, 99)( 12, 98)( 13, 93)( 14, 96)( 15, 95)
( 16, 94)( 17, 89)( 18, 92)( 19, 91)( 20, 90)( 21, 85)( 22, 88)( 23, 87)
( 24, 86)( 25, 81)( 26, 84)( 27, 83)( 28, 82)( 29, 77)( 30, 80)( 31, 79)
( 32, 78)( 33, 73)( 34, 76)( 35, 75)( 36, 74)( 37, 69)( 38, 72)( 39, 71)
( 40, 70)( 41, 65)( 42, 68)( 43, 67)( 44, 66)( 45, 61)( 46, 64)( 47, 63)
( 48, 62)( 49, 57)( 50, 60)( 51, 59)( 52, 58)(106,108)(109,153)(110,156)
(111,155)(112,154)(113,149)(114,152)(115,151)(116,150)(117,145)(118,148)
(119,147)(120,146)(121,141)(122,144)(123,143)(124,142)(125,137)(126,140)
(127,139)(128,138)(129,133)(130,136)(131,135)(132,134)(157,209)(158,212)
(159,211)(160,210)(161,257)(162,260)(163,259)(164,258)(165,253)(166,256)
(167,255)(168,254)(169,249)(170,252)(171,251)(172,250)(173,245)(174,248)
(175,247)(176,246)(177,241)(178,244)(179,243)(180,242)(181,237)(182,240)
(183,239)(184,238)(185,233)(186,236)(187,235)(188,234)(189,229)(190,232)
(191,231)(192,230)(193,225)(194,228)(195,227)(196,226)(197,221)(198,224)
(199,223)(200,222)(201,217)(202,220)(203,219)(204,218)(205,213)(206,216)
(207,215)(208,214)(262,264)(265,309)(266,312)(267,311)(268,310)(269,305)
(270,308)(271,307)(272,306)(273,301)(274,304)(275,303)(276,302)(277,297)
(278,300)(279,299)(280,298)(281,293)(282,296)(283,295)(284,294)(285,289)
(286,292)(287,291)(288,290);
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*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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