Polytope of Type {2,78,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,78,4}*1248b
if this polytope has a name.
Group : SmallGroup(1248,1441)
Rank : 4
Schlafli Type : {2,78,4}
Number of vertices, edges, etc : 2, 78, 156, 4
Order of s₀s₁s₂s₃ : 78
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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) :
   2-fold quotients : {2,39,4}*624
   13-fold quotients : {2,6,4}*96c
   26-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  7, 51)(  8, 53)(  9, 52)( 10, 54)( 11, 47)( 12, 49)( 13, 48)
( 14, 50)( 15, 43)( 16, 45)( 17, 44)( 18, 46)( 19, 39)( 20, 41)( 21, 40)
( 22, 42)( 23, 35)( 24, 37)( 25, 36)( 26, 38)( 27, 31)( 28, 33)( 29, 32)
( 30, 34)( 55,107)( 56,109)( 57,108)( 58,110)( 59,155)( 60,157)( 61,156)
( 62,158)( 63,151)( 64,153)( 65,152)( 66,154)( 67,147)( 68,149)( 69,148)
( 70,150)( 71,143)( 72,145)( 73,144)( 74,146)( 75,139)( 76,141)( 77,140)
( 78,142)( 79,135)( 80,137)( 81,136)( 82,138)( 83,131)( 84,133)( 85,132)
( 86,134)( 87,127)( 88,129)( 89,128)( 90,130)( 91,123)( 92,125)( 93,124)
( 94,126)( 95,119)( 96,121)( 97,120)( 98,122)( 99,115)(100,117)(101,116)
(102,118)(103,111)(104,113)(105,112)(106,114)(160,161)(163,207)(164,209)
(165,208)(166,210)(167,203)(168,205)(169,204)(170,206)(171,199)(172,201)
(173,200)(174,202)(175,195)(176,197)(177,196)(178,198)(179,191)(180,193)
(181,192)(182,194)(183,187)(184,189)(185,188)(186,190)(211,263)(212,265)
(213,264)(214,266)(215,311)(216,313)(217,312)(218,314)(219,307)(220,309)
(221,308)(222,310)(223,303)(224,305)(225,304)(226,306)(227,299)(228,301)
(229,300)(230,302)(231,295)(232,297)(233,296)(234,298)(235,291)(236,293)
(237,292)(238,294)(239,287)(240,289)(241,288)(242,290)(243,283)(244,285)
(245,284)(246,286)(247,279)(248,281)(249,280)(250,282)(251,275)(252,277)
(253,276)(254,278)(255,271)(256,273)(257,272)(258,274)(259,267)(260,269)
(261,268)(262,270);;
s2 := (  3,215)(  4,216)(  5,218)(  6,217)(  7,211)(  8,212)(  9,214)( 10,213)
( 11,259)( 12,260)( 13,262)( 14,261)( 15,255)( 16,256)( 17,258)( 18,257)
( 19,251)( 20,252)( 21,254)( 22,253)( 23,247)( 24,248)( 25,250)( 26,249)
( 27,243)( 28,244)( 29,246)( 30,245)( 31,239)( 32,240)( 33,242)( 34,241)
( 35,235)( 36,236)( 37,238)( 38,237)( 39,231)( 40,232)( 41,234)( 42,233)
( 43,227)( 44,228)( 45,230)( 46,229)( 47,223)( 48,224)( 49,226)( 50,225)
( 51,219)( 52,220)( 53,222)( 54,221)( 55,163)( 56,164)( 57,166)( 58,165)
( 59,159)( 60,160)( 61,162)( 62,161)( 63,207)( 64,208)( 65,210)( 66,209)
( 67,203)( 68,204)( 69,206)( 70,205)( 71,199)( 72,200)( 73,202)( 74,201)
( 75,195)( 76,196)( 77,198)( 78,197)( 79,191)( 80,192)( 81,194)( 82,193)
( 83,187)( 84,188)( 85,190)( 86,189)( 87,183)( 88,184)( 89,186)( 90,185)
( 91,179)( 92,180)( 93,182)( 94,181)( 95,175)( 96,176)( 97,178)( 98,177)
( 99,171)(100,172)(101,174)(102,173)(103,167)(104,168)(105,170)(106,169)
(107,267)(108,268)(109,270)(110,269)(111,263)(112,264)(113,266)(114,265)
(115,311)(116,312)(117,314)(118,313)(119,307)(120,308)(121,310)(122,309)
(123,303)(124,304)(125,306)(126,305)(127,299)(128,300)(129,302)(130,301)
(131,295)(132,296)(133,298)(134,297)(135,291)(136,292)(137,294)(138,293)
(139,287)(140,288)(141,290)(142,289)(143,283)(144,284)(145,286)(146,285)
(147,279)(148,280)(149,282)(150,281)(151,275)(152,276)(153,278)(154,277)
(155,271)(156,272)(157,274)(158,273);;
s3 := (  3,  6)(  4,  5)(  7, 10)(  8,  9)( 11, 14)( 12, 13)( 15, 18)( 16, 17)
( 19, 22)( 20, 21)( 23, 26)( 24, 25)( 27, 30)( 28, 29)( 31, 34)( 32, 33)
( 35, 38)( 36, 37)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)( 48, 49)
( 51, 54)( 52, 53)( 55, 58)( 56, 57)( 59, 62)( 60, 61)( 63, 66)( 64, 65)
( 67, 70)( 68, 69)( 71, 74)( 72, 73)( 75, 78)( 76, 77)( 79, 82)( 80, 81)
( 83, 86)( 84, 85)( 87, 90)( 88, 89)( 91, 94)( 92, 93)( 95, 98)( 96, 97)
( 99,102)(100,101)(103,106)(104,105)(107,110)(108,109)(111,114)(112,113)
(115,118)(116,117)(119,122)(120,121)(123,126)(124,125)(127,130)(128,129)
(131,134)(132,133)(135,138)(136,137)(139,142)(140,141)(143,146)(144,145)
(147,150)(148,149)(151,154)(152,153)(155,158)(156,157)(159,162)(160,161)
(163,166)(164,165)(167,170)(168,169)(171,174)(172,173)(175,178)(176,177)
(179,182)(180,181)(183,186)(184,185)(187,190)(188,189)(191,194)(192,193)
(195,198)(196,197)(199,202)(200,201)(203,206)(204,205)(207,210)(208,209)
(211,214)(212,213)(215,218)(216,217)(219,222)(220,221)(223,226)(224,225)
(227,230)(228,229)(231,234)(232,233)(235,238)(236,237)(239,242)(240,241)
(243,246)(244,245)(247,250)(248,249)(251,254)(252,253)(255,258)(256,257)
(259,262)(260,261)(263,266)(264,265)(267,270)(268,269)(271,274)(272,273)
(275,278)(276,277)(279,282)(280,281)(283,286)(284,285)(287,290)(288,289)
(291,294)(292,293)(295,298)(296,297)(299,302)(300,301)(303,306)(304,305)
(307,310)(308,309)(311,314)(312,313);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*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*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(314)!(1,2);
s1 := Sym(314)!(  4,  5)(  7, 51)(  8, 53)(  9, 52)( 10, 54)( 11, 47)( 12, 49)
( 13, 48)( 14, 50)( 15, 43)( 16, 45)( 17, 44)( 18, 46)( 19, 39)( 20, 41)
( 21, 40)( 22, 42)( 23, 35)( 24, 37)( 25, 36)( 26, 38)( 27, 31)( 28, 33)
( 29, 32)( 30, 34)( 55,107)( 56,109)( 57,108)( 58,110)( 59,155)( 60,157)
( 61,156)( 62,158)( 63,151)( 64,153)( 65,152)( 66,154)( 67,147)( 68,149)
( 69,148)( 70,150)( 71,143)( 72,145)( 73,144)( 74,146)( 75,139)( 76,141)
( 77,140)( 78,142)( 79,135)( 80,137)( 81,136)( 82,138)( 83,131)( 84,133)
( 85,132)( 86,134)( 87,127)( 88,129)( 89,128)( 90,130)( 91,123)( 92,125)
( 93,124)( 94,126)( 95,119)( 96,121)( 97,120)( 98,122)( 99,115)(100,117)
(101,116)(102,118)(103,111)(104,113)(105,112)(106,114)(160,161)(163,207)
(164,209)(165,208)(166,210)(167,203)(168,205)(169,204)(170,206)(171,199)
(172,201)(173,200)(174,202)(175,195)(176,197)(177,196)(178,198)(179,191)
(180,193)(181,192)(182,194)(183,187)(184,189)(185,188)(186,190)(211,263)
(212,265)(213,264)(214,266)(215,311)(216,313)(217,312)(218,314)(219,307)
(220,309)(221,308)(222,310)(223,303)(224,305)(225,304)(226,306)(227,299)
(228,301)(229,300)(230,302)(231,295)(232,297)(233,296)(234,298)(235,291)
(236,293)(237,292)(238,294)(239,287)(240,289)(241,288)(242,290)(243,283)
(244,285)(245,284)(246,286)(247,279)(248,281)(249,280)(250,282)(251,275)
(252,277)(253,276)(254,278)(255,271)(256,273)(257,272)(258,274)(259,267)
(260,269)(261,268)(262,270);
s2 := Sym(314)!(  3,215)(  4,216)(  5,218)(  6,217)(  7,211)(  8,212)(  9,214)
( 10,213)( 11,259)( 12,260)( 13,262)( 14,261)( 15,255)( 16,256)( 17,258)
( 18,257)( 19,251)( 20,252)( 21,254)( 22,253)( 23,247)( 24,248)( 25,250)
( 26,249)( 27,243)( 28,244)( 29,246)( 30,245)( 31,239)( 32,240)( 33,242)
( 34,241)( 35,235)( 36,236)( 37,238)( 38,237)( 39,231)( 40,232)( 41,234)
( 42,233)( 43,227)( 44,228)( 45,230)( 46,229)( 47,223)( 48,224)( 49,226)
( 50,225)( 51,219)( 52,220)( 53,222)( 54,221)( 55,163)( 56,164)( 57,166)
( 58,165)( 59,159)( 60,160)( 61,162)( 62,161)( 63,207)( 64,208)( 65,210)
( 66,209)( 67,203)( 68,204)( 69,206)( 70,205)( 71,199)( 72,200)( 73,202)
( 74,201)( 75,195)( 76,196)( 77,198)( 78,197)( 79,191)( 80,192)( 81,194)
( 82,193)( 83,187)( 84,188)( 85,190)( 86,189)( 87,183)( 88,184)( 89,186)
( 90,185)( 91,179)( 92,180)( 93,182)( 94,181)( 95,175)( 96,176)( 97,178)
( 98,177)( 99,171)(100,172)(101,174)(102,173)(103,167)(104,168)(105,170)
(106,169)(107,267)(108,268)(109,270)(110,269)(111,263)(112,264)(113,266)
(114,265)(115,311)(116,312)(117,314)(118,313)(119,307)(120,308)(121,310)
(122,309)(123,303)(124,304)(125,306)(126,305)(127,299)(128,300)(129,302)
(130,301)(131,295)(132,296)(133,298)(134,297)(135,291)(136,292)(137,294)
(138,293)(139,287)(140,288)(141,290)(142,289)(143,283)(144,284)(145,286)
(146,285)(147,279)(148,280)(149,282)(150,281)(151,275)(152,276)(153,278)
(154,277)(155,271)(156,272)(157,274)(158,273);
s3 := Sym(314)!(  3,  6)(  4,  5)(  7, 10)(  8,  9)( 11, 14)( 12, 13)( 15, 18)
( 16, 17)( 19, 22)( 20, 21)( 23, 26)( 24, 25)( 27, 30)( 28, 29)( 31, 34)
( 32, 33)( 35, 38)( 36, 37)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)
( 48, 49)( 51, 54)( 52, 53)( 55, 58)( 56, 57)( 59, 62)( 60, 61)( 63, 66)
( 64, 65)( 67, 70)( 68, 69)( 71, 74)( 72, 73)( 75, 78)( 76, 77)( 79, 82)
( 80, 81)( 83, 86)( 84, 85)( 87, 90)( 88, 89)( 91, 94)( 92, 93)( 95, 98)
( 96, 97)( 99,102)(100,101)(103,106)(104,105)(107,110)(108,109)(111,114)
(112,113)(115,118)(116,117)(119,122)(120,121)(123,126)(124,125)(127,130)
(128,129)(131,134)(132,133)(135,138)(136,137)(139,142)(140,141)(143,146)
(144,145)(147,150)(148,149)(151,154)(152,153)(155,158)(156,157)(159,162)
(160,161)(163,166)(164,165)(167,170)(168,169)(171,174)(172,173)(175,178)
(176,177)(179,182)(180,181)(183,186)(184,185)(187,190)(188,189)(191,194)
(192,193)(195,198)(196,197)(199,202)(200,201)(203,206)(204,205)(207,210)
(208,209)(211,214)(212,213)(215,218)(216,217)(219,222)(220,221)(223,226)
(224,225)(227,230)(228,229)(231,234)(232,233)(235,238)(236,237)(239,242)
(240,241)(243,246)(244,245)(247,250)(248,249)(251,254)(252,253)(255,258)
(256,257)(259,262)(260,261)(263,266)(264,265)(267,270)(268,269)(271,274)
(272,273)(275,278)(276,277)(279,282)(280,281)(283,286)(284,285)(287,290)
(288,289)(291,294)(292,293)(295,298)(296,297)(299,302)(300,301)(303,306)
(304,305)(307,310)(308,309)(311,314)(312,313);
poly := sub<Sym(314)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s2*s1*s3*s2*s3*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*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 >;

to this polytope