Polytope of Type {4,42,4} with group SmallGroup(1344,11399)

Overview

Group: SmallGroup(1344,11399)
Rank: 4
Schläfli Type: {4,42,4}
Vertices, edges, …: 4, 84, 84, 4
Order of s₀s₁s₂s₃: 84
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Non-Orientable
Flat

Quotients maximal quotients in bold

2-fold

{2,42,4}*672b

4-fold

{2,21,4}*336

7-fold

{4,6,4}*192b

14-fold

{2,6,4}*96c

28-fold

{2,3,4}*48

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

s0 := (169,253)(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)(176,260)(177,261)(178,262)(179,263)(180,264)(181,265)(182,266)(183,267)(184,268)(185,269)(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)(193,277)(194,278)(195,279)(196,280)(197,281)(198,282)(199,283)(200,284)(201,285)(202,286)(203,287)(204,288)(205,289)(206,290)(207,291)(208,292)(209,293)(210,294)(211,295)(212,296)(213,297)(214,298)(215,299)(216,300)(217,301)(218,302)(219,303)(220,304)(221,305)(222,306)(223,307)(224,308)(225,309)(226,310)(227,311)(228,312)(229,313)(230,314)(231,315)(232,316)(233,317)(234,318)(235,319)(236,320)(237,321)(238,322)(239,323)(240,324)(241,325)(242,326)(243,327)(244,328)(245,329)(246,330)(247,331)(248,332)(249,333)(250,334)(251,335)(252,336);;
s1 := (  1,169)(  2,170)(  3,172)(  4,171)(  5,193)(  6,194)(  7,196)(  8,195)(  9,189)( 10,190)( 11,192)( 12,191)( 13,185)( 14,186)( 15,188)( 16,187)( 17,181)( 18,182)( 19,184)( 20,183)( 21,177)( 22,178)( 23,180)( 24,179)( 25,173)( 26,174)( 27,176)( 28,175)( 29,225)( 30,226)( 31,228)( 32,227)( 33,249)( 34,250)( 35,252)( 36,251)( 37,245)( 38,246)( 39,248)( 40,247)( 41,241)( 42,242)( 43,244)( 44,243)( 45,237)( 46,238)( 47,240)( 48,239)( 49,233)( 50,234)( 51,236)( 52,235)( 53,229)( 54,230)( 55,232)( 56,231)( 57,197)( 58,198)( 59,200)( 60,199)( 61,221)( 62,222)( 63,224)( 64,223)( 65,217)( 66,218)( 67,220)( 68,219)( 69,213)( 70,214)( 71,216)( 72,215)( 73,209)( 74,210)( 75,212)( 76,211)( 77,205)( 78,206)( 79,208)( 80,207)( 81,201)( 82,202)( 83,204)( 84,203)( 85,253)( 86,254)( 87,256)( 88,255)( 89,277)( 90,278)( 91,280)( 92,279)( 93,273)( 94,274)( 95,276)( 96,275)( 97,269)( 98,270)( 99,272)(100,271)(101,265)(102,266)(103,268)(104,267)(105,261)(106,262)(107,264)(108,263)(109,257)(110,258)(111,260)(112,259)(113,309)(114,310)(115,312)(116,311)(117,333)(118,334)(119,336)(120,335)(121,329)(122,330)(123,332)(124,331)(125,325)(126,326)(127,328)(128,327)(129,321)(130,322)(131,324)(132,323)(133,317)(134,318)(135,320)(136,319)(137,313)(138,314)(139,316)(140,315)(141,281)(142,282)(143,284)(144,283)(145,305)(146,306)(147,308)(148,307)(149,301)(150,302)(151,304)(152,303)(153,297)(154,298)(155,300)(156,299)(157,293)(158,294)(159,296)(160,295)(161,289)(162,290)(163,292)(164,291)(165,285)(166,286)(167,288)(168,287);;
s2 := (  1, 33)(  2, 36)(  3, 35)(  4, 34)(  5, 29)(  6, 32)(  7, 31)(  8, 30)(  9, 53)( 10, 56)( 11, 55)( 12, 54)( 13, 49)( 14, 52)( 15, 51)( 16, 50)( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 21, 41)( 22, 44)( 23, 43)( 24, 42)( 25, 37)( 26, 40)( 27, 39)( 28, 38)( 57, 61)( 58, 64)( 59, 63)( 60, 62)( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 77)( 70, 80)( 71, 79)( 72, 78)( 74, 76)( 85,117)( 86,120)( 87,119)( 88,118)( 89,113)( 90,116)( 91,115)( 92,114)( 93,137)( 94,140)( 95,139)( 96,138)( 97,133)( 98,136)( 99,135)(100,134)(101,129)(102,132)(103,131)(104,130)(105,125)(106,128)(107,127)(108,126)(109,121)(110,124)(111,123)(112,122)(141,145)(142,148)(143,147)(144,146)(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)(155,163)(156,162)(158,160)(169,201)(170,204)(171,203)(172,202)(173,197)(174,200)(175,199)(176,198)(177,221)(178,224)(179,223)(180,222)(181,217)(182,220)(183,219)(184,218)(185,213)(186,216)(187,215)(188,214)(189,209)(190,212)(191,211)(192,210)(193,205)(194,208)(195,207)(196,206)(225,229)(226,232)(227,231)(228,230)(233,249)(234,252)(235,251)(236,250)(237,245)(238,248)(239,247)(240,246)(242,244)(253,285)(254,288)(255,287)(256,286)(257,281)(258,284)(259,283)(260,282)(261,305)(262,308)(263,307)(264,306)(265,301)(266,304)(267,303)(268,302)(269,297)(270,300)(271,299)(272,298)(273,293)(274,296)(275,295)(276,294)(277,289)(278,292)(279,291)(280,290)(309,313)(310,316)(311,315)(312,314)(317,333)(318,336)(319,335)(320,334)(321,329)(322,332)(323,331)(324,330)(326,328);;
s3 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288)(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)(303,304)(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)(319,320)(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)(335,336);;
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, 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 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(336)!(169,253)(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)(176,260)(177,261)(178,262)(179,263)(180,264)(181,265)(182,266)(183,267)(184,268)(185,269)(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)(193,277)(194,278)(195,279)(196,280)(197,281)(198,282)(199,283)(200,284)(201,285)(202,286)(203,287)(204,288)(205,289)(206,290)(207,291)(208,292)(209,293)(210,294)(211,295)(212,296)(213,297)(214,298)(215,299)(216,300)(217,301)(218,302)(219,303)(220,304)(221,305)(222,306)(223,307)(224,308)(225,309)(226,310)(227,311)(228,312)(229,313)(230,314)(231,315)(232,316)(233,317)(234,318)(235,319)(236,320)(237,321)(238,322)(239,323)(240,324)(241,325)(242,326)(243,327)(244,328)(245,329)(246,330)(247,331)(248,332)(249,333)(250,334)(251,335)(252,336);
s1 := Sym(336)!(  1,169)(  2,170)(  3,172)(  4,171)(  5,193)(  6,194)(  7,196)(  8,195)(  9,189)( 10,190)( 11,192)( 12,191)( 13,185)( 14,186)( 15,188)( 16,187)( 17,181)( 18,182)( 19,184)( 20,183)( 21,177)( 22,178)( 23,180)( 24,179)( 25,173)( 26,174)( 27,176)( 28,175)( 29,225)( 30,226)( 31,228)( 32,227)( 33,249)( 34,250)( 35,252)( 36,251)( 37,245)( 38,246)( 39,248)( 40,247)( 41,241)( 42,242)( 43,244)( 44,243)( 45,237)( 46,238)( 47,240)( 48,239)( 49,233)( 50,234)( 51,236)( 52,235)( 53,229)( 54,230)( 55,232)( 56,231)( 57,197)( 58,198)( 59,200)( 60,199)( 61,221)( 62,222)( 63,224)( 64,223)( 65,217)( 66,218)( 67,220)( 68,219)( 69,213)( 70,214)( 71,216)( 72,215)( 73,209)( 74,210)( 75,212)( 76,211)( 77,205)( 78,206)( 79,208)( 80,207)( 81,201)( 82,202)( 83,204)( 84,203)( 85,253)( 86,254)( 87,256)( 88,255)( 89,277)( 90,278)( 91,280)( 92,279)( 93,273)( 94,274)( 95,276)( 96,275)( 97,269)( 98,270)( 99,272)(100,271)(101,265)(102,266)(103,268)(104,267)(105,261)(106,262)(107,264)(108,263)(109,257)(110,258)(111,260)(112,259)(113,309)(114,310)(115,312)(116,311)(117,333)(118,334)(119,336)(120,335)(121,329)(122,330)(123,332)(124,331)(125,325)(126,326)(127,328)(128,327)(129,321)(130,322)(131,324)(132,323)(133,317)(134,318)(135,320)(136,319)(137,313)(138,314)(139,316)(140,315)(141,281)(142,282)(143,284)(144,283)(145,305)(146,306)(147,308)(148,307)(149,301)(150,302)(151,304)(152,303)(153,297)(154,298)(155,300)(156,299)(157,293)(158,294)(159,296)(160,295)(161,289)(162,290)(163,292)(164,291)(165,285)(166,286)(167,288)(168,287);
s2 := Sym(336)!(  1, 33)(  2, 36)(  3, 35)(  4, 34)(  5, 29)(  6, 32)(  7, 31)(  8, 30)(  9, 53)( 10, 56)( 11, 55)( 12, 54)( 13, 49)( 14, 52)( 15, 51)( 16, 50)( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 21, 41)( 22, 44)( 23, 43)( 24, 42)( 25, 37)( 26, 40)( 27, 39)( 28, 38)( 57, 61)( 58, 64)( 59, 63)( 60, 62)( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 77)( 70, 80)( 71, 79)( 72, 78)( 74, 76)( 85,117)( 86,120)( 87,119)( 88,118)( 89,113)( 90,116)( 91,115)( 92,114)( 93,137)( 94,140)( 95,139)( 96,138)( 97,133)( 98,136)( 99,135)(100,134)(101,129)(102,132)(103,131)(104,130)(105,125)(106,128)(107,127)(108,126)(109,121)(110,124)(111,123)(112,122)(141,145)(142,148)(143,147)(144,146)(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)(155,163)(156,162)(158,160)(169,201)(170,204)(171,203)(172,202)(173,197)(174,200)(175,199)(176,198)(177,221)(178,224)(179,223)(180,222)(181,217)(182,220)(183,219)(184,218)(185,213)(186,216)(187,215)(188,214)(189,209)(190,212)(191,211)(192,210)(193,205)(194,208)(195,207)(196,206)(225,229)(226,232)(227,231)(228,230)(233,249)(234,252)(235,251)(236,250)(237,245)(238,248)(239,247)(240,246)(242,244)(253,285)(254,288)(255,287)(256,286)(257,281)(258,284)(259,283)(260,282)(261,305)(262,308)(263,307)(264,306)(265,301)(266,304)(267,303)(268,302)(269,297)(270,300)(271,299)(272,298)(273,293)(274,296)(275,295)(276,294)(277,289)(278,292)(279,291)(280,290)(309,313)(310,316)(311,315)(312,314)(317,333)(318,336)(319,335)(320,334)(321,329)(322,332)(323,331)(324,330)(326,328);
s3 := Sym(336)!(  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288)(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)(303,304)(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)(319,320)(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)(335,336);
poly := sub<Sym(336)|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, 
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 >;

References