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Polytope of Type {12,56}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,56}*1344a
Also Known As : {12,56|2}. if this polytope has another name.
Group : SmallGroup(1344,2776)
Rank : 3
Schlafli Type : {12,56}
Number of vertices, edges, etc : 12, 336, 56
Order of s0s1s2 : 168
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {6,56}*672, {12,28}*672
   3-fold quotients : {4,56}*448a
   4-fold quotients : {12,14}*336, {6,28}*336a
   6-fold quotients : {4,28}*224, {2,56}*224
   7-fold quotients : {12,8}*192a
   8-fold quotients : {6,14}*168
   12-fold quotients : {2,28}*112, {4,14}*112
   14-fold quotients : {12,4}*96a, {6,8}*96
   21-fold quotients : {4,8}*64a
   24-fold quotients : {2,14}*56
   28-fold quotients : {12,2}*48, {6,4}*48a
   42-fold quotients : {4,4}*32, {2,8}*32
   48-fold quotients : {2,7}*28
   56-fold quotients : {6,2}*24
   84-fold quotients : {2,4}*16, {4,2}*16
   112-fold quotients : {3,2}*12
   168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  8, 15)(  9, 16)( 10, 17)( 11, 18)( 12, 19)( 13, 20)( 14, 21)( 29, 36)
( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 50, 57)( 51, 58)
( 52, 59)( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 71, 78)( 72, 79)( 73, 80)
( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 92, 99)( 93,100)( 94,101)( 95,102)
( 96,103)( 97,104)( 98,105)(113,120)(114,121)(115,122)(116,123)(117,124)
(118,125)(119,126)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)
(140,147)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)
(169,253)(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)(176,267)
(177,268)(178,269)(179,270)(180,271)(181,272)(182,273)(183,260)(184,261)
(185,262)(186,263)(187,264)(188,265)(189,266)(190,274)(191,275)(192,276)
(193,277)(194,278)(195,279)(196,280)(197,288)(198,289)(199,290)(200,291)
(201,292)(202,293)(203,294)(204,281)(205,282)(206,283)(207,284)(208,285)
(209,286)(210,287)(211,295)(212,296)(213,297)(214,298)(215,299)(216,300)
(217,301)(218,309)(219,310)(220,311)(221,312)(222,313)(223,314)(224,315)
(225,302)(226,303)(227,304)(228,305)(229,306)(230,307)(231,308)(232,316)
(233,317)(234,318)(235,319)(236,320)(237,321)(238,322)(239,330)(240,331)
(241,332)(242,333)(243,334)(244,335)(245,336)(246,323)(247,324)(248,325)
(249,326)(250,327)(251,328)(252,329);;
s1 := (  1,176)(  2,182)(  3,181)(  4,180)(  5,179)(  6,178)(  7,177)(  8,169)
(  9,175)( 10,174)( 11,173)( 12,172)( 13,171)( 14,170)( 15,183)( 16,189)
( 17,188)( 18,187)( 19,186)( 20,185)( 21,184)( 22,197)( 23,203)( 24,202)
( 25,201)( 26,200)( 27,199)( 28,198)( 29,190)( 30,196)( 31,195)( 32,194)
( 33,193)( 34,192)( 35,191)( 36,204)( 37,210)( 38,209)( 39,208)( 40,207)
( 41,206)( 42,205)( 43,239)( 44,245)( 45,244)( 46,243)( 47,242)( 48,241)
( 49,240)( 50,232)( 51,238)( 52,237)( 53,236)( 54,235)( 55,234)( 56,233)
( 57,246)( 58,252)( 59,251)( 60,250)( 61,249)( 62,248)( 63,247)( 64,218)
( 65,224)( 66,223)( 67,222)( 68,221)( 69,220)( 70,219)( 71,211)( 72,217)
( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,225)( 79,231)( 80,230)
( 81,229)( 82,228)( 83,227)( 84,226)( 85,260)( 86,266)( 87,265)( 88,264)
( 89,263)( 90,262)( 91,261)( 92,253)( 93,259)( 94,258)( 95,257)( 96,256)
( 97,255)( 98,254)( 99,267)(100,273)(101,272)(102,271)(103,270)(104,269)
(105,268)(106,281)(107,287)(108,286)(109,285)(110,284)(111,283)(112,282)
(113,274)(114,280)(115,279)(116,278)(117,277)(118,276)(119,275)(120,288)
(121,294)(122,293)(123,292)(124,291)(125,290)(126,289)(127,323)(128,329)
(129,328)(130,327)(131,326)(132,325)(133,324)(134,316)(135,322)(136,321)
(137,320)(138,319)(139,318)(140,317)(141,330)(142,336)(143,335)(144,334)
(145,333)(146,332)(147,331)(148,302)(149,308)(150,307)(151,306)(152,305)
(153,304)(154,303)(155,295)(156,301)(157,300)(158,299)(159,298)(160,297)
(161,296)(162,309)(163,315)(164,314)(165,313)(166,312)(167,311)(168,310);;
s2 := (  1,  2)(  3,  7)(  4,  6)(  8,  9)( 10, 14)( 11, 13)( 15, 16)( 17, 21)
( 18, 20)( 22, 23)( 24, 28)( 25, 27)( 29, 30)( 31, 35)( 32, 34)( 36, 37)
( 38, 42)( 39, 41)( 43, 65)( 44, 64)( 45, 70)( 46, 69)( 47, 68)( 48, 67)
( 49, 66)( 50, 72)( 51, 71)( 52, 77)( 53, 76)( 54, 75)( 55, 74)( 56, 73)
( 57, 79)( 58, 78)( 59, 84)( 60, 83)( 61, 82)( 62, 81)( 63, 80)( 85, 86)
( 87, 91)( 88, 90)( 92, 93)( 94, 98)( 95, 97)( 99,100)(101,105)(102,104)
(106,107)(108,112)(109,111)(113,114)(115,119)(116,118)(120,121)(122,126)
(123,125)(127,149)(128,148)(129,154)(130,153)(131,152)(132,151)(133,150)
(134,156)(135,155)(136,161)(137,160)(138,159)(139,158)(140,157)(141,163)
(142,162)(143,168)(144,167)(145,166)(146,165)(147,164)(169,212)(170,211)
(171,217)(172,216)(173,215)(174,214)(175,213)(176,219)(177,218)(178,224)
(179,223)(180,222)(181,221)(182,220)(183,226)(184,225)(185,231)(186,230)
(187,229)(188,228)(189,227)(190,233)(191,232)(192,238)(193,237)(194,236)
(195,235)(196,234)(197,240)(198,239)(199,245)(200,244)(201,243)(202,242)
(203,241)(204,247)(205,246)(206,252)(207,251)(208,250)(209,249)(210,248)
(253,296)(254,295)(255,301)(256,300)(257,299)(258,298)(259,297)(260,303)
(261,302)(262,308)(263,307)(264,306)(265,305)(266,304)(267,310)(268,309)
(269,315)(270,314)(271,313)(272,312)(273,311)(274,317)(275,316)(276,322)
(277,321)(278,320)(279,319)(280,318)(281,324)(282,323)(283,329)(284,328)
(285,327)(286,326)(287,325)(288,331)(289,330)(290,336)(291,335)(292,334)
(293,333)(294,332);;
poly := Group([s0,s1,s2]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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(336)!(  8, 15)(  9, 16)( 10, 17)( 11, 18)( 12, 19)( 13, 20)( 14, 21)
( 29, 36)( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 50, 57)
( 51, 58)( 52, 59)( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 71, 78)( 72, 79)
( 73, 80)( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 92, 99)( 93,100)( 94,101)
( 95,102)( 96,103)( 97,104)( 98,105)(113,120)(114,121)(115,122)(116,123)
(117,124)(118,125)(119,126)(134,141)(135,142)(136,143)(137,144)(138,145)
(139,146)(140,147)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)
(161,168)(169,253)(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)
(176,267)(177,268)(178,269)(179,270)(180,271)(181,272)(182,273)(183,260)
(184,261)(185,262)(186,263)(187,264)(188,265)(189,266)(190,274)(191,275)
(192,276)(193,277)(194,278)(195,279)(196,280)(197,288)(198,289)(199,290)
(200,291)(201,292)(202,293)(203,294)(204,281)(205,282)(206,283)(207,284)
(208,285)(209,286)(210,287)(211,295)(212,296)(213,297)(214,298)(215,299)
(216,300)(217,301)(218,309)(219,310)(220,311)(221,312)(222,313)(223,314)
(224,315)(225,302)(226,303)(227,304)(228,305)(229,306)(230,307)(231,308)
(232,316)(233,317)(234,318)(235,319)(236,320)(237,321)(238,322)(239,330)
(240,331)(241,332)(242,333)(243,334)(244,335)(245,336)(246,323)(247,324)
(248,325)(249,326)(250,327)(251,328)(252,329);
s1 := Sym(336)!(  1,176)(  2,182)(  3,181)(  4,180)(  5,179)(  6,178)(  7,177)
(  8,169)(  9,175)( 10,174)( 11,173)( 12,172)( 13,171)( 14,170)( 15,183)
( 16,189)( 17,188)( 18,187)( 19,186)( 20,185)( 21,184)( 22,197)( 23,203)
( 24,202)( 25,201)( 26,200)( 27,199)( 28,198)( 29,190)( 30,196)( 31,195)
( 32,194)( 33,193)( 34,192)( 35,191)( 36,204)( 37,210)( 38,209)( 39,208)
( 40,207)( 41,206)( 42,205)( 43,239)( 44,245)( 45,244)( 46,243)( 47,242)
( 48,241)( 49,240)( 50,232)( 51,238)( 52,237)( 53,236)( 54,235)( 55,234)
( 56,233)( 57,246)( 58,252)( 59,251)( 60,250)( 61,249)( 62,248)( 63,247)
( 64,218)( 65,224)( 66,223)( 67,222)( 68,221)( 69,220)( 70,219)( 71,211)
( 72,217)( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,225)( 79,231)
( 80,230)( 81,229)( 82,228)( 83,227)( 84,226)( 85,260)( 86,266)( 87,265)
( 88,264)( 89,263)( 90,262)( 91,261)( 92,253)( 93,259)( 94,258)( 95,257)
( 96,256)( 97,255)( 98,254)( 99,267)(100,273)(101,272)(102,271)(103,270)
(104,269)(105,268)(106,281)(107,287)(108,286)(109,285)(110,284)(111,283)
(112,282)(113,274)(114,280)(115,279)(116,278)(117,277)(118,276)(119,275)
(120,288)(121,294)(122,293)(123,292)(124,291)(125,290)(126,289)(127,323)
(128,329)(129,328)(130,327)(131,326)(132,325)(133,324)(134,316)(135,322)
(136,321)(137,320)(138,319)(139,318)(140,317)(141,330)(142,336)(143,335)
(144,334)(145,333)(146,332)(147,331)(148,302)(149,308)(150,307)(151,306)
(152,305)(153,304)(154,303)(155,295)(156,301)(157,300)(158,299)(159,298)
(160,297)(161,296)(162,309)(163,315)(164,314)(165,313)(166,312)(167,311)
(168,310);
s2 := Sym(336)!(  1,  2)(  3,  7)(  4,  6)(  8,  9)( 10, 14)( 11, 13)( 15, 16)
( 17, 21)( 18, 20)( 22, 23)( 24, 28)( 25, 27)( 29, 30)( 31, 35)( 32, 34)
( 36, 37)( 38, 42)( 39, 41)( 43, 65)( 44, 64)( 45, 70)( 46, 69)( 47, 68)
( 48, 67)( 49, 66)( 50, 72)( 51, 71)( 52, 77)( 53, 76)( 54, 75)( 55, 74)
( 56, 73)( 57, 79)( 58, 78)( 59, 84)( 60, 83)( 61, 82)( 62, 81)( 63, 80)
( 85, 86)( 87, 91)( 88, 90)( 92, 93)( 94, 98)( 95, 97)( 99,100)(101,105)
(102,104)(106,107)(108,112)(109,111)(113,114)(115,119)(116,118)(120,121)
(122,126)(123,125)(127,149)(128,148)(129,154)(130,153)(131,152)(132,151)
(133,150)(134,156)(135,155)(136,161)(137,160)(138,159)(139,158)(140,157)
(141,163)(142,162)(143,168)(144,167)(145,166)(146,165)(147,164)(169,212)
(170,211)(171,217)(172,216)(173,215)(174,214)(175,213)(176,219)(177,218)
(178,224)(179,223)(180,222)(181,221)(182,220)(183,226)(184,225)(185,231)
(186,230)(187,229)(188,228)(189,227)(190,233)(191,232)(192,238)(193,237)
(194,236)(195,235)(196,234)(197,240)(198,239)(199,245)(200,244)(201,243)
(202,242)(203,241)(204,247)(205,246)(206,252)(207,251)(208,250)(209,249)
(210,248)(253,296)(254,295)(255,301)(256,300)(257,299)(258,298)(259,297)
(260,303)(261,302)(262,308)(263,307)(264,306)(265,305)(266,304)(267,310)
(268,309)(269,315)(270,314)(271,313)(272,312)(273,311)(274,317)(275,316)
(276,322)(277,321)(278,320)(279,319)(280,318)(281,324)(282,323)(283,329)
(284,328)(285,327)(286,326)(287,325)(288,331)(289,330)(290,336)(291,335)
(292,334)(293,333)(294,332);
poly := sub<Sym(336)|s0,s1,s2>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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 >; 
 
References : None.
to this polytope