Polytope of Type {56,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {56,12}*1344a
Also Known As : {56,12|2}. if this polytope has another name.
Group : SmallGroup(1344,2776)
Rank : 3
Schlafli Type : {56,12}
Number of vertices, edges, etc : 56, 336, 12
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 : {56,6}*672, {28,12}*672
   3-fold quotients : {56,4}*448a
   4-fold quotients : {14,12}*336, {28,6}*336a
   6-fold quotients : {28,4}*224, {56,2}*224
   7-fold quotients : {8,12}*192a
   8-fold quotients : {14,6}*168
   12-fold quotients : {28,2}*112, {14,4}*112
   14-fold quotients : {4,12}*96a, {8,6}*96
   21-fold quotients : {8,4}*64a
   24-fold quotients : {14,2}*56
   28-fold quotients : {2,12}*48, {4,6}*48a
   42-fold quotients : {4,4}*32, {8,2}*32
   48-fold quotients : {7,2}*28
   56-fold quotients : {2,6}*24
   84-fold quotients : {2,4}*16, {4,2}*16
   112-fold quotients : {2,3}*12
   168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope