Polytope of Type {6,56}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,56}*1344c
if this polytope has a name.
Group : SmallGroup(1344,11355)
Rank : 3
Schlafli Type : {6,56}
Number of vertices, edges, etc : 12, 336, 112
Order of s₀s₁s₂ : 84
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {6,28}*672
   4-fold quotients : {6,28}*336b
   7-fold quotients : {6,8}*192b
   8-fold quotients : {6,14}*168
   14-fold quotients : {3,8}*96, {6,4}*96
   24-fold quotients : {2,14}*56
   28-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   48-fold quotients : {2,7}*28
   56-fold quotients : {3,4}*24, {6,2}*24
   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 := (  3,  4)(  5,  7)(  6,  8)( 11, 12)( 13, 15)( 14, 16)( 19, 20)( 21, 23)
( 22, 24)( 27, 28)( 29, 31)( 30, 32)( 35, 36)( 37, 39)( 38, 40)( 43, 44)
( 45, 47)( 46, 48)( 51, 52)( 53, 55)( 54, 56)( 57,113)( 58,114)( 59,116)
( 60,115)( 61,119)( 62,120)( 63,117)( 64,118)( 65,121)( 66,122)( 67,124)
( 68,123)( 69,127)( 70,128)( 71,125)( 72,126)( 73,129)( 74,130)( 75,132)
( 76,131)( 77,135)( 78,136)( 79,133)( 80,134)( 81,137)( 82,138)( 83,140)
( 84,139)( 85,143)( 86,144)( 87,141)( 88,142)( 89,145)( 90,146)( 91,148)
( 92,147)( 93,151)( 94,152)( 95,149)( 96,150)( 97,153)( 98,154)( 99,156)
(100,155)(101,159)(102,160)(103,157)(104,158)(105,161)(106,162)(107,164)
(108,163)(109,167)(110,168)(111,165)(112,166)(169,170)(173,176)(174,175)
(177,178)(181,184)(182,183)(185,186)(189,192)(190,191)(193,194)(197,200)
(198,199)(201,202)(205,208)(206,207)(209,210)(213,216)(214,215)(217,218)
(221,224)(222,223)(225,282)(226,281)(227,283)(228,284)(229,288)(230,287)
(231,286)(232,285)(233,290)(234,289)(235,291)(236,292)(237,296)(238,295)
(239,294)(240,293)(241,298)(242,297)(243,299)(244,300)(245,304)(246,303)
(247,302)(248,301)(249,306)(250,305)(251,307)(252,308)(253,312)(254,311)
(255,310)(256,309)(257,314)(258,313)(259,315)(260,316)(261,320)(262,319)
(263,318)(264,317)(265,322)(266,321)(267,323)(268,324)(269,328)(270,327)
(271,326)(272,325)(273,330)(274,329)(275,331)(276,332)(277,336)(278,335)
(279,334)(280,333);;
s1 := (  1, 57)(  2, 58)(  3, 63)(  4, 64)(  5, 62)(  6, 61)(  7, 59)(  8, 60)
(  9,105)( 10,106)( 11,111)( 12,112)( 13,110)( 14,109)( 15,107)( 16,108)
( 17, 97)( 18, 98)( 19,103)( 20,104)( 21,102)( 22,101)( 23, 99)( 24,100)
( 25, 89)( 26, 90)( 27, 95)( 28, 96)( 29, 94)( 30, 93)( 31, 91)( 32, 92)
( 33, 81)( 34, 82)( 35, 87)( 36, 88)( 37, 86)( 38, 85)( 39, 83)( 40, 84)
( 41, 73)( 42, 74)( 43, 79)( 44, 80)( 45, 78)( 46, 77)( 47, 75)( 48, 76)
( 49, 65)( 50, 66)( 51, 71)( 52, 72)( 53, 70)( 54, 69)( 55, 67)( 56, 68)
(115,119)(116,120)(117,118)(121,161)(122,162)(123,167)(124,168)(125,166)
(126,165)(127,163)(128,164)(129,153)(130,154)(131,159)(132,160)(133,158)
(134,157)(135,155)(136,156)(137,145)(138,146)(139,151)(140,152)(141,150)
(142,149)(143,147)(144,148)(169,226)(170,225)(171,232)(172,231)(173,229)
(174,230)(175,228)(176,227)(177,274)(178,273)(179,280)(180,279)(181,277)
(182,278)(183,276)(184,275)(185,266)(186,265)(187,272)(188,271)(189,269)
(190,270)(191,268)(192,267)(193,258)(194,257)(195,264)(196,263)(197,261)
(198,262)(199,260)(200,259)(201,250)(202,249)(203,256)(204,255)(205,253)
(206,254)(207,252)(208,251)(209,242)(210,241)(211,248)(212,247)(213,245)
(214,246)(215,244)(216,243)(217,234)(218,233)(219,240)(220,239)(221,237)
(222,238)(223,236)(224,235)(281,282)(283,288)(284,287)(289,330)(290,329)
(291,336)(292,335)(293,333)(294,334)(295,332)(296,331)(297,322)(298,321)
(299,328)(300,327)(301,325)(302,326)(303,324)(304,323)(305,314)(306,313)
(307,320)(308,319)(309,317)(310,318)(311,316)(312,315);;
s2 := (  1,179)(  2,180)(  3,178)(  4,177)(  5,183)(  6,184)(  7,182)(  8,181)
(  9,171)( 10,172)( 11,170)( 12,169)( 13,175)( 14,176)( 15,174)( 16,173)
( 17,219)( 18,220)( 19,218)( 20,217)( 21,223)( 22,224)( 23,222)( 24,221)
( 25,211)( 26,212)( 27,210)( 28,209)( 29,215)( 30,216)( 31,214)( 32,213)
( 33,203)( 34,204)( 35,202)( 36,201)( 37,207)( 38,208)( 39,206)( 40,205)
( 41,195)( 42,196)( 43,194)( 44,193)( 45,199)( 46,200)( 47,198)( 48,197)
( 49,187)( 50,188)( 51,186)( 52,185)( 53,191)( 54,192)( 55,190)( 56,189)
( 57,235)( 58,236)( 59,234)( 60,233)( 61,239)( 62,240)( 63,238)( 64,237)
( 65,227)( 66,228)( 67,226)( 68,225)( 69,231)( 70,232)( 71,230)( 72,229)
( 73,275)( 74,276)( 75,274)( 76,273)( 77,279)( 78,280)( 79,278)( 80,277)
( 81,267)( 82,268)( 83,266)( 84,265)( 85,271)( 86,272)( 87,270)( 88,269)
( 89,259)( 90,260)( 91,258)( 92,257)( 93,263)( 94,264)( 95,262)( 96,261)
( 97,251)( 98,252)( 99,250)(100,249)(101,255)(102,256)(103,254)(104,253)
(105,243)(106,244)(107,242)(108,241)(109,247)(110,248)(111,246)(112,245)
(113,291)(114,292)(115,290)(116,289)(117,295)(118,296)(119,294)(120,293)
(121,283)(122,284)(123,282)(124,281)(125,287)(126,288)(127,286)(128,285)
(129,331)(130,332)(131,330)(132,329)(133,335)(134,336)(135,334)(136,333)
(137,323)(138,324)(139,322)(140,321)(141,327)(142,328)(143,326)(144,325)
(145,315)(146,316)(147,314)(148,313)(149,319)(150,320)(151,318)(152,317)
(153,307)(154,308)(155,306)(156,305)(157,311)(158,312)(159,310)(160,309)
(161,299)(162,300)(163,298)(164,297)(165,303)(166,304)(167,302)(168,301);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope