Polytope of Type {56,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {56,6}*1344c
if this polytope has a name.
Group : SmallGroup(1344,11355)
Rank : 3
Schlafli Type : {56,6}
Number of vertices, edges, etc : 112, 336, 12
Order of s₀s₁s₂ : 84
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {28,6}*672
   4-fold quotients : {28,6}*336b
   7-fold quotients : {8,6}*192b
   8-fold quotients : {14,6}*168
   14-fold quotients : {8,3}*96, {4,6}*96
   24-fold quotients : {14,2}*56
   28-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   48-fold quotients : {7,2}*28
   56-fold quotients : {4,3}*24, {2,6}*24
   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 := (  1,173)(  2,174)(  3,176)(  4,175)(  5,170)(  6,169)(  7,171)(  8,172)
(  9,221)( 10,222)( 11,224)( 12,223)( 13,218)( 14,217)( 15,219)( 16,220)
( 17,213)( 18,214)( 19,216)( 20,215)( 21,210)( 22,209)( 23,211)( 24,212)
( 25,205)( 26,206)( 27,208)( 28,207)( 29,202)( 30,201)( 31,203)( 32,204)
( 33,197)( 34,198)( 35,200)( 36,199)( 37,194)( 38,193)( 39,195)( 40,196)
( 41,189)( 42,190)( 43,192)( 44,191)( 45,186)( 46,185)( 47,187)( 48,188)
( 49,181)( 50,182)( 51,184)( 52,183)( 53,178)( 54,177)( 55,179)( 56,180)
( 57,229)( 58,230)( 59,232)( 60,231)( 61,226)( 62,225)( 63,227)( 64,228)
( 65,277)( 66,278)( 67,280)( 68,279)( 69,274)( 70,273)( 71,275)( 72,276)
( 73,269)( 74,270)( 75,272)( 76,271)( 77,266)( 78,265)( 79,267)( 80,268)
( 81,261)( 82,262)( 83,264)( 84,263)( 85,258)( 86,257)( 87,259)( 88,260)
( 89,253)( 90,254)( 91,256)( 92,255)( 93,250)( 94,249)( 95,251)( 96,252)
( 97,245)( 98,246)( 99,248)(100,247)(101,242)(102,241)(103,243)(104,244)
(105,237)(106,238)(107,240)(108,239)(109,234)(110,233)(111,235)(112,236)
(113,285)(114,286)(115,288)(116,287)(117,282)(118,281)(119,283)(120,284)
(121,333)(122,334)(123,336)(124,335)(125,330)(126,329)(127,331)(128,332)
(129,325)(130,326)(131,328)(132,327)(133,322)(134,321)(135,323)(136,324)
(137,317)(138,318)(139,320)(140,319)(141,314)(142,313)(143,315)(144,316)
(145,309)(146,310)(147,312)(148,311)(149,306)(150,305)(151,307)(152,308)
(153,301)(154,302)(155,304)(156,303)(157,298)(158,297)(159,299)(160,300)
(161,293)(162,294)(163,296)(164,295)(165,290)(166,289)(167,291)(168,292);;
s1 := (  1,  9)(  2, 10)(  3, 12)(  4, 11)(  5, 15)(  6, 16)(  7, 13)(  8, 14)
( 17, 49)( 18, 50)( 19, 52)( 20, 51)( 21, 55)( 22, 56)( 23, 53)( 24, 54)
( 25, 41)( 26, 42)( 27, 44)( 28, 43)( 29, 47)( 30, 48)( 31, 45)( 32, 46)
( 35, 36)( 37, 39)( 38, 40)( 57,121)( 58,122)( 59,124)( 60,123)( 61,127)
( 62,128)( 63,125)( 64,126)( 65,113)( 66,114)( 67,116)( 68,115)( 69,119)
( 70,120)( 71,117)( 72,118)( 73,161)( 74,162)( 75,164)( 76,163)( 77,167)
( 78,168)( 79,165)( 80,166)( 81,153)( 82,154)( 83,156)( 84,155)( 85,159)
( 86,160)( 87,157)( 88,158)( 89,145)( 90,146)( 91,148)( 92,147)( 93,151)
( 94,152)( 95,149)( 96,150)( 97,137)( 98,138)( 99,140)(100,139)(101,143)
(102,144)(103,141)(104,142)(105,129)(106,130)(107,132)(108,131)(109,135)
(110,136)(111,133)(112,134)(169,178)(170,177)(171,179)(172,180)(173,184)
(174,183)(175,182)(176,181)(185,218)(186,217)(187,219)(188,220)(189,224)
(190,223)(191,222)(192,221)(193,210)(194,209)(195,211)(196,212)(197,216)
(198,215)(199,214)(200,213)(201,202)(205,208)(206,207)(225,290)(226,289)
(227,291)(228,292)(229,296)(230,295)(231,294)(232,293)(233,282)(234,281)
(235,283)(236,284)(237,288)(238,287)(239,286)(240,285)(241,330)(242,329)
(243,331)(244,332)(245,336)(246,335)(247,334)(248,333)(249,322)(250,321)
(251,323)(252,324)(253,328)(254,327)(255,326)(256,325)(257,314)(258,313)
(259,315)(260,316)(261,320)(262,319)(263,318)(264,317)(265,306)(266,305)
(267,307)(268,308)(269,312)(270,311)(271,310)(272,309)(273,298)(274,297)
(275,299)(276,300)(277,304)(278,303)(279,302)(280,301);;
s2 := (  1, 57)(  2, 58)(  3, 63)(  4, 64)(  5, 62)(  6, 61)(  7, 59)(  8, 60)
(  9, 65)( 10, 66)( 11, 71)( 12, 72)( 13, 70)( 14, 69)( 15, 67)( 16, 68)
( 17, 73)( 18, 74)( 19, 79)( 20, 80)( 21, 78)( 22, 77)( 23, 75)( 24, 76)
( 25, 81)( 26, 82)( 27, 87)( 28, 88)( 29, 86)( 30, 85)( 31, 83)( 32, 84)
( 33, 89)( 34, 90)( 35, 95)( 36, 96)( 37, 94)( 38, 93)( 39, 91)( 40, 92)
( 41, 97)( 42, 98)( 43,103)( 44,104)( 45,102)( 46,101)( 47, 99)( 48,100)
( 49,105)( 50,106)( 51,111)( 52,112)( 53,110)( 54,109)( 55,107)( 56,108)
(115,119)(116,120)(117,118)(123,127)(124,128)(125,126)(131,135)(132,136)
(133,134)(139,143)(140,144)(141,142)(147,151)(148,152)(149,150)(155,159)
(156,160)(157,158)(163,167)(164,168)(165,166)(169,226)(170,225)(171,232)
(172,231)(173,229)(174,230)(175,228)(176,227)(177,234)(178,233)(179,240)
(180,239)(181,237)(182,238)(183,236)(184,235)(185,242)(186,241)(187,248)
(188,247)(189,245)(190,246)(191,244)(192,243)(193,250)(194,249)(195,256)
(196,255)(197,253)(198,254)(199,252)(200,251)(201,258)(202,257)(203,264)
(204,263)(205,261)(206,262)(207,260)(208,259)(209,266)(210,265)(211,272)
(212,271)(213,269)(214,270)(215,268)(216,267)(217,274)(218,273)(219,280)
(220,279)(221,277)(222,278)(223,276)(224,275)(281,282)(283,288)(284,287)
(289,290)(291,296)(292,295)(297,298)(299,304)(300,303)(305,306)(307,312)
(308,311)(313,314)(315,320)(316,319)(321,322)(323,328)(324,327)(329,330)
(331,336)(332,335);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope