Polytope of Type {42,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,12}*1344b
if this polytope has a name.
Group : SmallGroup(1344,11370)
Rank : 3
Schlafli Type : {42,12}
Number of vertices, edges, etc : 56, 336, 16
Order of s₀s₁s₂ : 28
Order of s₀s₁s₂s₁ : 6
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 : {42,6}*672
   4-fold quotients : {21,6}*336
   7-fold quotients : {6,12}*192a
   12-fold quotients : {14,4}*112
   14-fold quotients : {6,6}*96
   24-fold quotients : {14,2}*56
   28-fold quotients : {3,6}*48, {6,3}*48
   48-fold quotients : {7,2}*28
   56-fold quotients : {3,3}*24
   84-fold quotients : {2,4}*16
   168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope