Polytope of Type {42,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,6}*1344
if this polytope has a name.
Group : SmallGroup(1344,11343)
Rank : 3
Schlafli Type : {42,6}
Number of vertices, edges, etc : 112, 336, 16
Order of s0s1s2 : 56
Order of s0s1s2s1 : 12
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 : {6,6}*192b
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
168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,169)( 2,170)( 3,172)( 4,171)( 5,175)( 6,176)( 7,173)( 8,174)
( 9,217)( 10,218)( 11,220)( 12,219)( 13,223)( 14,224)( 15,221)( 16,222)
( 17,209)( 18,210)( 19,212)( 20,211)( 21,215)( 22,216)( 23,213)( 24,214)
( 25,201)( 26,202)( 27,204)( 28,203)( 29,207)( 30,208)( 31,205)( 32,206)
( 33,193)( 34,194)( 35,196)( 36,195)( 37,199)( 38,200)( 39,197)( 40,198)
( 41,185)( 42,186)( 43,188)( 44,187)( 45,191)( 46,192)( 47,189)( 48,190)
( 49,177)( 50,178)( 51,180)( 52,179)( 53,183)( 54,184)( 55,181)( 56,182)
( 57,281)( 58,282)( 59,284)( 60,283)( 61,287)( 62,288)( 63,285)( 64,286)
( 65,329)( 66,330)( 67,332)( 68,331)( 69,335)( 70,336)( 71,333)( 72,334)
( 73,321)( 74,322)( 75,324)( 76,323)( 77,327)( 78,328)( 79,325)( 80,326)
( 81,313)( 82,314)( 83,316)( 84,315)( 85,319)( 86,320)( 87,317)( 88,318)
( 89,305)( 90,306)( 91,308)( 92,307)( 93,311)( 94,312)( 95,309)( 96,310)
( 97,297)( 98,298)( 99,300)(100,299)(101,303)(102,304)(103,301)(104,302)
(105,289)(106,290)(107,292)(108,291)(109,295)(110,296)(111,293)(112,294)
(113,225)(114,226)(115,228)(116,227)(117,231)(118,232)(119,229)(120,230)
(121,273)(122,274)(123,276)(124,275)(125,279)(126,280)(127,277)(128,278)
(129,265)(130,266)(131,268)(132,267)(133,271)(134,272)(135,269)(136,270)
(137,257)(138,258)(139,260)(140,259)(141,263)(142,264)(143,261)(144,262)
(145,249)(146,250)(147,252)(148,251)(149,255)(150,256)(151,253)(152,254)
(153,241)(154,242)(155,244)(156,243)(157,247)(158,248)(159,245)(160,246)
(161,233)(162,234)(163,236)(164,235)(165,239)(166,240)(167,237)(168,238);;
s1 := ( 1, 65)( 2, 66)( 3, 71)( 4, 72)( 5, 70)( 6, 69)( 7, 67)( 8, 68)
( 9, 57)( 10, 58)( 11, 63)( 12, 64)( 13, 62)( 14, 61)( 15, 59)( 16, 60)
( 17,105)( 18,106)( 19,111)( 20,112)( 21,110)( 22,109)( 23,107)( 24,108)
( 25, 97)( 26, 98)( 27,103)( 28,104)( 29,102)( 30,101)( 31, 99)( 32,100)
( 33, 89)( 34, 90)( 35, 95)( 36, 96)( 37, 94)( 38, 93)( 39, 91)( 40, 92)
( 41, 81)( 42, 82)( 43, 87)( 44, 88)( 45, 86)( 46, 85)( 47, 83)( 48, 84)
( 49, 73)( 50, 74)( 51, 79)( 52, 80)( 53, 78)( 54, 77)( 55, 75)( 56, 76)
(113,121)(114,122)(115,127)(116,128)(117,126)(118,125)(119,123)(120,124)
(129,161)(130,162)(131,167)(132,168)(133,166)(134,165)(135,163)(136,164)
(137,153)(138,154)(139,159)(140,160)(141,158)(142,157)(143,155)(144,156)
(147,151)(148,152)(149,150)(169,233)(170,234)(171,239)(172,240)(173,238)
(174,237)(175,235)(176,236)(177,225)(178,226)(179,231)(180,232)(181,230)
(182,229)(183,227)(184,228)(185,273)(186,274)(187,279)(188,280)(189,278)
(190,277)(191,275)(192,276)(193,265)(194,266)(195,271)(196,272)(197,270)
(198,269)(199,267)(200,268)(201,257)(202,258)(203,263)(204,264)(205,262)
(206,261)(207,259)(208,260)(209,249)(210,250)(211,255)(212,256)(213,254)
(214,253)(215,251)(216,252)(217,241)(218,242)(219,247)(220,248)(221,246)
(222,245)(223,243)(224,244)(281,289)(282,290)(283,295)(284,296)(285,294)
(286,293)(287,291)(288,292)(297,329)(298,330)(299,335)(300,336)(301,334)
(302,333)(303,331)(304,332)(305,321)(306,322)(307,327)(308,328)(309,326)
(310,325)(311,323)(312,324)(315,319)(316,320)(317,318);;
s2 := ( 1, 3)( 2, 4)( 5, 6)( 9, 11)( 10, 12)( 13, 14)( 17, 19)( 18, 20)
( 21, 22)( 25, 27)( 26, 28)( 29, 30)( 33, 35)( 34, 36)( 37, 38)( 41, 43)
( 42, 44)( 45, 46)( 49, 51)( 50, 52)( 53, 54)( 57,115)( 58,116)( 59,113)
( 60,114)( 61,118)( 62,117)( 63,119)( 64,120)( 65,123)( 66,124)( 67,121)
( 68,122)( 69,126)( 70,125)( 71,127)( 72,128)( 73,131)( 74,132)( 75,129)
( 76,130)( 77,134)( 78,133)( 79,135)( 80,136)( 81,139)( 82,140)( 83,137)
( 84,138)( 85,142)( 86,141)( 87,143)( 88,144)( 89,147)( 90,148)( 91,145)
( 92,146)( 93,150)( 94,149)( 95,151)( 96,152)( 97,155)( 98,156)( 99,153)
(100,154)(101,158)(102,157)(103,159)(104,160)(105,163)(106,164)(107,161)
(108,162)(109,166)(110,165)(111,167)(112,168)(169,172)(170,171)(175,176)
(177,180)(178,179)(183,184)(185,188)(186,187)(191,192)(193,196)(194,195)
(199,200)(201,204)(202,203)(207,208)(209,212)(210,211)(215,216)(217,220)
(218,219)(223,224)(225,284)(226,283)(227,282)(228,281)(229,285)(230,286)
(231,288)(232,287)(233,292)(234,291)(235,290)(236,289)(237,293)(238,294)
(239,296)(240,295)(241,300)(242,299)(243,298)(244,297)(245,301)(246,302)
(247,304)(248,303)(249,308)(250,307)(251,306)(252,305)(253,309)(254,310)
(255,312)(256,311)(257,316)(258,315)(259,314)(260,313)(261,317)(262,318)
(263,320)(264,319)(265,324)(266,323)(267,322)(268,321)(269,325)(270,326)
(271,328)(272,327)(273,332)(274,331)(275,330)(276,329)(277,333)(278,334)
(279,336)(280,335);;
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,
s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(336)!( 1,169)( 2,170)( 3,172)( 4,171)( 5,175)( 6,176)( 7,173)
( 8,174)( 9,217)( 10,218)( 11,220)( 12,219)( 13,223)( 14,224)( 15,221)
( 16,222)( 17,209)( 18,210)( 19,212)( 20,211)( 21,215)( 22,216)( 23,213)
( 24,214)( 25,201)( 26,202)( 27,204)( 28,203)( 29,207)( 30,208)( 31,205)
( 32,206)( 33,193)( 34,194)( 35,196)( 36,195)( 37,199)( 38,200)( 39,197)
( 40,198)( 41,185)( 42,186)( 43,188)( 44,187)( 45,191)( 46,192)( 47,189)
( 48,190)( 49,177)( 50,178)( 51,180)( 52,179)( 53,183)( 54,184)( 55,181)
( 56,182)( 57,281)( 58,282)( 59,284)( 60,283)( 61,287)( 62,288)( 63,285)
( 64,286)( 65,329)( 66,330)( 67,332)( 68,331)( 69,335)( 70,336)( 71,333)
( 72,334)( 73,321)( 74,322)( 75,324)( 76,323)( 77,327)( 78,328)( 79,325)
( 80,326)( 81,313)( 82,314)( 83,316)( 84,315)( 85,319)( 86,320)( 87,317)
( 88,318)( 89,305)( 90,306)( 91,308)( 92,307)( 93,311)( 94,312)( 95,309)
( 96,310)( 97,297)( 98,298)( 99,300)(100,299)(101,303)(102,304)(103,301)
(104,302)(105,289)(106,290)(107,292)(108,291)(109,295)(110,296)(111,293)
(112,294)(113,225)(114,226)(115,228)(116,227)(117,231)(118,232)(119,229)
(120,230)(121,273)(122,274)(123,276)(124,275)(125,279)(126,280)(127,277)
(128,278)(129,265)(130,266)(131,268)(132,267)(133,271)(134,272)(135,269)
(136,270)(137,257)(138,258)(139,260)(140,259)(141,263)(142,264)(143,261)
(144,262)(145,249)(146,250)(147,252)(148,251)(149,255)(150,256)(151,253)
(152,254)(153,241)(154,242)(155,244)(156,243)(157,247)(158,248)(159,245)
(160,246)(161,233)(162,234)(163,236)(164,235)(165,239)(166,240)(167,237)
(168,238);
s1 := Sym(336)!( 1, 65)( 2, 66)( 3, 71)( 4, 72)( 5, 70)( 6, 69)( 7, 67)
( 8, 68)( 9, 57)( 10, 58)( 11, 63)( 12, 64)( 13, 62)( 14, 61)( 15, 59)
( 16, 60)( 17,105)( 18,106)( 19,111)( 20,112)( 21,110)( 22,109)( 23,107)
( 24,108)( 25, 97)( 26, 98)( 27,103)( 28,104)( 29,102)( 30,101)( 31, 99)
( 32,100)( 33, 89)( 34, 90)( 35, 95)( 36, 96)( 37, 94)( 38, 93)( 39, 91)
( 40, 92)( 41, 81)( 42, 82)( 43, 87)( 44, 88)( 45, 86)( 46, 85)( 47, 83)
( 48, 84)( 49, 73)( 50, 74)( 51, 79)( 52, 80)( 53, 78)( 54, 77)( 55, 75)
( 56, 76)(113,121)(114,122)(115,127)(116,128)(117,126)(118,125)(119,123)
(120,124)(129,161)(130,162)(131,167)(132,168)(133,166)(134,165)(135,163)
(136,164)(137,153)(138,154)(139,159)(140,160)(141,158)(142,157)(143,155)
(144,156)(147,151)(148,152)(149,150)(169,233)(170,234)(171,239)(172,240)
(173,238)(174,237)(175,235)(176,236)(177,225)(178,226)(179,231)(180,232)
(181,230)(182,229)(183,227)(184,228)(185,273)(186,274)(187,279)(188,280)
(189,278)(190,277)(191,275)(192,276)(193,265)(194,266)(195,271)(196,272)
(197,270)(198,269)(199,267)(200,268)(201,257)(202,258)(203,263)(204,264)
(205,262)(206,261)(207,259)(208,260)(209,249)(210,250)(211,255)(212,256)
(213,254)(214,253)(215,251)(216,252)(217,241)(218,242)(219,247)(220,248)
(221,246)(222,245)(223,243)(224,244)(281,289)(282,290)(283,295)(284,296)
(285,294)(286,293)(287,291)(288,292)(297,329)(298,330)(299,335)(300,336)
(301,334)(302,333)(303,331)(304,332)(305,321)(306,322)(307,327)(308,328)
(309,326)(310,325)(311,323)(312,324)(315,319)(316,320)(317,318);
s2 := Sym(336)!( 1, 3)( 2, 4)( 5, 6)( 9, 11)( 10, 12)( 13, 14)( 17, 19)
( 18, 20)( 21, 22)( 25, 27)( 26, 28)( 29, 30)( 33, 35)( 34, 36)( 37, 38)
( 41, 43)( 42, 44)( 45, 46)( 49, 51)( 50, 52)( 53, 54)( 57,115)( 58,116)
( 59,113)( 60,114)( 61,118)( 62,117)( 63,119)( 64,120)( 65,123)( 66,124)
( 67,121)( 68,122)( 69,126)( 70,125)( 71,127)( 72,128)( 73,131)( 74,132)
( 75,129)( 76,130)( 77,134)( 78,133)( 79,135)( 80,136)( 81,139)( 82,140)
( 83,137)( 84,138)( 85,142)( 86,141)( 87,143)( 88,144)( 89,147)( 90,148)
( 91,145)( 92,146)( 93,150)( 94,149)( 95,151)( 96,152)( 97,155)( 98,156)
( 99,153)(100,154)(101,158)(102,157)(103,159)(104,160)(105,163)(106,164)
(107,161)(108,162)(109,166)(110,165)(111,167)(112,168)(169,172)(170,171)
(175,176)(177,180)(178,179)(183,184)(185,188)(186,187)(191,192)(193,196)
(194,195)(199,200)(201,204)(202,203)(207,208)(209,212)(210,211)(215,216)
(217,220)(218,219)(223,224)(225,284)(226,283)(227,282)(228,281)(229,285)
(230,286)(231,288)(232,287)(233,292)(234,291)(235,290)(236,289)(237,293)
(238,294)(239,296)(240,295)(241,300)(242,299)(243,298)(244,297)(245,301)
(246,302)(247,304)(248,303)(249,308)(250,307)(251,306)(252,305)(253,309)
(254,310)(255,312)(256,311)(257,316)(258,315)(259,314)(260,313)(261,317)
(262,318)(263,320)(264,319)(265,324)(266,323)(267,322)(268,321)(269,325)
(270,326)(271,328)(272,327)(273,332)(274,331)(275,330)(276,329)(277,333)
(278,334)(279,336)(280,335);
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,
s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope