Polytope of Type {4,42}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,42}*672
if this polytope has a name.
Group : SmallGroup(672,1263)
Rank : 3
Schlafli Type : {4,42}
Number of vertices, edges, etc : 8, 168, 84
Order of s0s1s2 : 42
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,42,2} of size 1344
Vertex Figure Of :
{2,4,42} of size 1344
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,21}*336, {4,42}*336b, {4,42}*336c
4-fold quotients : {4,21}*168, {2,42}*168
7-fold quotients : {4,6}*96
8-fold quotients : {2,21}*84
12-fold quotients : {2,14}*56
14-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
24-fold quotients : {2,7}*28
28-fold quotients : {4,3}*24, {2,6}*24
56-fold quotients : {2,3}*12
84-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,84}*1344b, {4,42}*1344b, {4,84}*1344c, {8,42}*1344b, {8,42}*1344c
Permutation Representation (GAP) :
s0 := ( 1,171)( 2,172)( 3,169)( 4,170)( 5,175)( 6,176)( 7,173)( 8,174)
( 9,179)( 10,180)( 11,177)( 12,178)( 13,183)( 14,184)( 15,181)( 16,182)
( 17,187)( 18,188)( 19,185)( 20,186)( 21,191)( 22,192)( 23,189)( 24,190)
( 25,195)( 26,196)( 27,193)( 28,194)( 29,199)( 30,200)( 31,197)( 32,198)
( 33,203)( 34,204)( 35,201)( 36,202)( 37,207)( 38,208)( 39,205)( 40,206)
( 41,211)( 42,212)( 43,209)( 44,210)( 45,215)( 46,216)( 47,213)( 48,214)
( 49,219)( 50,220)( 51,217)( 52,218)( 53,223)( 54,224)( 55,221)( 56,222)
( 57,227)( 58,228)( 59,225)( 60,226)( 61,231)( 62,232)( 63,229)( 64,230)
( 65,235)( 66,236)( 67,233)( 68,234)( 69,239)( 70,240)( 71,237)( 72,238)
( 73,243)( 74,244)( 75,241)( 76,242)( 77,247)( 78,248)( 79,245)( 80,246)
( 81,251)( 82,252)( 83,249)( 84,250)( 85,255)( 86,256)( 87,253)( 88,254)
( 89,259)( 90,260)( 91,257)( 92,258)( 93,263)( 94,264)( 95,261)( 96,262)
( 97,267)( 98,268)( 99,265)(100,266)(101,271)(102,272)(103,269)(104,270)
(105,275)(106,276)(107,273)(108,274)(109,279)(110,280)(111,277)(112,278)
(113,283)(114,284)(115,281)(116,282)(117,287)(118,288)(119,285)(120,286)
(121,291)(122,292)(123,289)(124,290)(125,295)(126,296)(127,293)(128,294)
(129,299)(130,300)(131,297)(132,298)(133,303)(134,304)(135,301)(136,302)
(137,307)(138,308)(139,305)(140,306)(141,311)(142,312)(143,309)(144,310)
(145,315)(146,316)(147,313)(148,314)(149,319)(150,320)(151,317)(152,318)
(153,323)(154,324)(155,321)(156,322)(157,327)(158,328)(159,325)(160,326)
(161,331)(162,332)(163,329)(164,330)(165,335)(166,336)(167,333)(168,334);;
s1 := ( 2, 3)( 5, 25)( 6, 27)( 7, 26)( 8, 28)( 9, 21)( 10, 23)( 11, 22)
( 12, 24)( 13, 17)( 14, 19)( 15, 18)( 16, 20)( 29, 57)( 30, 59)( 31, 58)
( 32, 60)( 33, 81)( 34, 83)( 35, 82)( 36, 84)( 37, 77)( 38, 79)( 39, 78)
( 40, 80)( 41, 73)( 42, 75)( 43, 74)( 44, 76)( 45, 69)( 46, 71)( 47, 70)
( 48, 72)( 49, 65)( 50, 67)( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)
( 56, 64)( 86, 87)( 89,109)( 90,111)( 91,110)( 92,112)( 93,105)( 94,107)
( 95,106)( 96,108)( 97,101)( 98,103)( 99,102)(100,104)(113,141)(114,143)
(115,142)(116,144)(117,165)(118,167)(119,166)(120,168)(121,161)(122,163)
(123,162)(124,164)(125,157)(126,159)(127,158)(128,160)(129,153)(130,155)
(131,154)(132,156)(133,149)(134,151)(135,150)(136,152)(137,145)(138,147)
(139,146)(140,148)(170,171)(173,193)(174,195)(175,194)(176,196)(177,189)
(178,191)(179,190)(180,192)(181,185)(182,187)(183,186)(184,188)(197,225)
(198,227)(199,226)(200,228)(201,249)(202,251)(203,250)(204,252)(205,245)
(206,247)(207,246)(208,248)(209,241)(210,243)(211,242)(212,244)(213,237)
(214,239)(215,238)(216,240)(217,233)(218,235)(219,234)(220,236)(221,229)
(222,231)(223,230)(224,232)(254,255)(257,277)(258,279)(259,278)(260,280)
(261,273)(262,275)(263,274)(264,276)(265,269)(266,271)(267,270)(268,272)
(281,309)(282,311)(283,310)(284,312)(285,333)(286,335)(287,334)(288,336)
(289,329)(290,331)(291,330)(292,332)(293,325)(294,327)(295,326)(296,328)
(297,321)(298,323)(299,322)(300,324)(301,317)(302,319)(303,318)(304,320)
(305,313)(306,315)(307,314)(308,316);;
s2 := ( 1,145)( 2,148)( 3,147)( 4,146)( 5,141)( 6,144)( 7,143)( 8,142)
( 9,165)( 10,168)( 11,167)( 12,166)( 13,161)( 14,164)( 15,163)( 16,162)
( 17,157)( 18,160)( 19,159)( 20,158)( 21,153)( 22,156)( 23,155)( 24,154)
( 25,149)( 26,152)( 27,151)( 28,150)( 29,117)( 30,120)( 31,119)( 32,118)
( 33,113)( 34,116)( 35,115)( 36,114)( 37,137)( 38,140)( 39,139)( 40,138)
( 41,133)( 42,136)( 43,135)( 44,134)( 45,129)( 46,132)( 47,131)( 48,130)
( 49,125)( 50,128)( 51,127)( 52,126)( 53,121)( 54,124)( 55,123)( 56,122)
( 57, 89)( 58, 92)( 59, 91)( 60, 90)( 61, 85)( 62, 88)( 63, 87)( 64, 86)
( 65,109)( 66,112)( 67,111)( 68,110)( 69,105)( 70,108)( 71,107)( 72,106)
( 73,101)( 74,104)( 75,103)( 76,102)( 77, 97)( 78,100)( 79, 99)( 80, 98)
( 81, 93)( 82, 96)( 83, 95)( 84, 94)(169,313)(170,316)(171,315)(172,314)
(173,309)(174,312)(175,311)(176,310)(177,333)(178,336)(179,335)(180,334)
(181,329)(182,332)(183,331)(184,330)(185,325)(186,328)(187,327)(188,326)
(189,321)(190,324)(191,323)(192,322)(193,317)(194,320)(195,319)(196,318)
(197,285)(198,288)(199,287)(200,286)(201,281)(202,284)(203,283)(204,282)
(205,305)(206,308)(207,307)(208,306)(209,301)(210,304)(211,303)(212,302)
(213,297)(214,300)(215,299)(216,298)(217,293)(218,296)(219,295)(220,294)
(221,289)(222,292)(223,291)(224,290)(225,257)(226,260)(227,259)(228,258)
(229,253)(230,256)(231,255)(232,254)(233,277)(234,280)(235,279)(236,278)
(237,273)(238,276)(239,275)(240,274)(241,269)(242,272)(243,271)(244,270)
(245,265)(246,268)(247,267)(248,266)(249,261)(250,264)(251,263)(252,262);;
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*s2*s1*s2*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(336)!( 1,171)( 2,172)( 3,169)( 4,170)( 5,175)( 6,176)( 7,173)
( 8,174)( 9,179)( 10,180)( 11,177)( 12,178)( 13,183)( 14,184)( 15,181)
( 16,182)( 17,187)( 18,188)( 19,185)( 20,186)( 21,191)( 22,192)( 23,189)
( 24,190)( 25,195)( 26,196)( 27,193)( 28,194)( 29,199)( 30,200)( 31,197)
( 32,198)( 33,203)( 34,204)( 35,201)( 36,202)( 37,207)( 38,208)( 39,205)
( 40,206)( 41,211)( 42,212)( 43,209)( 44,210)( 45,215)( 46,216)( 47,213)
( 48,214)( 49,219)( 50,220)( 51,217)( 52,218)( 53,223)( 54,224)( 55,221)
( 56,222)( 57,227)( 58,228)( 59,225)( 60,226)( 61,231)( 62,232)( 63,229)
( 64,230)( 65,235)( 66,236)( 67,233)( 68,234)( 69,239)( 70,240)( 71,237)
( 72,238)( 73,243)( 74,244)( 75,241)( 76,242)( 77,247)( 78,248)( 79,245)
( 80,246)( 81,251)( 82,252)( 83,249)( 84,250)( 85,255)( 86,256)( 87,253)
( 88,254)( 89,259)( 90,260)( 91,257)( 92,258)( 93,263)( 94,264)( 95,261)
( 96,262)( 97,267)( 98,268)( 99,265)(100,266)(101,271)(102,272)(103,269)
(104,270)(105,275)(106,276)(107,273)(108,274)(109,279)(110,280)(111,277)
(112,278)(113,283)(114,284)(115,281)(116,282)(117,287)(118,288)(119,285)
(120,286)(121,291)(122,292)(123,289)(124,290)(125,295)(126,296)(127,293)
(128,294)(129,299)(130,300)(131,297)(132,298)(133,303)(134,304)(135,301)
(136,302)(137,307)(138,308)(139,305)(140,306)(141,311)(142,312)(143,309)
(144,310)(145,315)(146,316)(147,313)(148,314)(149,319)(150,320)(151,317)
(152,318)(153,323)(154,324)(155,321)(156,322)(157,327)(158,328)(159,325)
(160,326)(161,331)(162,332)(163,329)(164,330)(165,335)(166,336)(167,333)
(168,334);
s1 := Sym(336)!( 2, 3)( 5, 25)( 6, 27)( 7, 26)( 8, 28)( 9, 21)( 10, 23)
( 11, 22)( 12, 24)( 13, 17)( 14, 19)( 15, 18)( 16, 20)( 29, 57)( 30, 59)
( 31, 58)( 32, 60)( 33, 81)( 34, 83)( 35, 82)( 36, 84)( 37, 77)( 38, 79)
( 39, 78)( 40, 80)( 41, 73)( 42, 75)( 43, 74)( 44, 76)( 45, 69)( 46, 71)
( 47, 70)( 48, 72)( 49, 65)( 50, 67)( 51, 66)( 52, 68)( 53, 61)( 54, 63)
( 55, 62)( 56, 64)( 86, 87)( 89,109)( 90,111)( 91,110)( 92,112)( 93,105)
( 94,107)( 95,106)( 96,108)( 97,101)( 98,103)( 99,102)(100,104)(113,141)
(114,143)(115,142)(116,144)(117,165)(118,167)(119,166)(120,168)(121,161)
(122,163)(123,162)(124,164)(125,157)(126,159)(127,158)(128,160)(129,153)
(130,155)(131,154)(132,156)(133,149)(134,151)(135,150)(136,152)(137,145)
(138,147)(139,146)(140,148)(170,171)(173,193)(174,195)(175,194)(176,196)
(177,189)(178,191)(179,190)(180,192)(181,185)(182,187)(183,186)(184,188)
(197,225)(198,227)(199,226)(200,228)(201,249)(202,251)(203,250)(204,252)
(205,245)(206,247)(207,246)(208,248)(209,241)(210,243)(211,242)(212,244)
(213,237)(214,239)(215,238)(216,240)(217,233)(218,235)(219,234)(220,236)
(221,229)(222,231)(223,230)(224,232)(254,255)(257,277)(258,279)(259,278)
(260,280)(261,273)(262,275)(263,274)(264,276)(265,269)(266,271)(267,270)
(268,272)(281,309)(282,311)(283,310)(284,312)(285,333)(286,335)(287,334)
(288,336)(289,329)(290,331)(291,330)(292,332)(293,325)(294,327)(295,326)
(296,328)(297,321)(298,323)(299,322)(300,324)(301,317)(302,319)(303,318)
(304,320)(305,313)(306,315)(307,314)(308,316);
s2 := Sym(336)!( 1,145)( 2,148)( 3,147)( 4,146)( 5,141)( 6,144)( 7,143)
( 8,142)( 9,165)( 10,168)( 11,167)( 12,166)( 13,161)( 14,164)( 15,163)
( 16,162)( 17,157)( 18,160)( 19,159)( 20,158)( 21,153)( 22,156)( 23,155)
( 24,154)( 25,149)( 26,152)( 27,151)( 28,150)( 29,117)( 30,120)( 31,119)
( 32,118)( 33,113)( 34,116)( 35,115)( 36,114)( 37,137)( 38,140)( 39,139)
( 40,138)( 41,133)( 42,136)( 43,135)( 44,134)( 45,129)( 46,132)( 47,131)
( 48,130)( 49,125)( 50,128)( 51,127)( 52,126)( 53,121)( 54,124)( 55,123)
( 56,122)( 57, 89)( 58, 92)( 59, 91)( 60, 90)( 61, 85)( 62, 88)( 63, 87)
( 64, 86)( 65,109)( 66,112)( 67,111)( 68,110)( 69,105)( 70,108)( 71,107)
( 72,106)( 73,101)( 74,104)( 75,103)( 76,102)( 77, 97)( 78,100)( 79, 99)
( 80, 98)( 81, 93)( 82, 96)( 83, 95)( 84, 94)(169,313)(170,316)(171,315)
(172,314)(173,309)(174,312)(175,311)(176,310)(177,333)(178,336)(179,335)
(180,334)(181,329)(182,332)(183,331)(184,330)(185,325)(186,328)(187,327)
(188,326)(189,321)(190,324)(191,323)(192,322)(193,317)(194,320)(195,319)
(196,318)(197,285)(198,288)(199,287)(200,286)(201,281)(202,284)(203,283)
(204,282)(205,305)(206,308)(207,307)(208,306)(209,301)(210,304)(211,303)
(212,302)(213,297)(214,300)(215,299)(216,298)(217,293)(218,296)(219,295)
(220,294)(221,289)(222,292)(223,291)(224,290)(225,257)(226,260)(227,259)
(228,258)(229,253)(230,256)(231,255)(232,254)(233,277)(234,280)(235,279)
(236,278)(237,273)(238,276)(239,275)(240,274)(241,269)(242,272)(243,271)
(244,270)(245,265)(246,268)(247,267)(248,266)(249,261)(250,264)(251,263)
(252,262);
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*s2*s1*s2*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope