Polytope of Type {16,42}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {16,42}*1344
Also Known As : {16,42|2}. if this polytope has another name.
Group : SmallGroup(1344,5836)
Rank : 3
Schlafli Type : {16,42}
Number of vertices, edges, etc : 16, 336, 42
Order of s0s1s2 : 336
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
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 : {8,42}*672
3-fold quotients : {16,14}*448
4-fold quotients : {4,42}*336a
6-fold quotients : {8,14}*224
7-fold quotients : {16,6}*192
8-fold quotients : {2,42}*168
12-fold quotients : {4,14}*112
14-fold quotients : {8,6}*96
16-fold quotients : {2,21}*84
21-fold quotients : {16,2}*64
24-fold quotients : {2,14}*56
28-fold quotients : {4,6}*48a
42-fold quotients : {8,2}*32
48-fold quotients : {2,7}*28
56-fold quotients : {2,6}*24
84-fold quotients : {4,2}*16
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 := ( 43, 64)( 44, 65)( 45, 66)( 46, 67)( 47, 68)( 48, 69)( 49, 70)( 50, 71)
( 51, 72)( 52, 73)( 53, 74)( 54, 75)( 55, 76)( 56, 77)( 57, 78)( 58, 79)
( 59, 80)( 60, 81)( 61, 82)( 62, 83)( 63, 84)( 85,127)( 86,128)( 87,129)
( 88,130)( 89,131)( 90,132)( 91,133)( 92,134)( 93,135)( 94,136)( 95,137)
( 96,138)( 97,139)( 98,140)( 99,141)(100,142)(101,143)(102,144)(103,145)
(104,146)(105,147)(106,148)(107,149)(108,150)(109,151)(110,152)(111,153)
(112,154)(113,155)(114,156)(115,157)(116,158)(117,159)(118,160)(119,161)
(120,162)(121,163)(122,164)(123,165)(124,166)(125,167)(126,168)(169,253)
(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)(176,260)(177,261)
(178,262)(179,263)(180,264)(181,265)(182,266)(183,267)(184,268)(185,269)
(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)(193,277)
(194,278)(195,279)(196,280)(197,281)(198,282)(199,283)(200,284)(201,285)
(202,286)(203,287)(204,288)(205,289)(206,290)(207,291)(208,292)(209,293)
(210,294)(211,316)(212,317)(213,318)(214,319)(215,320)(216,321)(217,322)
(218,323)(219,324)(220,325)(221,326)(222,327)(223,328)(224,329)(225,330)
(226,331)(227,332)(228,333)(229,334)(230,335)(231,336)(232,295)(233,296)
(234,297)(235,298)(236,299)(237,300)(238,301)(239,302)(240,303)(241,304)
(242,305)(243,306)(244,307)(245,308)(246,309)(247,310)(248,311)(249,312)
(250,313)(251,314)(252,315);;
s1 := ( 1,169)( 2,175)( 3,174)( 4,173)( 5,172)( 6,171)( 7,170)( 8,183)
( 9,189)( 10,188)( 11,187)( 12,186)( 13,185)( 14,184)( 15,176)( 16,182)
( 17,181)( 18,180)( 19,179)( 20,178)( 21,177)( 22,190)( 23,196)( 24,195)
( 25,194)( 26,193)( 27,192)( 28,191)( 29,204)( 30,210)( 31,209)( 32,208)
( 33,207)( 34,206)( 35,205)( 36,197)( 37,203)( 38,202)( 39,201)( 40,200)
( 41,199)( 42,198)( 43,232)( 44,238)( 45,237)( 46,236)( 47,235)( 48,234)
( 49,233)( 50,246)( 51,252)( 52,251)( 53,250)( 54,249)( 55,248)( 56,247)
( 57,239)( 58,245)( 59,244)( 60,243)( 61,242)( 62,241)( 63,240)( 64,211)
( 65,217)( 66,216)( 67,215)( 68,214)( 69,213)( 70,212)( 71,225)( 72,231)
( 73,230)( 74,229)( 75,228)( 76,227)( 77,226)( 78,218)( 79,224)( 80,223)
( 81,222)( 82,221)( 83,220)( 84,219)( 85,295)( 86,301)( 87,300)( 88,299)
( 89,298)( 90,297)( 91,296)( 92,309)( 93,315)( 94,314)( 95,313)( 96,312)
( 97,311)( 98,310)( 99,302)(100,308)(101,307)(102,306)(103,305)(104,304)
(105,303)(106,316)(107,322)(108,321)(109,320)(110,319)(111,318)(112,317)
(113,330)(114,336)(115,335)(116,334)(117,333)(118,332)(119,331)(120,323)
(121,329)(122,328)(123,327)(124,326)(125,325)(126,324)(127,253)(128,259)
(129,258)(130,257)(131,256)(132,255)(133,254)(134,267)(135,273)(136,272)
(137,271)(138,270)(139,269)(140,268)(141,260)(142,266)(143,265)(144,264)
(145,263)(146,262)(147,261)(148,274)(149,280)(150,279)(151,278)(152,277)
(153,276)(154,275)(155,288)(156,294)(157,293)(158,292)(159,291)(160,290)
(161,289)(162,281)(163,287)(164,286)(165,285)(166,284)(167,283)(168,282);;
s2 := ( 1, 9)( 2, 8)( 3, 14)( 4, 13)( 5, 12)( 6, 11)( 7, 10)( 15, 16)
( 17, 21)( 18, 20)( 22, 30)( 23, 29)( 24, 35)( 25, 34)( 26, 33)( 27, 32)
( 28, 31)( 36, 37)( 38, 42)( 39, 41)( 43, 51)( 44, 50)( 45, 56)( 46, 55)
( 47, 54)( 48, 53)( 49, 52)( 57, 58)( 59, 63)( 60, 62)( 64, 72)( 65, 71)
( 66, 77)( 67, 76)( 68, 75)( 69, 74)( 70, 73)( 78, 79)( 80, 84)( 81, 83)
( 85, 93)( 86, 92)( 87, 98)( 88, 97)( 89, 96)( 90, 95)( 91, 94)( 99,100)
(101,105)(102,104)(106,114)(107,113)(108,119)(109,118)(110,117)(111,116)
(112,115)(120,121)(122,126)(123,125)(127,135)(128,134)(129,140)(130,139)
(131,138)(132,137)(133,136)(141,142)(143,147)(144,146)(148,156)(149,155)
(150,161)(151,160)(152,159)(153,158)(154,157)(162,163)(164,168)(165,167)
(169,177)(170,176)(171,182)(172,181)(173,180)(174,179)(175,178)(183,184)
(185,189)(186,188)(190,198)(191,197)(192,203)(193,202)(194,201)(195,200)
(196,199)(204,205)(206,210)(207,209)(211,219)(212,218)(213,224)(214,223)
(215,222)(216,221)(217,220)(225,226)(227,231)(228,230)(232,240)(233,239)
(234,245)(235,244)(236,243)(237,242)(238,241)(246,247)(248,252)(249,251)
(253,261)(254,260)(255,266)(256,265)(257,264)(258,263)(259,262)(267,268)
(269,273)(270,272)(274,282)(275,281)(276,287)(277,286)(278,285)(279,284)
(280,283)(288,289)(290,294)(291,293)(295,303)(296,302)(297,308)(298,307)
(299,306)(300,305)(301,304)(309,310)(311,315)(312,314)(316,324)(317,323)
(318,329)(319,328)(320,327)(321,326)(322,325)(330,331)(332,336)(333,335);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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)!( 43, 64)( 44, 65)( 45, 66)( 46, 67)( 47, 68)( 48, 69)( 49, 70)
( 50, 71)( 51, 72)( 52, 73)( 53, 74)( 54, 75)( 55, 76)( 56, 77)( 57, 78)
( 58, 79)( 59, 80)( 60, 81)( 61, 82)( 62, 83)( 63, 84)( 85,127)( 86,128)
( 87,129)( 88,130)( 89,131)( 90,132)( 91,133)( 92,134)( 93,135)( 94,136)
( 95,137)( 96,138)( 97,139)( 98,140)( 99,141)(100,142)(101,143)(102,144)
(103,145)(104,146)(105,147)(106,148)(107,149)(108,150)(109,151)(110,152)
(111,153)(112,154)(113,155)(114,156)(115,157)(116,158)(117,159)(118,160)
(119,161)(120,162)(121,163)(122,164)(123,165)(124,166)(125,167)(126,168)
(169,253)(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)(176,260)
(177,261)(178,262)(179,263)(180,264)(181,265)(182,266)(183,267)(184,268)
(185,269)(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)
(193,277)(194,278)(195,279)(196,280)(197,281)(198,282)(199,283)(200,284)
(201,285)(202,286)(203,287)(204,288)(205,289)(206,290)(207,291)(208,292)
(209,293)(210,294)(211,316)(212,317)(213,318)(214,319)(215,320)(216,321)
(217,322)(218,323)(219,324)(220,325)(221,326)(222,327)(223,328)(224,329)
(225,330)(226,331)(227,332)(228,333)(229,334)(230,335)(231,336)(232,295)
(233,296)(234,297)(235,298)(236,299)(237,300)(238,301)(239,302)(240,303)
(241,304)(242,305)(243,306)(244,307)(245,308)(246,309)(247,310)(248,311)
(249,312)(250,313)(251,314)(252,315);
s1 := Sym(336)!( 1,169)( 2,175)( 3,174)( 4,173)( 5,172)( 6,171)( 7,170)
( 8,183)( 9,189)( 10,188)( 11,187)( 12,186)( 13,185)( 14,184)( 15,176)
( 16,182)( 17,181)( 18,180)( 19,179)( 20,178)( 21,177)( 22,190)( 23,196)
( 24,195)( 25,194)( 26,193)( 27,192)( 28,191)( 29,204)( 30,210)( 31,209)
( 32,208)( 33,207)( 34,206)( 35,205)( 36,197)( 37,203)( 38,202)( 39,201)
( 40,200)( 41,199)( 42,198)( 43,232)( 44,238)( 45,237)( 46,236)( 47,235)
( 48,234)( 49,233)( 50,246)( 51,252)( 52,251)( 53,250)( 54,249)( 55,248)
( 56,247)( 57,239)( 58,245)( 59,244)( 60,243)( 61,242)( 62,241)( 63,240)
( 64,211)( 65,217)( 66,216)( 67,215)( 68,214)( 69,213)( 70,212)( 71,225)
( 72,231)( 73,230)( 74,229)( 75,228)( 76,227)( 77,226)( 78,218)( 79,224)
( 80,223)( 81,222)( 82,221)( 83,220)( 84,219)( 85,295)( 86,301)( 87,300)
( 88,299)( 89,298)( 90,297)( 91,296)( 92,309)( 93,315)( 94,314)( 95,313)
( 96,312)( 97,311)( 98,310)( 99,302)(100,308)(101,307)(102,306)(103,305)
(104,304)(105,303)(106,316)(107,322)(108,321)(109,320)(110,319)(111,318)
(112,317)(113,330)(114,336)(115,335)(116,334)(117,333)(118,332)(119,331)
(120,323)(121,329)(122,328)(123,327)(124,326)(125,325)(126,324)(127,253)
(128,259)(129,258)(130,257)(131,256)(132,255)(133,254)(134,267)(135,273)
(136,272)(137,271)(138,270)(139,269)(140,268)(141,260)(142,266)(143,265)
(144,264)(145,263)(146,262)(147,261)(148,274)(149,280)(150,279)(151,278)
(152,277)(153,276)(154,275)(155,288)(156,294)(157,293)(158,292)(159,291)
(160,290)(161,289)(162,281)(163,287)(164,286)(165,285)(166,284)(167,283)
(168,282);
s2 := Sym(336)!( 1, 9)( 2, 8)( 3, 14)( 4, 13)( 5, 12)( 6, 11)( 7, 10)
( 15, 16)( 17, 21)( 18, 20)( 22, 30)( 23, 29)( 24, 35)( 25, 34)( 26, 33)
( 27, 32)( 28, 31)( 36, 37)( 38, 42)( 39, 41)( 43, 51)( 44, 50)( 45, 56)
( 46, 55)( 47, 54)( 48, 53)( 49, 52)( 57, 58)( 59, 63)( 60, 62)( 64, 72)
( 65, 71)( 66, 77)( 67, 76)( 68, 75)( 69, 74)( 70, 73)( 78, 79)( 80, 84)
( 81, 83)( 85, 93)( 86, 92)( 87, 98)( 88, 97)( 89, 96)( 90, 95)( 91, 94)
( 99,100)(101,105)(102,104)(106,114)(107,113)(108,119)(109,118)(110,117)
(111,116)(112,115)(120,121)(122,126)(123,125)(127,135)(128,134)(129,140)
(130,139)(131,138)(132,137)(133,136)(141,142)(143,147)(144,146)(148,156)
(149,155)(150,161)(151,160)(152,159)(153,158)(154,157)(162,163)(164,168)
(165,167)(169,177)(170,176)(171,182)(172,181)(173,180)(174,179)(175,178)
(183,184)(185,189)(186,188)(190,198)(191,197)(192,203)(193,202)(194,201)
(195,200)(196,199)(204,205)(206,210)(207,209)(211,219)(212,218)(213,224)
(214,223)(215,222)(216,221)(217,220)(225,226)(227,231)(228,230)(232,240)
(233,239)(234,245)(235,244)(236,243)(237,242)(238,241)(246,247)(248,252)
(249,251)(253,261)(254,260)(255,266)(256,265)(257,264)(258,263)(259,262)
(267,268)(269,273)(270,272)(274,282)(275,281)(276,287)(277,286)(278,285)
(279,284)(280,283)(288,289)(290,294)(291,293)(295,303)(296,302)(297,308)
(298,307)(299,306)(300,305)(301,304)(309,310)(311,315)(312,314)(316,324)
(317,323)(318,329)(319,328)(320,327)(321,326)(322,325)(330,331)(332,336)
(333,335);
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, s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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