Polytope of Type {4,84}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,84}*1344c
if this polytope has a name.
Group : SmallGroup(1344,11399)
Rank : 3
Schlafli Type : {4,84}
Number of vertices, edges, etc : 8, 336, 168
Order of s₀s₁s₂ : 42
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {4,42}*672
   4-fold quotients : {4,21}*336, {4,42}*336b, {4,42}*336c
   7-fold quotients : {4,12}*192c
   8-fold quotients : {4,21}*168, {2,42}*168
   14-fold quotients : {4,6}*96
   16-fold quotients : {2,21}*84
   24-fold quotients : {2,14}*56
   28-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   48-fold quotients : {2,7}*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,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,255)(170,256)(171,253)(172,254)
(173,259)(174,260)(175,257)(176,258)(177,263)(178,264)(179,261)(180,262)
(181,267)(182,268)(183,265)(184,266)(185,271)(186,272)(187,269)(188,270)
(189,275)(190,276)(191,273)(192,274)(193,279)(194,280)(195,277)(196,278)
(197,283)(198,284)(199,281)(200,282)(201,287)(202,288)(203,285)(204,286)
(205,291)(206,292)(207,289)(208,290)(209,295)(210,296)(211,293)(212,294)
(213,299)(214,300)(215,297)(216,298)(217,303)(218,304)(219,301)(220,302)
(221,307)(222,308)(223,305)(224,306)(225,311)(226,312)(227,309)(228,310)
(229,315)(230,316)(231,313)(232,314)(233,319)(234,320)(235,317)(236,318)
(237,323)(238,324)(239,321)(240,322)(241,327)(242,328)(243,325)(244,326)
(245,331)(246,332)(247,329)(248,330)(249,335)(250,336)(251,333)(252,334);;
s1 := (  1,169)(  2,170)(  3,172)(  4,171)(  5,193)(  6,194)(  7,196)(  8,195)
(  9,189)( 10,190)( 11,192)( 12,191)( 13,185)( 14,186)( 15,188)( 16,187)
( 17,181)( 18,182)( 19,184)( 20,183)( 21,177)( 22,178)( 23,180)( 24,179)
( 25,173)( 26,174)( 27,176)( 28,175)( 29,225)( 30,226)( 31,228)( 32,227)
( 33,249)( 34,250)( 35,252)( 36,251)( 37,245)( 38,246)( 39,248)( 40,247)
( 41,241)( 42,242)( 43,244)( 44,243)( 45,237)( 46,238)( 47,240)( 48,239)
( 49,233)( 50,234)( 51,236)( 52,235)( 53,229)( 54,230)( 55,232)( 56,231)
( 57,197)( 58,198)( 59,200)( 60,199)( 61,221)( 62,222)( 63,224)( 64,223)
( 65,217)( 66,218)( 67,220)( 68,219)( 69,213)( 70,214)( 71,216)( 72,215)
( 73,209)( 74,210)( 75,212)( 76,211)( 77,205)( 78,206)( 79,208)( 80,207)
( 81,201)( 82,202)( 83,204)( 84,203)( 85,253)( 86,254)( 87,256)( 88,255)
( 89,277)( 90,278)( 91,280)( 92,279)( 93,273)( 94,274)( 95,276)( 96,275)
( 97,269)( 98,270)( 99,272)(100,271)(101,265)(102,266)(103,268)(104,267)
(105,261)(106,262)(107,264)(108,263)(109,257)(110,258)(111,260)(112,259)
(113,309)(114,310)(115,312)(116,311)(117,333)(118,334)(119,336)(120,335)
(121,329)(122,330)(123,332)(124,331)(125,325)(126,326)(127,328)(128,327)
(129,321)(130,322)(131,324)(132,323)(133,317)(134,318)(135,320)(136,319)
(137,313)(138,314)(139,316)(140,315)(141,281)(142,282)(143,284)(144,283)
(145,305)(146,306)(147,308)(148,307)(149,301)(150,302)(151,304)(152,303)
(153,297)(154,298)(155,300)(156,299)(157,293)(158,294)(159,296)(160,295)
(161,289)(162,290)(163,292)(164,291)(165,285)(166,286)(167,288)(168,287);;
s2 := (  1, 33)(  2, 36)(  3, 35)(  4, 34)(  5, 29)(  6, 32)(  7, 31)(  8, 30)
(  9, 53)( 10, 56)( 11, 55)( 12, 54)( 13, 49)( 14, 52)( 15, 51)( 16, 50)
( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 21, 41)( 22, 44)( 23, 43)( 24, 42)
( 25, 37)( 26, 40)( 27, 39)( 28, 38)( 57, 61)( 58, 64)( 59, 63)( 60, 62)
( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 77)( 70, 80)( 71, 79)( 72, 78)
( 74, 76)( 85,117)( 86,120)( 87,119)( 88,118)( 89,113)( 90,116)( 91,115)
( 92,114)( 93,137)( 94,140)( 95,139)( 96,138)( 97,133)( 98,136)( 99,135)
(100,134)(101,129)(102,132)(103,131)(104,130)(105,125)(106,128)(107,127)
(108,126)(109,121)(110,124)(111,123)(112,122)(141,145)(142,148)(143,147)
(144,146)(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)(155,163)
(156,162)(158,160)(169,285)(170,288)(171,287)(172,286)(173,281)(174,284)
(175,283)(176,282)(177,305)(178,308)(179,307)(180,306)(181,301)(182,304)
(183,303)(184,302)(185,297)(186,300)(187,299)(188,298)(189,293)(190,296)
(191,295)(192,294)(193,289)(194,292)(195,291)(196,290)(197,257)(198,260)
(199,259)(200,258)(201,253)(202,256)(203,255)(204,254)(205,277)(206,280)
(207,279)(208,278)(209,273)(210,276)(211,275)(212,274)(213,269)(214,272)
(215,271)(216,270)(217,265)(218,268)(219,267)(220,266)(221,261)(222,264)
(223,263)(224,262)(225,313)(226,316)(227,315)(228,314)(229,309)(230,312)
(231,311)(232,310)(233,333)(234,336)(235,335)(236,334)(237,329)(238,332)
(239,331)(240,330)(241,325)(242,328)(243,327)(244,326)(245,321)(246,324)
(247,323)(248,322)(249,317)(250,320)(251,319)(252,318);;
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*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1, 
s0*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*s0*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope