Polytope of Type {162,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {162,4}*1296a
Also Known As : {162,4|2}. if this polytope has another name.
Group : SmallGroup(1296,67)
Rank : 3
Schlafli Type : {162,4}
Number of vertices, edges, etc : 162, 324, 4
Order of s0s1s2 : 324
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 : {162,2}*648
3-fold quotients : {54,4}*432a
4-fold quotients : {81,2}*324
6-fold quotients : {54,2}*216
9-fold quotients : {18,4}*144a
12-fold quotients : {27,2}*108
18-fold quotients : {18,2}*72
27-fold quotients : {6,4}*48a
36-fold quotients : {9,2}*36
54-fold quotients : {6,2}*24
81-fold quotients : {2,4}*16
108-fold quotients : {3,2}*12
162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 9)( 5, 8)( 6, 7)( 10, 27)( 11, 26)( 12, 25)( 13, 24)
( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 28, 81)( 29, 80)( 30, 79)
( 31, 78)( 32, 77)( 33, 76)( 34, 75)( 35, 74)( 36, 73)( 37, 72)( 38, 71)
( 39, 70)( 40, 69)( 41, 68)( 42, 67)( 43, 66)( 44, 65)( 45, 64)( 46, 63)
( 47, 62)( 48, 61)( 49, 60)( 50, 59)( 51, 58)( 52, 57)( 53, 56)( 54, 55)
( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 91,108)( 92,107)( 93,106)( 94,105)
( 95,104)( 96,103)( 97,102)( 98,101)( 99,100)(109,162)(110,161)(111,160)
(112,159)(113,158)(114,157)(115,156)(116,155)(117,154)(118,153)(119,152)
(120,151)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,144)
(128,143)(129,142)(130,141)(131,140)(132,139)(133,138)(134,137)(135,136)
(164,165)(166,171)(167,170)(168,169)(172,189)(173,188)(174,187)(175,186)
(176,185)(177,184)(178,183)(179,182)(180,181)(190,243)(191,242)(192,241)
(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,234)(200,233)
(201,232)(202,231)(203,230)(204,229)(205,228)(206,227)(207,226)(208,225)
(209,224)(210,223)(211,222)(212,221)(213,220)(214,219)(215,218)(216,217)
(245,246)(247,252)(248,251)(249,250)(253,270)(254,269)(255,268)(256,267)
(257,266)(258,265)(259,264)(260,263)(261,262)(271,324)(272,323)(273,322)
(274,321)(275,320)(276,319)(277,318)(278,317)(279,316)(280,315)(281,314)
(282,313)(283,312)(284,311)(285,310)(286,309)(287,308)(288,307)(289,306)
(290,305)(291,304)(292,303)(293,302)(294,301)(295,300)(296,299)(297,298);;
s1 := ( 1, 28)( 2, 30)( 3, 29)( 4, 36)( 5, 35)( 6, 34)( 7, 33)( 8, 32)
( 9, 31)( 10, 54)( 11, 53)( 12, 52)( 13, 51)( 14, 50)( 15, 49)( 16, 48)
( 17, 47)( 18, 46)( 19, 45)( 20, 44)( 21, 43)( 22, 42)( 23, 41)( 24, 40)
( 25, 39)( 26, 38)( 27, 37)( 55, 81)( 56, 80)( 57, 79)( 58, 78)( 59, 77)
( 60, 76)( 61, 75)( 62, 74)( 63, 73)( 64, 72)( 65, 71)( 66, 70)( 67, 69)
( 82,109)( 83,111)( 84,110)( 85,117)( 86,116)( 87,115)( 88,114)( 89,113)
( 90,112)( 91,135)( 92,134)( 93,133)( 94,132)( 95,131)( 96,130)( 97,129)
( 98,128)( 99,127)(100,126)(101,125)(102,124)(103,123)(104,122)(105,121)
(106,120)(107,119)(108,118)(136,162)(137,161)(138,160)(139,159)(140,158)
(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150)
(163,271)(164,273)(165,272)(166,279)(167,278)(168,277)(169,276)(170,275)
(171,274)(172,297)(173,296)(174,295)(175,294)(176,293)(177,292)(178,291)
(179,290)(180,289)(181,288)(182,287)(183,286)(184,285)(185,284)(186,283)
(187,282)(188,281)(189,280)(190,244)(191,246)(192,245)(193,252)(194,251)
(195,250)(196,249)(197,248)(198,247)(199,270)(200,269)(201,268)(202,267)
(203,266)(204,265)(205,264)(206,263)(207,262)(208,261)(209,260)(210,259)
(211,258)(212,257)(213,256)(214,255)(215,254)(216,253)(217,324)(218,323)
(219,322)(220,321)(221,320)(222,319)(223,318)(224,317)(225,316)(226,315)
(227,314)(228,313)(229,312)(230,311)(231,310)(232,309)(233,308)(234,307)
(235,306)(236,305)(237,304)(238,303)(239,302)(240,301)(241,300)(242,299)
(243,298);;
s2 := ( 1,163)( 2,164)( 3,165)( 4,166)( 5,167)( 6,168)( 7,169)( 8,170)
( 9,171)( 10,172)( 11,173)( 12,174)( 13,175)( 14,176)( 15,177)( 16,178)
( 17,179)( 18,180)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)( 24,186)
( 25,187)( 26,188)( 27,189)( 28,190)( 29,191)( 30,192)( 31,193)( 32,194)
( 33,195)( 34,196)( 35,197)( 36,198)( 37,199)( 38,200)( 39,201)( 40,202)
( 41,203)( 42,204)( 43,205)( 44,206)( 45,207)( 46,208)( 47,209)( 48,210)
( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,217)( 56,218)
( 57,219)( 58,220)( 59,221)( 60,222)( 61,223)( 62,224)( 63,225)( 64,226)
( 65,227)( 66,228)( 67,229)( 68,230)( 69,231)( 70,232)( 71,233)( 72,234)
( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)( 80,242)
( 81,243)( 82,244)( 83,245)( 84,246)( 85,247)( 86,248)( 87,249)( 88,250)
( 89,251)( 90,252)( 91,253)( 92,254)( 93,255)( 94,256)( 95,257)( 96,258)
( 97,259)( 98,260)( 99,261)(100,262)(101,263)(102,264)(103,265)(104,266)
(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)(112,274)
(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)(120,282)
(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)(128,290)
(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,298)
(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)(144,306)
(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)(152,314)
(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)
(161,323)(162,324);;
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,
s1*s2*s1*s2*s1*s2*s1*s2, 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*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*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*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*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*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*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*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*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*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*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(324)!( 2, 3)( 4, 9)( 5, 8)( 6, 7)( 10, 27)( 11, 26)( 12, 25)
( 13, 24)( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 28, 81)( 29, 80)
( 30, 79)( 31, 78)( 32, 77)( 33, 76)( 34, 75)( 35, 74)( 36, 73)( 37, 72)
( 38, 71)( 39, 70)( 40, 69)( 41, 68)( 42, 67)( 43, 66)( 44, 65)( 45, 64)
( 46, 63)( 47, 62)( 48, 61)( 49, 60)( 50, 59)( 51, 58)( 52, 57)( 53, 56)
( 54, 55)( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 91,108)( 92,107)( 93,106)
( 94,105)( 95,104)( 96,103)( 97,102)( 98,101)( 99,100)(109,162)(110,161)
(111,160)(112,159)(113,158)(114,157)(115,156)(116,155)(117,154)(118,153)
(119,152)(120,151)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)
(127,144)(128,143)(129,142)(130,141)(131,140)(132,139)(133,138)(134,137)
(135,136)(164,165)(166,171)(167,170)(168,169)(172,189)(173,188)(174,187)
(175,186)(176,185)(177,184)(178,183)(179,182)(180,181)(190,243)(191,242)
(192,241)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,234)
(200,233)(201,232)(202,231)(203,230)(204,229)(205,228)(206,227)(207,226)
(208,225)(209,224)(210,223)(211,222)(212,221)(213,220)(214,219)(215,218)
(216,217)(245,246)(247,252)(248,251)(249,250)(253,270)(254,269)(255,268)
(256,267)(257,266)(258,265)(259,264)(260,263)(261,262)(271,324)(272,323)
(273,322)(274,321)(275,320)(276,319)(277,318)(278,317)(279,316)(280,315)
(281,314)(282,313)(283,312)(284,311)(285,310)(286,309)(287,308)(288,307)
(289,306)(290,305)(291,304)(292,303)(293,302)(294,301)(295,300)(296,299)
(297,298);
s1 := Sym(324)!( 1, 28)( 2, 30)( 3, 29)( 4, 36)( 5, 35)( 6, 34)( 7, 33)
( 8, 32)( 9, 31)( 10, 54)( 11, 53)( 12, 52)( 13, 51)( 14, 50)( 15, 49)
( 16, 48)( 17, 47)( 18, 46)( 19, 45)( 20, 44)( 21, 43)( 22, 42)( 23, 41)
( 24, 40)( 25, 39)( 26, 38)( 27, 37)( 55, 81)( 56, 80)( 57, 79)( 58, 78)
( 59, 77)( 60, 76)( 61, 75)( 62, 74)( 63, 73)( 64, 72)( 65, 71)( 66, 70)
( 67, 69)( 82,109)( 83,111)( 84,110)( 85,117)( 86,116)( 87,115)( 88,114)
( 89,113)( 90,112)( 91,135)( 92,134)( 93,133)( 94,132)( 95,131)( 96,130)
( 97,129)( 98,128)( 99,127)(100,126)(101,125)(102,124)(103,123)(104,122)
(105,121)(106,120)(107,119)(108,118)(136,162)(137,161)(138,160)(139,159)
(140,158)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)
(148,150)(163,271)(164,273)(165,272)(166,279)(167,278)(168,277)(169,276)
(170,275)(171,274)(172,297)(173,296)(174,295)(175,294)(176,293)(177,292)
(178,291)(179,290)(180,289)(181,288)(182,287)(183,286)(184,285)(185,284)
(186,283)(187,282)(188,281)(189,280)(190,244)(191,246)(192,245)(193,252)
(194,251)(195,250)(196,249)(197,248)(198,247)(199,270)(200,269)(201,268)
(202,267)(203,266)(204,265)(205,264)(206,263)(207,262)(208,261)(209,260)
(210,259)(211,258)(212,257)(213,256)(214,255)(215,254)(216,253)(217,324)
(218,323)(219,322)(220,321)(221,320)(222,319)(223,318)(224,317)(225,316)
(226,315)(227,314)(228,313)(229,312)(230,311)(231,310)(232,309)(233,308)
(234,307)(235,306)(236,305)(237,304)(238,303)(239,302)(240,301)(241,300)
(242,299)(243,298);
s2 := Sym(324)!( 1,163)( 2,164)( 3,165)( 4,166)( 5,167)( 6,168)( 7,169)
( 8,170)( 9,171)( 10,172)( 11,173)( 12,174)( 13,175)( 14,176)( 15,177)
( 16,178)( 17,179)( 18,180)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)
( 24,186)( 25,187)( 26,188)( 27,189)( 28,190)( 29,191)( 30,192)( 31,193)
( 32,194)( 33,195)( 34,196)( 35,197)( 36,198)( 37,199)( 38,200)( 39,201)
( 40,202)( 41,203)( 42,204)( 43,205)( 44,206)( 45,207)( 46,208)( 47,209)
( 48,210)( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,217)
( 56,218)( 57,219)( 58,220)( 59,221)( 60,222)( 61,223)( 62,224)( 63,225)
( 64,226)( 65,227)( 66,228)( 67,229)( 68,230)( 69,231)( 70,232)( 71,233)
( 72,234)( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)
( 80,242)( 81,243)( 82,244)( 83,245)( 84,246)( 85,247)( 86,248)( 87,249)
( 88,250)( 89,251)( 90,252)( 91,253)( 92,254)( 93,255)( 94,256)( 95,257)
( 96,258)( 97,259)( 98,260)( 99,261)(100,262)(101,263)(102,264)(103,265)
(104,266)(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)
(112,274)(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)
(120,282)(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)
(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)
(136,298)(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)
(144,306)(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)
(152,314)(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)
(160,322)(161,323)(162,324);
poly := sub<Sym(324)|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,
s1*s2*s1*s2*s1*s2*s1*s2, 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*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*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*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*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*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*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*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*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*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*s0*s1*s0*s1 >;
References : None.
to this polytope