Polytope of Type {3,6,36}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,36}*1296a
if this polytope has a name.
Group : SmallGroup(1296,812)
Rank : 4
Schlafli Type : {3,6,36}
Number of vertices, edges, etc : 3, 9, 108, 36
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 6
Special Properties :
Universal
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 : {3,6,18}*648a
3-fold quotients : {3,2,36}*432, {3,6,12}*432a
4-fold quotients : {3,6,9}*324
6-fold quotients : {3,2,18}*216, {3,6,6}*216a
9-fold quotients : {3,2,12}*144
12-fold quotients : {3,2,9}*108, {3,6,3}*108
18-fold quotients : {3,2,6}*72
27-fold quotients : {3,2,4}*48
36-fold quotients : {3,2,3}*36
54-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 28, 55)( 29, 57)( 30, 56)( 31, 58)( 32, 60)( 33, 59)( 34, 61)
( 35, 63)( 36, 62)( 37, 64)( 38, 66)( 39, 65)( 40, 67)( 41, 69)( 42, 68)
( 43, 70)( 44, 72)( 45, 71)( 46, 73)( 47, 75)( 48, 74)( 49, 76)( 50, 78)
( 51, 77)( 52, 79)( 53, 81)( 54, 80)( 83, 84)( 86, 87)( 89, 90)( 92, 93)
( 95, 96)( 98, 99)(101,102)(104,105)(107,108)(109,136)(110,138)(111,137)
(112,139)(113,141)(114,140)(115,142)(116,144)(117,143)(118,145)(119,147)
(120,146)(121,148)(122,150)(123,149)(124,151)(125,153)(126,152)(127,154)
(128,156)(129,155)(130,157)(131,159)(132,158)(133,160)(134,162)(135,161)
(164,165)(167,168)(170,171)(173,174)(176,177)(179,180)(182,183)(185,186)
(188,189)(190,217)(191,219)(192,218)(193,220)(194,222)(195,221)(196,223)
(197,225)(198,224)(199,226)(200,228)(201,227)(202,229)(203,231)(204,230)
(205,232)(206,234)(207,233)(208,235)(209,237)(210,236)(211,238)(212,240)
(213,239)(214,241)(215,243)(216,242)(245,246)(248,249)(251,252)(254,255)
(257,258)(260,261)(263,264)(266,267)(269,270)(271,298)(272,300)(273,299)
(274,301)(275,303)(276,302)(277,304)(278,306)(279,305)(280,307)(281,309)
(282,308)(283,310)(284,312)(285,311)(286,313)(287,315)(288,314)(289,316)
(290,318)(291,317)(292,319)(293,321)(294,320)(295,322)(296,324)(297,323);;
s1 := ( 1, 28)( 2, 30)( 3, 29)( 4, 31)( 5, 33)( 6, 32)( 7, 34)( 8, 36)
( 9, 35)( 10, 38)( 11, 37)( 12, 39)( 13, 41)( 14, 40)( 15, 42)( 16, 44)
( 17, 43)( 18, 45)( 19, 48)( 20, 47)( 21, 46)( 22, 51)( 23, 50)( 24, 49)
( 25, 54)( 26, 53)( 27, 52)( 56, 57)( 59, 60)( 62, 63)( 64, 65)( 67, 68)
( 70, 71)( 73, 75)( 76, 78)( 79, 81)( 82,109)( 83,111)( 84,110)( 85,112)
( 86,114)( 87,113)( 88,115)( 89,117)( 90,116)( 91,119)( 92,118)( 93,120)
( 94,122)( 95,121)( 96,123)( 97,125)( 98,124)( 99,126)(100,129)(101,128)
(102,127)(103,132)(104,131)(105,130)(106,135)(107,134)(108,133)(137,138)
(140,141)(143,144)(145,146)(148,149)(151,152)(154,156)(157,159)(160,162)
(163,190)(164,192)(165,191)(166,193)(167,195)(168,194)(169,196)(170,198)
(171,197)(172,200)(173,199)(174,201)(175,203)(176,202)(177,204)(178,206)
(179,205)(180,207)(181,210)(182,209)(183,208)(184,213)(185,212)(186,211)
(187,216)(188,215)(189,214)(218,219)(221,222)(224,225)(226,227)(229,230)
(232,233)(235,237)(238,240)(241,243)(244,271)(245,273)(246,272)(247,274)
(248,276)(249,275)(250,277)(251,279)(252,278)(253,281)(254,280)(255,282)
(256,284)(257,283)(258,285)(259,287)(260,286)(261,288)(262,291)(263,290)
(264,289)(265,294)(266,293)(267,292)(268,297)(269,296)(270,295)(299,300)
(302,303)(305,306)(307,308)(310,311)(313,314)(316,318)(319,321)(322,324);;
s2 := ( 1, 10)( 2, 12)( 3, 11)( 4, 16)( 5, 18)( 6, 17)( 7, 13)( 8, 15)
( 9, 14)( 19, 25)( 20, 27)( 21, 26)( 23, 24)( 28, 37)( 29, 39)( 30, 38)
( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 52)( 47, 54)
( 48, 53)( 50, 51)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)
( 61, 67)( 62, 69)( 63, 68)( 73, 79)( 74, 81)( 75, 80)( 77, 78)( 82, 91)
( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)( 90, 95)
(100,106)(101,108)(102,107)(104,105)(109,118)(110,120)(111,119)(112,124)
(113,126)(114,125)(115,121)(116,123)(117,122)(127,133)(128,135)(129,134)
(131,132)(136,145)(137,147)(138,146)(139,151)(140,153)(141,152)(142,148)
(143,150)(144,149)(154,160)(155,162)(156,161)(158,159)(163,253)(164,255)
(165,254)(166,259)(167,261)(168,260)(169,256)(170,258)(171,257)(172,244)
(173,246)(174,245)(175,250)(176,252)(177,251)(178,247)(179,249)(180,248)
(181,268)(182,270)(183,269)(184,265)(185,267)(186,266)(187,262)(188,264)
(189,263)(190,280)(191,282)(192,281)(193,286)(194,288)(195,287)(196,283)
(197,285)(198,284)(199,271)(200,273)(201,272)(202,277)(203,279)(204,278)
(205,274)(206,276)(207,275)(208,295)(209,297)(210,296)(211,292)(212,294)
(213,293)(214,289)(215,291)(216,290)(217,307)(218,309)(219,308)(220,313)
(221,315)(222,314)(223,310)(224,312)(225,311)(226,298)(227,300)(228,299)
(229,304)(230,306)(231,305)(232,301)(233,303)(234,302)(235,322)(236,324)
(237,323)(238,319)(239,321)(240,320)(241,316)(242,318)(243,317);;
s3 := ( 1,163)( 2,165)( 3,164)( 4,169)( 5,171)( 6,170)( 7,166)( 8,168)
( 9,167)( 10,187)( 11,189)( 12,188)( 13,184)( 14,186)( 15,185)( 16,181)
( 17,183)( 18,182)( 19,178)( 20,180)( 21,179)( 22,175)( 23,177)( 24,176)
( 25,172)( 26,174)( 27,173)( 28,190)( 29,192)( 30,191)( 31,196)( 32,198)
( 33,197)( 34,193)( 35,195)( 36,194)( 37,214)( 38,216)( 39,215)( 40,211)
( 41,213)( 42,212)( 43,208)( 44,210)( 45,209)( 46,205)( 47,207)( 48,206)
( 49,202)( 50,204)( 51,203)( 52,199)( 53,201)( 54,200)( 55,217)( 56,219)
( 57,218)( 58,223)( 59,225)( 60,224)( 61,220)( 62,222)( 63,221)( 64,241)
( 65,243)( 66,242)( 67,238)( 68,240)( 69,239)( 70,235)( 71,237)( 72,236)
( 73,232)( 74,234)( 75,233)( 76,229)( 77,231)( 78,230)( 79,226)( 80,228)
( 81,227)( 82,244)( 83,246)( 84,245)( 85,250)( 86,252)( 87,251)( 88,247)
( 89,249)( 90,248)( 91,268)( 92,270)( 93,269)( 94,265)( 95,267)( 96,266)
( 97,262)( 98,264)( 99,263)(100,259)(101,261)(102,260)(103,256)(104,258)
(105,257)(106,253)(107,255)(108,254)(109,271)(110,273)(111,272)(112,277)
(113,279)(114,278)(115,274)(116,276)(117,275)(118,295)(119,297)(120,296)
(121,292)(122,294)(123,293)(124,289)(125,291)(126,290)(127,286)(128,288)
(129,287)(130,283)(131,285)(132,284)(133,280)(134,282)(135,281)(136,298)
(137,300)(138,299)(139,304)(140,306)(141,305)(142,301)(143,303)(144,302)
(145,322)(146,324)(147,323)(148,319)(149,321)(150,320)(151,316)(152,318)
(153,317)(154,313)(155,315)(156,314)(157,310)(158,312)(159,311)(160,307)
(161,309)(162,308);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(324)!( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 23, 24)( 26, 27)( 28, 55)( 29, 57)( 30, 56)( 31, 58)( 32, 60)( 33, 59)
( 34, 61)( 35, 63)( 36, 62)( 37, 64)( 38, 66)( 39, 65)( 40, 67)( 41, 69)
( 42, 68)( 43, 70)( 44, 72)( 45, 71)( 46, 73)( 47, 75)( 48, 74)( 49, 76)
( 50, 78)( 51, 77)( 52, 79)( 53, 81)( 54, 80)( 83, 84)( 86, 87)( 89, 90)
( 92, 93)( 95, 96)( 98, 99)(101,102)(104,105)(107,108)(109,136)(110,138)
(111,137)(112,139)(113,141)(114,140)(115,142)(116,144)(117,143)(118,145)
(119,147)(120,146)(121,148)(122,150)(123,149)(124,151)(125,153)(126,152)
(127,154)(128,156)(129,155)(130,157)(131,159)(132,158)(133,160)(134,162)
(135,161)(164,165)(167,168)(170,171)(173,174)(176,177)(179,180)(182,183)
(185,186)(188,189)(190,217)(191,219)(192,218)(193,220)(194,222)(195,221)
(196,223)(197,225)(198,224)(199,226)(200,228)(201,227)(202,229)(203,231)
(204,230)(205,232)(206,234)(207,233)(208,235)(209,237)(210,236)(211,238)
(212,240)(213,239)(214,241)(215,243)(216,242)(245,246)(248,249)(251,252)
(254,255)(257,258)(260,261)(263,264)(266,267)(269,270)(271,298)(272,300)
(273,299)(274,301)(275,303)(276,302)(277,304)(278,306)(279,305)(280,307)
(281,309)(282,308)(283,310)(284,312)(285,311)(286,313)(287,315)(288,314)
(289,316)(290,318)(291,317)(292,319)(293,321)(294,320)(295,322)(296,324)
(297,323);
s1 := Sym(324)!( 1, 28)( 2, 30)( 3, 29)( 4, 31)( 5, 33)( 6, 32)( 7, 34)
( 8, 36)( 9, 35)( 10, 38)( 11, 37)( 12, 39)( 13, 41)( 14, 40)( 15, 42)
( 16, 44)( 17, 43)( 18, 45)( 19, 48)( 20, 47)( 21, 46)( 22, 51)( 23, 50)
( 24, 49)( 25, 54)( 26, 53)( 27, 52)( 56, 57)( 59, 60)( 62, 63)( 64, 65)
( 67, 68)( 70, 71)( 73, 75)( 76, 78)( 79, 81)( 82,109)( 83,111)( 84,110)
( 85,112)( 86,114)( 87,113)( 88,115)( 89,117)( 90,116)( 91,119)( 92,118)
( 93,120)( 94,122)( 95,121)( 96,123)( 97,125)( 98,124)( 99,126)(100,129)
(101,128)(102,127)(103,132)(104,131)(105,130)(106,135)(107,134)(108,133)
(137,138)(140,141)(143,144)(145,146)(148,149)(151,152)(154,156)(157,159)
(160,162)(163,190)(164,192)(165,191)(166,193)(167,195)(168,194)(169,196)
(170,198)(171,197)(172,200)(173,199)(174,201)(175,203)(176,202)(177,204)
(178,206)(179,205)(180,207)(181,210)(182,209)(183,208)(184,213)(185,212)
(186,211)(187,216)(188,215)(189,214)(218,219)(221,222)(224,225)(226,227)
(229,230)(232,233)(235,237)(238,240)(241,243)(244,271)(245,273)(246,272)
(247,274)(248,276)(249,275)(250,277)(251,279)(252,278)(253,281)(254,280)
(255,282)(256,284)(257,283)(258,285)(259,287)(260,286)(261,288)(262,291)
(263,290)(264,289)(265,294)(266,293)(267,292)(268,297)(269,296)(270,295)
(299,300)(302,303)(305,306)(307,308)(310,311)(313,314)(316,318)(319,321)
(322,324);
s2 := Sym(324)!( 1, 10)( 2, 12)( 3, 11)( 4, 16)( 5, 18)( 6, 17)( 7, 13)
( 8, 15)( 9, 14)( 19, 25)( 20, 27)( 21, 26)( 23, 24)( 28, 37)( 29, 39)
( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 52)
( 47, 54)( 48, 53)( 50, 51)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)
( 60, 71)( 61, 67)( 62, 69)( 63, 68)( 73, 79)( 74, 81)( 75, 80)( 77, 78)
( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)
( 90, 95)(100,106)(101,108)(102,107)(104,105)(109,118)(110,120)(111,119)
(112,124)(113,126)(114,125)(115,121)(116,123)(117,122)(127,133)(128,135)
(129,134)(131,132)(136,145)(137,147)(138,146)(139,151)(140,153)(141,152)
(142,148)(143,150)(144,149)(154,160)(155,162)(156,161)(158,159)(163,253)
(164,255)(165,254)(166,259)(167,261)(168,260)(169,256)(170,258)(171,257)
(172,244)(173,246)(174,245)(175,250)(176,252)(177,251)(178,247)(179,249)
(180,248)(181,268)(182,270)(183,269)(184,265)(185,267)(186,266)(187,262)
(188,264)(189,263)(190,280)(191,282)(192,281)(193,286)(194,288)(195,287)
(196,283)(197,285)(198,284)(199,271)(200,273)(201,272)(202,277)(203,279)
(204,278)(205,274)(206,276)(207,275)(208,295)(209,297)(210,296)(211,292)
(212,294)(213,293)(214,289)(215,291)(216,290)(217,307)(218,309)(219,308)
(220,313)(221,315)(222,314)(223,310)(224,312)(225,311)(226,298)(227,300)
(228,299)(229,304)(230,306)(231,305)(232,301)(233,303)(234,302)(235,322)
(236,324)(237,323)(238,319)(239,321)(240,320)(241,316)(242,318)(243,317);
s3 := Sym(324)!( 1,163)( 2,165)( 3,164)( 4,169)( 5,171)( 6,170)( 7,166)
( 8,168)( 9,167)( 10,187)( 11,189)( 12,188)( 13,184)( 14,186)( 15,185)
( 16,181)( 17,183)( 18,182)( 19,178)( 20,180)( 21,179)( 22,175)( 23,177)
( 24,176)( 25,172)( 26,174)( 27,173)( 28,190)( 29,192)( 30,191)( 31,196)
( 32,198)( 33,197)( 34,193)( 35,195)( 36,194)( 37,214)( 38,216)( 39,215)
( 40,211)( 41,213)( 42,212)( 43,208)( 44,210)( 45,209)( 46,205)( 47,207)
( 48,206)( 49,202)( 50,204)( 51,203)( 52,199)( 53,201)( 54,200)( 55,217)
( 56,219)( 57,218)( 58,223)( 59,225)( 60,224)( 61,220)( 62,222)( 63,221)
( 64,241)( 65,243)( 66,242)( 67,238)( 68,240)( 69,239)( 70,235)( 71,237)
( 72,236)( 73,232)( 74,234)( 75,233)( 76,229)( 77,231)( 78,230)( 79,226)
( 80,228)( 81,227)( 82,244)( 83,246)( 84,245)( 85,250)( 86,252)( 87,251)
( 88,247)( 89,249)( 90,248)( 91,268)( 92,270)( 93,269)( 94,265)( 95,267)
( 96,266)( 97,262)( 98,264)( 99,263)(100,259)(101,261)(102,260)(103,256)
(104,258)(105,257)(106,253)(107,255)(108,254)(109,271)(110,273)(111,272)
(112,277)(113,279)(114,278)(115,274)(116,276)(117,275)(118,295)(119,297)
(120,296)(121,292)(122,294)(123,293)(124,289)(125,291)(126,290)(127,286)
(128,288)(129,287)(130,283)(131,285)(132,284)(133,280)(134,282)(135,281)
(136,298)(137,300)(138,299)(139,304)(140,306)(141,305)(142,301)(143,303)
(144,302)(145,322)(146,324)(147,323)(148,319)(149,321)(150,320)(151,316)
(152,318)(153,317)(154,313)(155,315)(156,314)(157,310)(158,312)(159,311)
(160,307)(161,309)(162,308);
poly := sub<Sym(324)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope