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