Polytope of Type {36,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,18}*1296a
Also Known As : {36,18|2}. if this polytope has another name.
Group : SmallGroup(1296,798)
Rank : 3
Schlafli Type : {36,18}
Number of vertices, edges, etc : 36, 324, 18
Order of s0s1s2 : 36
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {18,18}*648a
3-fold quotients : {36,6}*432a, {12,18}*432a
6-fold quotients : {6,18}*216a, {18,6}*216a
9-fold quotients : {36,2}*144, {4,18}*144a, {12,6}*144a
18-fold quotients : {2,18}*72, {18,2}*72, {6,6}*72a
27-fold quotients : {12,2}*48, {4,6}*48a
36-fold quotients : {2,9}*36, {9,2}*36
54-fold quotients : {2,6}*24, {6,2}*24
81-fold quotients : {4,2}*16
108-fold quotients : {2,3}*12, {3,2}*12
162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 10, 21)( 11, 20)( 12, 19)( 13, 24)( 14, 23)
( 15, 22)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 32, 33)( 35, 36)( 37, 48)
( 38, 47)( 39, 46)( 40, 51)( 41, 50)( 42, 49)( 43, 54)( 44, 53)( 45, 52)
( 56, 57)( 59, 60)( 62, 63)( 64, 75)( 65, 74)( 66, 73)( 67, 78)( 68, 77)
( 69, 76)( 70, 81)( 71, 80)( 72, 79)( 83, 84)( 86, 87)( 89, 90)( 91,102)
( 92,101)( 93,100)( 94,105)( 95,104)( 96,103)( 97,108)( 98,107)( 99,106)
(110,111)(113,114)(116,117)(118,129)(119,128)(120,127)(121,132)(122,131)
(123,130)(124,135)(125,134)(126,133)(137,138)(140,141)(143,144)(145,156)
(146,155)(147,154)(148,159)(149,158)(150,157)(151,162)(152,161)(153,160)
(163,244)(164,246)(165,245)(166,247)(167,249)(168,248)(169,250)(170,252)
(171,251)(172,264)(173,263)(174,262)(175,267)(176,266)(177,265)(178,270)
(179,269)(180,268)(181,255)(182,254)(183,253)(184,258)(185,257)(186,256)
(187,261)(188,260)(189,259)(190,271)(191,273)(192,272)(193,274)(194,276)
(195,275)(196,277)(197,279)(198,278)(199,291)(200,290)(201,289)(202,294)
(203,293)(204,292)(205,297)(206,296)(207,295)(208,282)(209,281)(210,280)
(211,285)(212,284)(213,283)(214,288)(215,287)(216,286)(217,298)(218,300)
(219,299)(220,301)(221,303)(222,302)(223,304)(224,306)(225,305)(226,318)
(227,317)(228,316)(229,321)(230,320)(231,319)(232,324)(233,323)(234,322)
(235,309)(236,308)(237,307)(238,312)(239,311)(240,310)(241,315)(242,314)
(243,313);;
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)( 19,183)( 20,182)( 21,181)( 22,189)( 23,188)( 24,187)
( 25,186)( 26,185)( 27,184)( 28,232)( 29,234)( 30,233)( 31,229)( 32,231)
( 33,230)( 34,226)( 35,228)( 36,227)( 37,223)( 38,225)( 39,224)( 40,220)
( 41,222)( 42,221)( 43,217)( 44,219)( 45,218)( 46,243)( 47,242)( 48,241)
( 49,240)( 50,239)( 51,238)( 52,237)( 53,236)( 54,235)( 55,205)( 56,207)
( 57,206)( 58,202)( 59,204)( 60,203)( 61,199)( 62,201)( 63,200)( 64,196)
( 65,198)( 66,197)( 67,193)( 68,195)( 69,194)( 70,190)( 71,192)( 72,191)
( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,211)( 79,210)( 80,209)
( 81,208)( 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)(100,264)(101,263)(102,262)(103,270)(104,269)
(105,268)(106,267)(107,266)(108,265)(109,313)(110,315)(111,314)(112,310)
(113,312)(114,311)(115,307)(116,309)(117,308)(118,304)(119,306)(120,305)
(121,301)(122,303)(123,302)(124,298)(125,300)(126,299)(127,324)(128,323)
(129,322)(130,321)(131,320)(132,319)(133,318)(134,317)(135,316)(136,286)
(137,288)(138,287)(139,283)(140,285)(141,284)(142,280)(143,282)(144,281)
(145,277)(146,279)(147,278)(148,274)(149,276)(150,275)(151,271)(152,273)
(153,272)(154,297)(155,296)(156,295)(157,294)(158,293)(159,292)(160,291)
(161,290)(162,289);;
s2 := ( 1, 28)( 2, 29)( 3, 30)( 4, 34)( 5, 35)( 6, 36)( 7, 31)( 8, 32)
( 9, 33)( 10, 37)( 11, 38)( 12, 39)( 13, 43)( 14, 44)( 15, 45)( 16, 40)
( 17, 41)( 18, 42)( 19, 46)( 20, 47)( 21, 48)( 22, 52)( 23, 53)( 24, 54)
( 25, 49)( 26, 50)( 27, 51)( 55, 61)( 56, 62)( 57, 63)( 64, 70)( 65, 71)
( 66, 72)( 73, 79)( 74, 80)( 75, 81)( 82,109)( 83,110)( 84,111)( 85,115)
( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)( 91,118)( 92,119)( 93,120)
( 94,124)( 95,125)( 96,126)( 97,121)( 98,122)( 99,123)(100,127)(101,128)
(102,129)(103,133)(104,134)(105,135)(106,130)(107,131)(108,132)(136,142)
(137,143)(138,144)(145,151)(146,152)(147,153)(154,160)(155,161)(156,162)
(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)
(171,195)(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)
(179,203)(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)
(187,211)(188,212)(189,213)(217,223)(218,224)(219,225)(226,232)(227,233)
(228,234)(235,241)(236,242)(237,243)(244,271)(245,272)(246,273)(247,277)
(248,278)(249,279)(250,274)(251,275)(252,276)(253,280)(254,281)(255,282)
(256,286)(257,287)(258,288)(259,283)(260,284)(261,285)(262,289)(263,290)
(264,291)(265,295)(266,296)(267,297)(268,292)(269,293)(270,294)(298,304)
(299,305)(300,306)(307,313)(308,314)(309,315)(316,322)(317,323)(318,324);;
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*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)( 5, 6)( 8, 9)( 10, 21)( 11, 20)( 12, 19)( 13, 24)
( 14, 23)( 15, 22)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 32, 33)( 35, 36)
( 37, 48)( 38, 47)( 39, 46)( 40, 51)( 41, 50)( 42, 49)( 43, 54)( 44, 53)
( 45, 52)( 56, 57)( 59, 60)( 62, 63)( 64, 75)( 65, 74)( 66, 73)( 67, 78)
( 68, 77)( 69, 76)( 70, 81)( 71, 80)( 72, 79)( 83, 84)( 86, 87)( 89, 90)
( 91,102)( 92,101)( 93,100)( 94,105)( 95,104)( 96,103)( 97,108)( 98,107)
( 99,106)(110,111)(113,114)(116,117)(118,129)(119,128)(120,127)(121,132)
(122,131)(123,130)(124,135)(125,134)(126,133)(137,138)(140,141)(143,144)
(145,156)(146,155)(147,154)(148,159)(149,158)(150,157)(151,162)(152,161)
(153,160)(163,244)(164,246)(165,245)(166,247)(167,249)(168,248)(169,250)
(170,252)(171,251)(172,264)(173,263)(174,262)(175,267)(176,266)(177,265)
(178,270)(179,269)(180,268)(181,255)(182,254)(183,253)(184,258)(185,257)
(186,256)(187,261)(188,260)(189,259)(190,271)(191,273)(192,272)(193,274)
(194,276)(195,275)(196,277)(197,279)(198,278)(199,291)(200,290)(201,289)
(202,294)(203,293)(204,292)(205,297)(206,296)(207,295)(208,282)(209,281)
(210,280)(211,285)(212,284)(213,283)(214,288)(215,287)(216,286)(217,298)
(218,300)(219,299)(220,301)(221,303)(222,302)(223,304)(224,306)(225,305)
(226,318)(227,317)(228,316)(229,321)(230,320)(231,319)(232,324)(233,323)
(234,322)(235,309)(236,308)(237,307)(238,312)(239,311)(240,310)(241,315)
(242,314)(243,313);
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)( 19,183)( 20,182)( 21,181)( 22,189)( 23,188)
( 24,187)( 25,186)( 26,185)( 27,184)( 28,232)( 29,234)( 30,233)( 31,229)
( 32,231)( 33,230)( 34,226)( 35,228)( 36,227)( 37,223)( 38,225)( 39,224)
( 40,220)( 41,222)( 42,221)( 43,217)( 44,219)( 45,218)( 46,243)( 47,242)
( 48,241)( 49,240)( 50,239)( 51,238)( 52,237)( 53,236)( 54,235)( 55,205)
( 56,207)( 57,206)( 58,202)( 59,204)( 60,203)( 61,199)( 62,201)( 63,200)
( 64,196)( 65,198)( 66,197)( 67,193)( 68,195)( 69,194)( 70,190)( 71,192)
( 72,191)( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,211)( 79,210)
( 80,209)( 81,208)( 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)(100,264)(101,263)(102,262)(103,270)
(104,269)(105,268)(106,267)(107,266)(108,265)(109,313)(110,315)(111,314)
(112,310)(113,312)(114,311)(115,307)(116,309)(117,308)(118,304)(119,306)
(120,305)(121,301)(122,303)(123,302)(124,298)(125,300)(126,299)(127,324)
(128,323)(129,322)(130,321)(131,320)(132,319)(133,318)(134,317)(135,316)
(136,286)(137,288)(138,287)(139,283)(140,285)(141,284)(142,280)(143,282)
(144,281)(145,277)(146,279)(147,278)(148,274)(149,276)(150,275)(151,271)
(152,273)(153,272)(154,297)(155,296)(156,295)(157,294)(158,293)(159,292)
(160,291)(161,290)(162,289);
s2 := Sym(324)!( 1, 28)( 2, 29)( 3, 30)( 4, 34)( 5, 35)( 6, 36)( 7, 31)
( 8, 32)( 9, 33)( 10, 37)( 11, 38)( 12, 39)( 13, 43)( 14, 44)( 15, 45)
( 16, 40)( 17, 41)( 18, 42)( 19, 46)( 20, 47)( 21, 48)( 22, 52)( 23, 53)
( 24, 54)( 25, 49)( 26, 50)( 27, 51)( 55, 61)( 56, 62)( 57, 63)( 64, 70)
( 65, 71)( 66, 72)( 73, 79)( 74, 80)( 75, 81)( 82,109)( 83,110)( 84,111)
( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)( 91,118)( 92,119)
( 93,120)( 94,124)( 95,125)( 96,126)( 97,121)( 98,122)( 99,123)(100,127)
(101,128)(102,129)(103,133)(104,134)(105,135)(106,130)(107,131)(108,132)
(136,142)(137,143)(138,144)(145,151)(146,152)(147,153)(154,160)(155,161)
(156,162)(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)
(170,194)(171,195)(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)
(178,202)(179,203)(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)
(186,216)(187,211)(188,212)(189,213)(217,223)(218,224)(219,225)(226,232)
(227,233)(228,234)(235,241)(236,242)(237,243)(244,271)(245,272)(246,273)
(247,277)(248,278)(249,279)(250,274)(251,275)(252,276)(253,280)(254,281)
(255,282)(256,286)(257,287)(258,288)(259,283)(260,284)(261,285)(262,289)
(263,290)(264,291)(265,295)(266,296)(267,297)(268,292)(269,293)(270,294)
(298,304)(299,305)(300,306)(307,313)(308,314)(309,315)(316,322)(317,323)
(318,324);
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*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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