Polytope of Type {36,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,6}*1296f
if this polytope has a name.
Group : SmallGroup(1296,854)
Rank : 3
Schlafli Type : {36,6}
Number of vertices, edges, etc : 108, 324, 18
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {18,6}*648f
   3-fold quotients : {12,6}*432b
   4-fold quotients : {18,6}*324b
   6-fold quotients : {6,6}*216b
   9-fold quotients : {12,6}*144a
   12-fold quotients : {6,6}*108
   18-fold quotients : {6,6}*72a
   27-fold quotients : {12,2}*48, {4,6}*48a
   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)(  4,  7)(  5,  9)(  6,  8)( 10, 21)( 11, 20)( 12, 19)( 13, 27)
( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 48)( 38, 47)( 39, 46)( 40, 54)( 41, 53)( 42, 52)( 43, 51)
( 44, 50)( 45, 49)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 75)( 65, 74)
( 66, 73)( 67, 81)( 68, 80)( 69, 79)( 70, 78)( 71, 77)( 72, 76)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 91,102)( 92,101)( 93,100)( 94,108)( 95,107)
( 96,106)( 97,105)( 98,104)( 99,103)(110,111)(112,115)(113,117)(114,116)
(118,129)(119,128)(120,127)(121,135)(122,134)(123,133)(124,132)(125,131)
(126,130)(137,138)(139,142)(140,144)(141,143)(145,156)(146,155)(147,154)
(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(163,244)(164,246)
(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)(171,248)(172,264)
(173,263)(174,262)(175,270)(176,269)(177,268)(178,267)(179,266)(180,265)
(181,255)(182,254)(183,253)(184,261)(185,260)(186,259)(187,258)(188,257)
(189,256)(190,271)(191,273)(192,272)(193,277)(194,279)(195,278)(196,274)
(197,276)(198,275)(199,291)(200,290)(201,289)(202,297)(203,296)(204,295)
(205,294)(206,293)(207,292)(208,282)(209,281)(210,280)(211,288)(212,287)
(213,286)(214,285)(215,284)(216,283)(217,298)(218,300)(219,299)(220,304)
(221,306)(222,305)(223,301)(224,303)(225,302)(226,318)(227,317)(228,316)
(229,324)(230,323)(231,322)(232,321)(233,320)(234,319)(235,309)(236,308)
(237,307)(238,315)(239,314)(240,313)(241,312)(242,311)(243,310);;
s1 := (  1,172)(  2,174)(  3,173)(  4,177)(  5,176)(  6,175)(  7,179)(  8,178)
(  9,180)( 10,163)( 11,165)( 12,164)( 13,168)( 14,167)( 15,166)( 16,170)
( 17,169)( 18,171)( 19,183)( 20,182)( 21,181)( 22,185)( 23,184)( 24,186)
( 25,187)( 26,189)( 27,188)( 28,226)( 29,228)( 30,227)( 31,231)( 32,230)
( 33,229)( 34,233)( 35,232)( 36,234)( 37,217)( 38,219)( 39,218)( 40,222)
( 41,221)( 42,220)( 43,224)( 44,223)( 45,225)( 46,237)( 47,236)( 48,235)
( 49,239)( 50,238)( 51,240)( 52,241)( 53,243)( 54,242)( 55,199)( 56,201)
( 57,200)( 58,204)( 59,203)( 60,202)( 61,206)( 62,205)( 63,207)( 64,190)
( 65,192)( 66,191)( 67,195)( 68,194)( 69,193)( 70,197)( 71,196)( 72,198)
( 73,210)( 74,209)( 75,208)( 76,212)( 77,211)( 78,213)( 79,214)( 80,216)
( 81,215)( 82,253)( 83,255)( 84,254)( 85,258)( 86,257)( 87,256)( 88,260)
( 89,259)( 90,261)( 91,244)( 92,246)( 93,245)( 94,249)( 95,248)( 96,247)
( 97,251)( 98,250)( 99,252)(100,264)(101,263)(102,262)(103,266)(104,265)
(105,267)(106,268)(107,270)(108,269)(109,307)(110,309)(111,308)(112,312)
(113,311)(114,310)(115,314)(116,313)(117,315)(118,298)(119,300)(120,299)
(121,303)(122,302)(123,301)(124,305)(125,304)(126,306)(127,318)(128,317)
(129,316)(130,320)(131,319)(132,321)(133,322)(134,324)(135,323)(136,280)
(137,282)(138,281)(139,285)(140,284)(141,283)(142,287)(143,286)(144,288)
(145,271)(146,273)(147,272)(148,276)(149,275)(150,274)(151,278)(152,277)
(153,279)(154,291)(155,290)(156,289)(157,293)(158,292)(159,294)(160,295)
(161,297)(162,296);;
s2 := (  1, 28)(  2, 29)(  3, 30)(  4, 34)(  5, 35)(  6, 36)(  7, 31)(  8, 32)
(  9, 33)( 10, 40)( 11, 41)( 12, 42)( 13, 37)( 14, 38)( 15, 39)( 16, 43)
( 17, 44)( 18, 45)( 19, 52)( 20, 53)( 21, 54)( 22, 49)( 23, 50)( 24, 51)
( 25, 46)( 26, 47)( 27, 48)( 58, 61)( 59, 62)( 60, 63)( 64, 67)( 65, 68)
( 66, 69)( 73, 79)( 74, 80)( 75, 81)( 82,109)( 83,110)( 84,111)( 85,115)
( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)( 91,121)( 92,122)( 93,123)
( 94,118)( 95,119)( 96,120)( 97,124)( 98,125)( 99,126)(100,133)(101,134)
(102,135)(103,130)(104,131)(105,132)(106,127)(107,128)(108,129)(139,142)
(140,143)(141,144)(145,148)(146,149)(147,150)(154,160)(155,161)(156,162)
(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)
(171,195)(172,202)(173,203)(174,204)(175,199)(176,200)(177,201)(178,205)
(179,206)(180,207)(181,214)(182,215)(183,216)(184,211)(185,212)(186,213)
(187,208)(188,209)(189,210)(220,223)(221,224)(222,225)(226,229)(227,230)
(228,231)(235,241)(236,242)(237,243)(244,271)(245,272)(246,273)(247,277)
(248,278)(249,279)(250,274)(251,275)(252,276)(253,283)(254,284)(255,285)
(256,280)(257,281)(258,282)(259,286)(260,287)(261,288)(262,295)(263,296)
(264,297)(265,292)(266,293)(267,294)(268,289)(269,290)(270,291)(301,304)
(302,305)(303,306)(307,310)(308,311)(309,312)(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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*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, 21)( 11, 20)( 12, 19)
( 13, 27)( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 34)
( 32, 36)( 33, 35)( 37, 48)( 38, 47)( 39, 46)( 40, 54)( 41, 53)( 42, 52)
( 43, 51)( 44, 50)( 45, 49)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 75)
( 65, 74)( 66, 73)( 67, 81)( 68, 80)( 69, 79)( 70, 78)( 71, 77)( 72, 76)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 91,102)( 92,101)( 93,100)( 94,108)
( 95,107)( 96,106)( 97,105)( 98,104)( 99,103)(110,111)(112,115)(113,117)
(114,116)(118,129)(119,128)(120,127)(121,135)(122,134)(123,133)(124,132)
(125,131)(126,130)(137,138)(139,142)(140,144)(141,143)(145,156)(146,155)
(147,154)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(163,244)
(164,246)(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)(171,248)
(172,264)(173,263)(174,262)(175,270)(176,269)(177,268)(178,267)(179,266)
(180,265)(181,255)(182,254)(183,253)(184,261)(185,260)(186,259)(187,258)
(188,257)(189,256)(190,271)(191,273)(192,272)(193,277)(194,279)(195,278)
(196,274)(197,276)(198,275)(199,291)(200,290)(201,289)(202,297)(203,296)
(204,295)(205,294)(206,293)(207,292)(208,282)(209,281)(210,280)(211,288)
(212,287)(213,286)(214,285)(215,284)(216,283)(217,298)(218,300)(219,299)
(220,304)(221,306)(222,305)(223,301)(224,303)(225,302)(226,318)(227,317)
(228,316)(229,324)(230,323)(231,322)(232,321)(233,320)(234,319)(235,309)
(236,308)(237,307)(238,315)(239,314)(240,313)(241,312)(242,311)(243,310);
s1 := Sym(324)!(  1,172)(  2,174)(  3,173)(  4,177)(  5,176)(  6,175)(  7,179)
(  8,178)(  9,180)( 10,163)( 11,165)( 12,164)( 13,168)( 14,167)( 15,166)
( 16,170)( 17,169)( 18,171)( 19,183)( 20,182)( 21,181)( 22,185)( 23,184)
( 24,186)( 25,187)( 26,189)( 27,188)( 28,226)( 29,228)( 30,227)( 31,231)
( 32,230)( 33,229)( 34,233)( 35,232)( 36,234)( 37,217)( 38,219)( 39,218)
( 40,222)( 41,221)( 42,220)( 43,224)( 44,223)( 45,225)( 46,237)( 47,236)
( 48,235)( 49,239)( 50,238)( 51,240)( 52,241)( 53,243)( 54,242)( 55,199)
( 56,201)( 57,200)( 58,204)( 59,203)( 60,202)( 61,206)( 62,205)( 63,207)
( 64,190)( 65,192)( 66,191)( 67,195)( 68,194)( 69,193)( 70,197)( 71,196)
( 72,198)( 73,210)( 74,209)( 75,208)( 76,212)( 77,211)( 78,213)( 79,214)
( 80,216)( 81,215)( 82,253)( 83,255)( 84,254)( 85,258)( 86,257)( 87,256)
( 88,260)( 89,259)( 90,261)( 91,244)( 92,246)( 93,245)( 94,249)( 95,248)
( 96,247)( 97,251)( 98,250)( 99,252)(100,264)(101,263)(102,262)(103,266)
(104,265)(105,267)(106,268)(107,270)(108,269)(109,307)(110,309)(111,308)
(112,312)(113,311)(114,310)(115,314)(116,313)(117,315)(118,298)(119,300)
(120,299)(121,303)(122,302)(123,301)(124,305)(125,304)(126,306)(127,318)
(128,317)(129,316)(130,320)(131,319)(132,321)(133,322)(134,324)(135,323)
(136,280)(137,282)(138,281)(139,285)(140,284)(141,283)(142,287)(143,286)
(144,288)(145,271)(146,273)(147,272)(148,276)(149,275)(150,274)(151,278)
(152,277)(153,279)(154,291)(155,290)(156,289)(157,293)(158,292)(159,294)
(160,295)(161,297)(162,296);
s2 := Sym(324)!(  1, 28)(  2, 29)(  3, 30)(  4, 34)(  5, 35)(  6, 36)(  7, 31)
(  8, 32)(  9, 33)( 10, 40)( 11, 41)( 12, 42)( 13, 37)( 14, 38)( 15, 39)
( 16, 43)( 17, 44)( 18, 45)( 19, 52)( 20, 53)( 21, 54)( 22, 49)( 23, 50)
( 24, 51)( 25, 46)( 26, 47)( 27, 48)( 58, 61)( 59, 62)( 60, 63)( 64, 67)
( 65, 68)( 66, 69)( 73, 79)( 74, 80)( 75, 81)( 82,109)( 83,110)( 84,111)
( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)( 91,121)( 92,122)
( 93,123)( 94,118)( 95,119)( 96,120)( 97,124)( 98,125)( 99,126)(100,133)
(101,134)(102,135)(103,130)(104,131)(105,132)(106,127)(107,128)(108,129)
(139,142)(140,143)(141,144)(145,148)(146,149)(147,150)(154,160)(155,161)
(156,162)(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)
(170,194)(171,195)(172,202)(173,203)(174,204)(175,199)(176,200)(177,201)
(178,205)(179,206)(180,207)(181,214)(182,215)(183,216)(184,211)(185,212)
(186,213)(187,208)(188,209)(189,210)(220,223)(221,224)(222,225)(226,229)
(227,230)(228,231)(235,241)(236,242)(237,243)(244,271)(245,272)(246,273)
(247,277)(248,278)(249,279)(250,274)(251,275)(252,276)(253,283)(254,284)
(255,285)(256,280)(257,281)(258,282)(259,286)(260,287)(261,288)(262,295)
(263,296)(264,297)(265,292)(266,293)(267,294)(268,289)(269,290)(270,291)
(301,304)(302,305)(303,306)(307,310)(308,311)(309,312)(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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope