Polytope of Type {6,108}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,108}*1296b
if this polytope has a name.
Group : SmallGroup(1296,826)
Rank : 3
Schlafli Type : {6,108}
Number of vertices, edges, etc : 6, 324, 108
Order of s₀s₁s₂ : 108
Order of s₀s₁s₂s₁ : 6
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 : {6,54}*648b
   3-fold quotients : {2,108}*432, {6,36}*432b
   4-fold quotients : {6,27}*324
   6-fold quotients : {2,54}*216, {6,18}*216b
   9-fold quotients : {2,36}*144, {6,12}*144b
   12-fold quotients : {2,27}*108, {6,9}*108
   18-fold quotients : {2,18}*72, {6,6}*72b
   27-fold quotients : {2,12}*48
   36-fold quotients : {2,9}*36, {6,3}*36
   54-fold quotients : {2,6}*24
   81-fold quotients : {2,4}*16
   108-fold quotients : {2,3}*12
   162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope