Polytope of Type {108,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {108,6}*1296b
if this polytope has a name.
Group : SmallGroup(1296,826)
Rank : 3
Schlafli Type : {108,6}
Number of vertices, edges, etc : 108, 324, 6
Order of s₀s₁s₂ : 108
Order of s₀s₁s₂s₁ : 6
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 : {54,6}*648b
   3-fold quotients : {108,2}*432, {36,6}*432b
   4-fold quotients : {27,6}*324
   6-fold quotients : {54,2}*216, {18,6}*216b
   9-fold quotients : {36,2}*144, {12,6}*144b
   12-fold quotients : {27,2}*108, {9,6}*108
   18-fold quotients : {18,2}*72, {6,6}*72c
   27-fold quotients : {12,2}*48
   36-fold quotients : {9,2}*36, {3,6}*36
   54-fold quotients : {6,2}*24
   81-fold quotients : {4,2}*16
   108-fold quotients : {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,  9)(  5,  8)(  6,  7)( 10, 27)( 11, 26)( 12, 25)( 13, 24)
( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 28, 55)( 29, 57)( 30, 56)
( 31, 63)( 32, 62)( 33, 61)( 34, 60)( 35, 59)( 36, 58)( 37, 81)( 38, 80)
( 39, 79)( 40, 78)( 41, 77)( 42, 76)( 43, 75)( 44, 74)( 45, 73)( 46, 72)
( 47, 71)( 48, 70)( 49, 69)( 50, 68)( 51, 67)( 52, 66)( 53, 65)( 54, 64)
( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 91,108)( 92,107)( 93,106)( 94,105)
( 95,104)( 96,103)( 97,102)( 98,101)( 99,100)(109,136)(110,138)(111,137)
(112,144)(113,143)(114,142)(115,141)(116,140)(117,139)(118,162)(119,161)
(120,160)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154)(127,153)
(128,152)(129,151)(130,150)(131,149)(132,148)(133,147)(134,146)(135,145)
(163,244)(164,246)(165,245)(166,252)(167,251)(168,250)(169,249)(170,248)
(171,247)(172,270)(173,269)(174,268)(175,267)(176,266)(177,265)(178,264)
(179,263)(180,262)(181,261)(182,260)(183,259)(184,258)(185,257)(186,256)
(187,255)(188,254)(189,253)(190,298)(191,300)(192,299)(193,306)(194,305)
(195,304)(196,303)(197,302)(198,301)(199,324)(200,323)(201,322)(202,321)
(203,320)(204,319)(205,318)(206,317)(207,316)(208,315)(209,314)(210,313)
(211,312)(212,311)(213,310)(214,309)(215,308)(216,307)(217,271)(218,273)
(219,272)(220,279)(221,278)(222,277)(223,276)(224,275)(225,274)(226,297)
(227,296)(228,295)(229,294)(230,293)(231,292)(232,291)(233,290)(234,289)
(235,288)(236,287)(237,286)(238,285)(239,284)(240,283)(241,282)(242,281)
(243,280);;
s1 := (  1,199)(  2,201)(  3,200)(  4,207)(  5,206)(  6,205)(  7,204)(  8,203)
(  9,202)( 10,190)( 11,192)( 12,191)( 13,198)( 14,197)( 15,196)( 16,195)
( 17,194)( 18,193)( 19,216)( 20,215)( 21,214)( 22,213)( 23,212)( 24,211)
( 25,210)( 26,209)( 27,208)( 28,172)( 29,174)( 30,173)( 31,180)( 32,179)
( 33,178)( 34,177)( 35,176)( 36,175)( 37,163)( 38,165)( 39,164)( 40,171)
( 41,170)( 42,169)( 43,168)( 44,167)( 45,166)( 46,189)( 47,188)( 48,187)
( 49,186)( 50,185)( 51,184)( 52,183)( 53,182)( 54,181)( 55,226)( 56,228)
( 57,227)( 58,234)( 59,233)( 60,232)( 61,231)( 62,230)( 63,229)( 64,217)
( 65,219)( 66,218)( 67,225)( 68,224)( 69,223)( 70,222)( 71,221)( 72,220)
( 73,243)( 74,242)( 75,241)( 76,240)( 77,239)( 78,238)( 79,237)( 80,236)
( 81,235)( 82,280)( 83,282)( 84,281)( 85,288)( 86,287)( 87,286)( 88,285)
( 89,284)( 90,283)( 91,271)( 92,273)( 93,272)( 94,279)( 95,278)( 96,277)
( 97,276)( 98,275)( 99,274)(100,297)(101,296)(102,295)(103,294)(104,293)
(105,292)(106,291)(107,290)(108,289)(109,253)(110,255)(111,254)(112,261)
(113,260)(114,259)(115,258)(116,257)(117,256)(118,244)(119,246)(120,245)
(121,252)(122,251)(123,250)(124,249)(125,248)(126,247)(127,270)(128,269)
(129,268)(130,267)(131,266)(132,265)(133,264)(134,263)(135,262)(136,307)
(137,309)(138,308)(139,315)(140,314)(141,313)(142,312)(143,311)(144,310)
(145,298)(146,300)(147,299)(148,306)(149,305)(150,304)(151,303)(152,302)
(153,301)(154,324)(155,323)(156,322)(157,321)(158,320)(159,319)(160,318)
(161,317)(162,316);;
s2 := ( 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);;
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*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*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*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*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*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope