Polytope of Type {6,108}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,108}*1296a
Also Known As : {6,108|2}. if this polytope has another 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 : 2
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}*648a
3-fold quotients : {2,108}*432, {6,36}*432a
6-fold quotients : {2,54}*216, {6,18}*216a
9-fold quotients : {2,36}*144, {6,12}*144a
12-fold quotients : {2,27}*108
18-fold quotients : {2,18}*72, {6,6}*72a
27-fold quotients : {2,12}*48, {6,4}*48a
36-fold quotients : {2,9}*36
54-fold quotients : {2,6}*24, {6,2}*24
81-fold quotients : {2,4}*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 := ( 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)( 28,199)( 29,201)( 30,200)( 31,207)( 32,206)
( 33,205)( 34,204)( 35,203)( 36,202)( 37,190)( 38,192)( 39,191)( 40,198)
( 41,197)( 42,196)( 43,195)( 44,194)( 45,193)( 46,216)( 47,215)( 48,214)
( 49,213)( 50,212)( 51,211)( 52,210)( 53,209)( 54,208)( 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)( 82,253)( 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)(109,280)(110,282)(111,281)(112,288)
(113,287)(114,286)(115,285)(116,284)(117,283)(118,271)(119,273)(120,272)
(121,279)(122,278)(123,277)(124,276)(125,275)(126,274)(127,297)(128,296)
(129,295)(130,294)(131,293)(132,292)(133,291)(134,290)(135,289)(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);;
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,
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)( 28,199)( 29,201)( 30,200)( 31,207)
( 32,206)( 33,205)( 34,204)( 35,203)( 36,202)( 37,190)( 38,192)( 39,191)
( 40,198)( 41,197)( 42,196)( 43,195)( 44,194)( 45,193)( 46,216)( 47,215)
( 48,214)( 49,213)( 50,212)( 51,211)( 52,210)( 53,209)( 54,208)( 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)( 82,253)( 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)(109,280)(110,282)(111,281)
(112,288)(113,287)(114,286)(115,285)(116,284)(117,283)(118,271)(119,273)
(120,272)(121,279)(122,278)(123,277)(124,276)(125,275)(126,274)(127,297)
(128,296)(129,295)(130,294)(131,293)(132,292)(133,291)(134,290)(135,289)
(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);
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,
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