Polytope of Type {6,105}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,105}*1260
if this polytope has a name.
Group : SmallGroup(1260,112)
Rank : 3
Schlafli Type : {6,105}
Number of vertices, edges, etc : 6, 315, 105
Order of s₀s₁s₂ : 210
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) :
   3-fold quotients : {2,105}*420
   5-fold quotients : {6,21}*252
   7-fold quotients : {6,15}*180
   9-fold quotients : {2,35}*140
   15-fold quotients : {2,21}*84
   21-fold quotients : {2,15}*60
   35-fold quotients : {6,3}*36
   45-fold quotients : {2,7}*28
   63-fold quotients : {2,5}*20
   105-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope