Polytope of Type {105,6}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope