Polytope of Type {111,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {111,6}*1332
if this polytope has a name.
Group : SmallGroup(1332,51)
Rank : 3
Schlafli Type : {111,6}
Number of vertices, edges, etc : 111, 333, 6
Order of s₀s₁s₂ : 222
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 : {111,2}*444
   9-fold quotients : {37,2}*148
   37-fold quotients : {3,6}*36
   111-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2, 37)(  3, 36)(  4, 35)(  5, 34)(  6, 33)(  7, 32)(  8, 31)(  9, 30)
( 10, 29)( 11, 28)( 12, 27)( 13, 26)( 14, 25)( 15, 24)( 16, 23)( 17, 22)
( 18, 21)( 19, 20)( 38, 75)( 39,111)( 40,110)( 41,109)( 42,108)( 43,107)
( 44,106)( 45,105)( 46,104)( 47,103)( 48,102)( 49,101)( 50,100)( 51, 99)
( 52, 98)( 53, 97)( 54, 96)( 55, 95)( 56, 94)( 57, 93)( 58, 92)( 59, 91)
( 60, 90)( 61, 89)( 62, 88)( 63, 87)( 64, 86)( 65, 85)( 66, 84)( 67, 83)
( 68, 82)( 69, 81)( 70, 80)( 71, 79)( 72, 78)( 73, 77)( 74, 76)(112,223)
(113,259)(114,258)(115,257)(116,256)(117,255)(118,254)(119,253)(120,252)
(121,251)(122,250)(123,249)(124,248)(125,247)(126,246)(127,245)(128,244)
(129,243)(130,242)(131,241)(132,240)(133,239)(134,238)(135,237)(136,236)
(137,235)(138,234)(139,233)(140,232)(141,231)(142,230)(143,229)(144,228)
(145,227)(146,226)(147,225)(148,224)(149,297)(150,333)(151,332)(152,331)
(153,330)(154,329)(155,328)(156,327)(157,326)(158,325)(159,324)(160,323)
(161,322)(162,321)(163,320)(164,319)(165,318)(166,317)(167,316)(168,315)
(169,314)(170,313)(171,312)(172,311)(173,310)(174,309)(175,308)(176,307)
(177,306)(178,305)(179,304)(180,303)(181,302)(182,301)(183,300)(184,299)
(185,298)(186,260)(187,296)(188,295)(189,294)(190,293)(191,292)(192,291)
(193,290)(194,289)(195,288)(196,287)(197,286)(198,285)(199,284)(200,283)
(201,282)(202,281)(203,280)(204,279)(205,278)(206,277)(207,276)(208,275)
(209,274)(210,273)(211,272)(212,271)(213,270)(214,269)(215,268)(216,267)
(217,266)(218,265)(219,264)(220,263)(221,262)(222,261);;
s1 := (  1,150)(  2,149)(  3,185)(  4,184)(  5,183)(  6,182)(  7,181)(  8,180)
(  9,179)( 10,178)( 11,177)( 12,176)( 13,175)( 14,174)( 15,173)( 16,172)
( 17,171)( 18,170)( 19,169)( 20,168)( 21,167)( 22,166)( 23,165)( 24,164)
( 25,163)( 26,162)( 27,161)( 28,160)( 29,159)( 30,158)( 31,157)( 32,156)
( 33,155)( 34,154)( 35,153)( 36,152)( 37,151)( 38,113)( 39,112)( 40,148)
( 41,147)( 42,146)( 43,145)( 44,144)( 45,143)( 46,142)( 47,141)( 48,140)
( 49,139)( 50,138)( 51,137)( 52,136)( 53,135)( 54,134)( 55,133)( 56,132)
( 57,131)( 58,130)( 59,129)( 60,128)( 61,127)( 62,126)( 63,125)( 64,124)
( 65,123)( 66,122)( 67,121)( 68,120)( 69,119)( 70,118)( 71,117)( 72,116)
( 73,115)( 74,114)( 75,187)( 76,186)( 77,222)( 78,221)( 79,220)( 80,219)
( 81,218)( 82,217)( 83,216)( 84,215)( 85,214)( 86,213)( 87,212)( 88,211)
( 89,210)( 90,209)( 91,208)( 92,207)( 93,206)( 94,205)( 95,204)( 96,203)
( 97,202)( 98,201)( 99,200)(100,199)(101,198)(102,197)(103,196)(104,195)
(105,194)(106,193)(107,192)(108,191)(109,190)(110,189)(111,188)(223,261)
(224,260)(225,296)(226,295)(227,294)(228,293)(229,292)(230,291)(231,290)
(232,289)(233,288)(234,287)(235,286)(236,285)(237,284)(238,283)(239,282)
(240,281)(241,280)(242,279)(243,278)(244,277)(245,276)(246,275)(247,274)
(248,273)(249,272)(250,271)(251,270)(252,269)(253,268)(254,267)(255,266)
(256,265)(257,264)(258,263)(259,262)(297,298)(299,333)(300,332)(301,331)
(302,330)(303,329)(304,328)(305,327)(306,326)(307,325)(308,324)(309,323)
(310,322)(311,321)(312,320)(313,319)(314,318)(315,317);;
s2 := ( 38, 75)( 39, 76)( 40, 77)( 41, 78)( 42, 79)( 43, 80)( 44, 81)( 45, 82)
( 46, 83)( 47, 84)( 48, 85)( 49, 86)( 50, 87)( 51, 88)( 52, 89)( 53, 90)
( 54, 91)( 55, 92)( 56, 93)( 57, 94)( 58, 95)( 59, 96)( 60, 97)( 61, 98)
( 62, 99)( 63,100)( 64,101)( 65,102)( 66,103)( 67,104)( 68,105)( 69,106)
( 70,107)( 71,108)( 72,109)( 73,110)( 74,111)(149,186)(150,187)(151,188)
(152,189)(153,190)(154,191)(155,192)(156,193)(157,194)(158,195)(159,196)
(160,197)(161,198)(162,199)(163,200)(164,201)(165,202)(166,203)(167,204)
(168,205)(169,206)(170,207)(171,208)(172,209)(173,210)(174,211)(175,212)
(176,213)(177,214)(178,215)(179,216)(180,217)(181,218)(182,219)(183,220)
(184,221)(185,222)(260,297)(261,298)(262,299)(263,300)(264,301)(265,302)
(266,303)(267,304)(268,305)(269,306)(270,307)(271,308)(272,309)(273,310)
(274,311)(275,312)(276,313)(277,314)(278,315)(279,316)(280,317)(281,318)
(282,319)(283,320)(284,321)(285,322)(286,323)(287,324)(288,325)(289,326)
(290,327)(291,328)(292,329)(293,330)(294,331)(295,332)(296,333);;
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*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(333)!(  2, 37)(  3, 36)(  4, 35)(  5, 34)(  6, 33)(  7, 32)(  8, 31)
(  9, 30)( 10, 29)( 11, 28)( 12, 27)( 13, 26)( 14, 25)( 15, 24)( 16, 23)
( 17, 22)( 18, 21)( 19, 20)( 38, 75)( 39,111)( 40,110)( 41,109)( 42,108)
( 43,107)( 44,106)( 45,105)( 46,104)( 47,103)( 48,102)( 49,101)( 50,100)
( 51, 99)( 52, 98)( 53, 97)( 54, 96)( 55, 95)( 56, 94)( 57, 93)( 58, 92)
( 59, 91)( 60, 90)( 61, 89)( 62, 88)( 63, 87)( 64, 86)( 65, 85)( 66, 84)
( 67, 83)( 68, 82)( 69, 81)( 70, 80)( 71, 79)( 72, 78)( 73, 77)( 74, 76)
(112,223)(113,259)(114,258)(115,257)(116,256)(117,255)(118,254)(119,253)
(120,252)(121,251)(122,250)(123,249)(124,248)(125,247)(126,246)(127,245)
(128,244)(129,243)(130,242)(131,241)(132,240)(133,239)(134,238)(135,237)
(136,236)(137,235)(138,234)(139,233)(140,232)(141,231)(142,230)(143,229)
(144,228)(145,227)(146,226)(147,225)(148,224)(149,297)(150,333)(151,332)
(152,331)(153,330)(154,329)(155,328)(156,327)(157,326)(158,325)(159,324)
(160,323)(161,322)(162,321)(163,320)(164,319)(165,318)(166,317)(167,316)
(168,315)(169,314)(170,313)(171,312)(172,311)(173,310)(174,309)(175,308)
(176,307)(177,306)(178,305)(179,304)(180,303)(181,302)(182,301)(183,300)
(184,299)(185,298)(186,260)(187,296)(188,295)(189,294)(190,293)(191,292)
(192,291)(193,290)(194,289)(195,288)(196,287)(197,286)(198,285)(199,284)
(200,283)(201,282)(202,281)(203,280)(204,279)(205,278)(206,277)(207,276)
(208,275)(209,274)(210,273)(211,272)(212,271)(213,270)(214,269)(215,268)
(216,267)(217,266)(218,265)(219,264)(220,263)(221,262)(222,261);
s1 := Sym(333)!(  1,150)(  2,149)(  3,185)(  4,184)(  5,183)(  6,182)(  7,181)
(  8,180)(  9,179)( 10,178)( 11,177)( 12,176)( 13,175)( 14,174)( 15,173)
( 16,172)( 17,171)( 18,170)( 19,169)( 20,168)( 21,167)( 22,166)( 23,165)
( 24,164)( 25,163)( 26,162)( 27,161)( 28,160)( 29,159)( 30,158)( 31,157)
( 32,156)( 33,155)( 34,154)( 35,153)( 36,152)( 37,151)( 38,113)( 39,112)
( 40,148)( 41,147)( 42,146)( 43,145)( 44,144)( 45,143)( 46,142)( 47,141)
( 48,140)( 49,139)( 50,138)( 51,137)( 52,136)( 53,135)( 54,134)( 55,133)
( 56,132)( 57,131)( 58,130)( 59,129)( 60,128)( 61,127)( 62,126)( 63,125)
( 64,124)( 65,123)( 66,122)( 67,121)( 68,120)( 69,119)( 70,118)( 71,117)
( 72,116)( 73,115)( 74,114)( 75,187)( 76,186)( 77,222)( 78,221)( 79,220)
( 80,219)( 81,218)( 82,217)( 83,216)( 84,215)( 85,214)( 86,213)( 87,212)
( 88,211)( 89,210)( 90,209)( 91,208)( 92,207)( 93,206)( 94,205)( 95,204)
( 96,203)( 97,202)( 98,201)( 99,200)(100,199)(101,198)(102,197)(103,196)
(104,195)(105,194)(106,193)(107,192)(108,191)(109,190)(110,189)(111,188)
(223,261)(224,260)(225,296)(226,295)(227,294)(228,293)(229,292)(230,291)
(231,290)(232,289)(233,288)(234,287)(235,286)(236,285)(237,284)(238,283)
(239,282)(240,281)(241,280)(242,279)(243,278)(244,277)(245,276)(246,275)
(247,274)(248,273)(249,272)(250,271)(251,270)(252,269)(253,268)(254,267)
(255,266)(256,265)(257,264)(258,263)(259,262)(297,298)(299,333)(300,332)
(301,331)(302,330)(303,329)(304,328)(305,327)(306,326)(307,325)(308,324)
(309,323)(310,322)(311,321)(312,320)(313,319)(314,318)(315,317);
s2 := Sym(333)!( 38, 75)( 39, 76)( 40, 77)( 41, 78)( 42, 79)( 43, 80)( 44, 81)
( 45, 82)( 46, 83)( 47, 84)( 48, 85)( 49, 86)( 50, 87)( 51, 88)( 52, 89)
( 53, 90)( 54, 91)( 55, 92)( 56, 93)( 57, 94)( 58, 95)( 59, 96)( 60, 97)
( 61, 98)( 62, 99)( 63,100)( 64,101)( 65,102)( 66,103)( 67,104)( 68,105)
( 69,106)( 70,107)( 71,108)( 72,109)( 73,110)( 74,111)(149,186)(150,187)
(151,188)(152,189)(153,190)(154,191)(155,192)(156,193)(157,194)(158,195)
(159,196)(160,197)(161,198)(162,199)(163,200)(164,201)(165,202)(166,203)
(167,204)(168,205)(169,206)(170,207)(171,208)(172,209)(173,210)(174,211)
(175,212)(176,213)(177,214)(178,215)(179,216)(180,217)(181,218)(182,219)
(183,220)(184,221)(185,222)(260,297)(261,298)(262,299)(263,300)(264,301)
(265,302)(266,303)(267,304)(268,305)(269,306)(270,307)(271,308)(272,309)
(273,310)(274,311)(275,312)(276,313)(277,314)(278,315)(279,316)(280,317)
(281,318)(282,319)(283,320)(284,321)(285,322)(286,323)(287,324)(288,325)
(289,326)(290,327)(291,328)(292,329)(293,330)(294,331)(295,332)(296,333);
poly := sub<Sym(333)|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*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope