Polytope of Type {6,111} with group SmallGroup(1332,51)

Overview

Group: SmallGroup(1332,51)
Rank: 3
Schläfli Type: {6,111}
Vertices, edges, …: 6, 333, 111
Order of s₀s₁s₂: 222
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat

Quotients maximal quotients in bold

3-fold

{2,111}*444

9-fold

{2,37}*148

37-fold

{6,3}*36

111-fold

{2,3}*12

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References