Polytope of Type {2,9,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,6}*1944d
if this polytope has a name.
Group : SmallGroup(1944,949)
Rank : 4
Schlafli Type : {2,9,6}
Number of vertices, edges, etc : 2, 81, 243, 54
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
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 : {2,9,6}*648a, {2,9,6}*648b
9-fold quotients : {2,9,6}*216, {2,3,6}*216
27-fold quotients : {2,9,2}*72, {2,3,6}*72
81-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 6, 8)( 9, 10)( 12, 21)( 13, 23)( 14, 22)( 15, 26)( 16, 25)
( 17, 24)( 18, 28)( 19, 27)( 20, 29)( 30, 57)( 31, 59)( 32, 58)( 33, 62)
( 34, 61)( 35, 60)( 36, 64)( 37, 63)( 38, 65)( 39, 75)( 40, 77)( 41, 76)
( 42, 80)( 43, 79)( 44, 78)( 45, 82)( 46, 81)( 47, 83)( 48, 66)( 49, 68)
( 50, 67)( 51, 71)( 52, 70)( 53, 69)( 54, 73)( 55, 72)( 56, 74)( 84,183)
( 85,185)( 86,184)( 87,188)( 88,187)( 89,186)( 90,190)( 91,189)( 92,191)
( 93,174)( 94,176)( 95,175)( 96,179)( 97,178)( 98,177)( 99,181)(100,180)
(101,182)(102,165)(103,167)(104,166)(105,170)(106,169)(107,168)(108,172)
(109,171)(110,173)(111,237)(112,239)(113,238)(114,242)(115,241)(116,240)
(117,244)(118,243)(119,245)(120,228)(121,230)(122,229)(123,233)(124,232)
(125,231)(126,235)(127,234)(128,236)(129,219)(130,221)(131,220)(132,224)
(133,223)(134,222)(135,226)(136,225)(137,227)(138,210)(139,212)(140,211)
(141,215)(142,214)(143,213)(144,217)(145,216)(146,218)(147,201)(148,203)
(149,202)(150,206)(151,205)(152,204)(153,208)(154,207)(155,209)(156,192)
(157,194)(158,193)(159,197)(160,196)(161,195)(162,199)(163,198)(164,200);;
s2 := ( 3,195)( 4,197)( 5,196)( 6,199)( 7,198)( 8,200)( 9,194)( 10,193)
( 11,192)( 12,213)( 13,215)( 14,214)( 15,217)( 16,216)( 17,218)( 18,212)
( 19,211)( 20,210)( 21,204)( 22,206)( 23,205)( 24,208)( 25,207)( 26,209)
( 27,203)( 28,202)( 29,201)( 30,173)( 31,172)( 32,171)( 33,165)( 34,167)
( 35,166)( 36,169)( 37,168)( 38,170)( 39,191)( 40,190)( 41,189)( 42,183)
( 43,185)( 44,184)( 45,187)( 46,186)( 47,188)( 48,182)( 49,181)( 50,180)
( 51,174)( 52,176)( 53,175)( 54,178)( 55,177)( 56,179)( 57,220)( 58,219)
( 59,221)( 60,224)( 61,223)( 62,222)( 63,225)( 64,227)( 65,226)( 66,238)
( 67,237)( 68,239)( 69,242)( 70,241)( 71,240)( 72,243)( 73,245)( 74,244)
( 75,229)( 76,228)( 77,230)( 78,233)( 79,232)( 80,231)( 81,234)( 82,236)
( 83,235)( 84,114)( 85,116)( 86,115)( 87,118)( 88,117)( 89,119)( 90,113)
( 91,112)( 92,111)( 93,132)( 94,134)( 95,133)( 96,136)( 97,135)( 98,137)
( 99,131)(100,130)(101,129)(102,123)(103,125)(104,124)(105,127)(106,126)
(107,128)(108,122)(109,121)(110,120)(138,139)(141,143)(145,146)(147,157)
(148,156)(149,158)(150,161)(151,160)(152,159)(153,162)(154,164)(155,163);;
s3 := ( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 57)( 31, 59)( 32, 58)( 33, 63)
( 34, 65)( 35, 64)( 36, 60)( 37, 62)( 38, 61)( 39, 66)( 40, 68)( 41, 67)
( 42, 72)( 43, 74)( 44, 73)( 45, 69)( 46, 71)( 47, 70)( 48, 75)( 49, 77)
( 50, 76)( 51, 81)( 52, 83)( 53, 82)( 54, 78)( 55, 80)( 56, 79)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)
(105,108)(106,110)(107,109)(111,138)(112,140)(113,139)(114,144)(115,146)
(116,145)(117,141)(118,143)(119,142)(120,147)(121,149)(122,148)(123,153)
(124,155)(125,154)(126,150)(127,152)(128,151)(129,156)(130,158)(131,157)
(132,162)(133,164)(134,163)(135,159)(136,161)(137,160)(166,167)(168,171)
(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)(184,185)(186,189)
(187,191)(188,190)(192,219)(193,221)(194,220)(195,225)(196,227)(197,226)
(198,222)(199,224)(200,223)(201,228)(202,230)(203,229)(204,234)(205,236)
(206,235)(207,231)(208,233)(209,232)(210,237)(211,239)(212,238)(213,243)
(214,245)(215,244)(216,240)(217,242)(218,241);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s2*s1*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*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(245)!(1,2);
s1 := Sym(245)!( 4, 5)( 6, 8)( 9, 10)( 12, 21)( 13, 23)( 14, 22)( 15, 26)
( 16, 25)( 17, 24)( 18, 28)( 19, 27)( 20, 29)( 30, 57)( 31, 59)( 32, 58)
( 33, 62)( 34, 61)( 35, 60)( 36, 64)( 37, 63)( 38, 65)( 39, 75)( 40, 77)
( 41, 76)( 42, 80)( 43, 79)( 44, 78)( 45, 82)( 46, 81)( 47, 83)( 48, 66)
( 49, 68)( 50, 67)( 51, 71)( 52, 70)( 53, 69)( 54, 73)( 55, 72)( 56, 74)
( 84,183)( 85,185)( 86,184)( 87,188)( 88,187)( 89,186)( 90,190)( 91,189)
( 92,191)( 93,174)( 94,176)( 95,175)( 96,179)( 97,178)( 98,177)( 99,181)
(100,180)(101,182)(102,165)(103,167)(104,166)(105,170)(106,169)(107,168)
(108,172)(109,171)(110,173)(111,237)(112,239)(113,238)(114,242)(115,241)
(116,240)(117,244)(118,243)(119,245)(120,228)(121,230)(122,229)(123,233)
(124,232)(125,231)(126,235)(127,234)(128,236)(129,219)(130,221)(131,220)
(132,224)(133,223)(134,222)(135,226)(136,225)(137,227)(138,210)(139,212)
(140,211)(141,215)(142,214)(143,213)(144,217)(145,216)(146,218)(147,201)
(148,203)(149,202)(150,206)(151,205)(152,204)(153,208)(154,207)(155,209)
(156,192)(157,194)(158,193)(159,197)(160,196)(161,195)(162,199)(163,198)
(164,200);
s2 := Sym(245)!( 3,195)( 4,197)( 5,196)( 6,199)( 7,198)( 8,200)( 9,194)
( 10,193)( 11,192)( 12,213)( 13,215)( 14,214)( 15,217)( 16,216)( 17,218)
( 18,212)( 19,211)( 20,210)( 21,204)( 22,206)( 23,205)( 24,208)( 25,207)
( 26,209)( 27,203)( 28,202)( 29,201)( 30,173)( 31,172)( 32,171)( 33,165)
( 34,167)( 35,166)( 36,169)( 37,168)( 38,170)( 39,191)( 40,190)( 41,189)
( 42,183)( 43,185)( 44,184)( 45,187)( 46,186)( 47,188)( 48,182)( 49,181)
( 50,180)( 51,174)( 52,176)( 53,175)( 54,178)( 55,177)( 56,179)( 57,220)
( 58,219)( 59,221)( 60,224)( 61,223)( 62,222)( 63,225)( 64,227)( 65,226)
( 66,238)( 67,237)( 68,239)( 69,242)( 70,241)( 71,240)( 72,243)( 73,245)
( 74,244)( 75,229)( 76,228)( 77,230)( 78,233)( 79,232)( 80,231)( 81,234)
( 82,236)( 83,235)( 84,114)( 85,116)( 86,115)( 87,118)( 88,117)( 89,119)
( 90,113)( 91,112)( 92,111)( 93,132)( 94,134)( 95,133)( 96,136)( 97,135)
( 98,137)( 99,131)(100,130)(101,129)(102,123)(103,125)(104,124)(105,127)
(106,126)(107,128)(108,122)(109,121)(110,120)(138,139)(141,143)(145,146)
(147,157)(148,156)(149,158)(150,161)(151,160)(152,159)(153,162)(154,164)
(155,163);
s3 := Sym(245)!( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 57)( 31, 59)( 32, 58)
( 33, 63)( 34, 65)( 35, 64)( 36, 60)( 37, 62)( 38, 61)( 39, 66)( 40, 68)
( 41, 67)( 42, 72)( 43, 74)( 44, 73)( 45, 69)( 46, 71)( 47, 70)( 48, 75)
( 49, 77)( 50, 76)( 51, 81)( 52, 83)( 53, 82)( 54, 78)( 55, 80)( 56, 79)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)
(103,104)(105,108)(106,110)(107,109)(111,138)(112,140)(113,139)(114,144)
(115,146)(116,145)(117,141)(118,143)(119,142)(120,147)(121,149)(122,148)
(123,153)(124,155)(125,154)(126,150)(127,152)(128,151)(129,156)(130,158)
(131,157)(132,162)(133,164)(134,163)(135,159)(136,161)(137,160)(166,167)
(168,171)(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)(184,185)
(186,189)(187,191)(188,190)(192,219)(193,221)(194,220)(195,225)(196,227)
(197,226)(198,222)(199,224)(200,223)(201,228)(202,230)(203,229)(204,234)
(205,236)(206,235)(207,231)(208,233)(209,232)(210,237)(211,239)(212,238)
(213,243)(214,245)(215,244)(216,240)(217,242)(218,241);
poly := sub<Sym(245)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s2*s1*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope