Polytope of Type {2,9,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,6}*1944e
if this polytope has a name.
Group : SmallGroup(1944,952)
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}*648c, {2,9,6}*648d, {2,3,6}*648
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)( 6,198)( 7,200)( 8,199)( 9,192)( 10,194)
( 11,193)( 12,213)( 13,215)( 14,214)( 15,216)( 16,218)( 17,217)( 18,210)
( 19,212)( 20,211)( 21,204)( 22,206)( 23,205)( 24,207)( 25,209)( 26,208)
( 27,201)( 28,203)( 29,202)( 30,171)( 31,173)( 32,172)( 33,165)( 34,167)
( 35,166)( 36,168)( 37,170)( 38,169)( 39,189)( 40,191)( 41,190)( 42,183)
( 43,185)( 44,184)( 45,186)( 46,188)( 47,187)( 48,180)( 49,182)( 50,181)
( 51,174)( 52,176)( 53,175)( 54,177)( 55,179)( 56,178)( 57,221)( 58,220)
( 59,219)( 60,224)( 61,223)( 62,222)( 63,227)( 64,226)( 65,225)( 66,239)
( 67,238)( 68,237)( 69,242)( 70,241)( 71,240)( 72,245)( 73,244)( 74,243)
( 75,230)( 76,229)( 77,228)( 78,233)( 79,232)( 80,231)( 81,236)( 82,235)
( 83,234)( 84,114)( 85,116)( 86,115)( 87,117)( 88,119)( 89,118)( 90,111)
( 91,113)( 92,112)( 93,132)( 94,134)( 95,133)( 96,135)( 97,137)( 98,136)
( 99,129)(100,131)(101,130)(102,123)(103,125)(104,124)(105,126)(106,128)
(107,127)(108,120)(109,122)(110,121)(138,140)(141,143)(144,146)(147,158)
(148,157)(149,156)(150,161)(151,160)(152,159)(153,164)(154,163)(155,162);;
s3 := ( 6, 10)( 7, 11)( 8, 9)( 15, 19)( 16, 20)( 17, 18)( 24, 28)( 25, 29)
( 26, 27)( 30, 57)( 31, 58)( 32, 59)( 33, 64)( 34, 65)( 35, 63)( 36, 62)
( 37, 60)( 38, 61)( 39, 66)( 40, 67)( 41, 68)( 42, 73)( 43, 74)( 44, 72)
( 45, 71)( 46, 69)( 47, 70)( 48, 75)( 49, 76)( 50, 77)( 51, 82)( 52, 83)
( 53, 81)( 54, 80)( 55, 78)( 56, 79)( 87, 91)( 88, 92)( 89, 90)( 96,100)
( 97,101)( 98, 99)(105,109)(106,110)(107,108)(111,138)(112,139)(113,140)
(114,145)(115,146)(116,144)(117,143)(118,141)(119,142)(120,147)(121,148)
(122,149)(123,154)(124,155)(125,153)(126,152)(127,150)(128,151)(129,156)
(130,157)(131,158)(132,163)(133,164)(134,162)(135,161)(136,159)(137,160)
(168,172)(169,173)(170,171)(177,181)(178,182)(179,180)(186,190)(187,191)
(188,189)(192,219)(193,220)(194,221)(195,226)(196,227)(197,225)(198,224)
(199,222)(200,223)(201,228)(202,229)(203,230)(204,235)(205,236)(206,234)
(207,233)(208,231)(209,232)(210,237)(211,238)(212,239)(213,244)(214,245)
(215,243)(216,242)(217,240)(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,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*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)( 6,198)( 7,200)( 8,199)( 9,192)
( 10,194)( 11,193)( 12,213)( 13,215)( 14,214)( 15,216)( 16,218)( 17,217)
( 18,210)( 19,212)( 20,211)( 21,204)( 22,206)( 23,205)( 24,207)( 25,209)
( 26,208)( 27,201)( 28,203)( 29,202)( 30,171)( 31,173)( 32,172)( 33,165)
( 34,167)( 35,166)( 36,168)( 37,170)( 38,169)( 39,189)( 40,191)( 41,190)
( 42,183)( 43,185)( 44,184)( 45,186)( 46,188)( 47,187)( 48,180)( 49,182)
( 50,181)( 51,174)( 52,176)( 53,175)( 54,177)( 55,179)( 56,178)( 57,221)
( 58,220)( 59,219)( 60,224)( 61,223)( 62,222)( 63,227)( 64,226)( 65,225)
( 66,239)( 67,238)( 68,237)( 69,242)( 70,241)( 71,240)( 72,245)( 73,244)
( 74,243)( 75,230)( 76,229)( 77,228)( 78,233)( 79,232)( 80,231)( 81,236)
( 82,235)( 83,234)( 84,114)( 85,116)( 86,115)( 87,117)( 88,119)( 89,118)
( 90,111)( 91,113)( 92,112)( 93,132)( 94,134)( 95,133)( 96,135)( 97,137)
( 98,136)( 99,129)(100,131)(101,130)(102,123)(103,125)(104,124)(105,126)
(106,128)(107,127)(108,120)(109,122)(110,121)(138,140)(141,143)(144,146)
(147,158)(148,157)(149,156)(150,161)(151,160)(152,159)(153,164)(154,163)
(155,162);
s3 := Sym(245)!( 6, 10)( 7, 11)( 8, 9)( 15, 19)( 16, 20)( 17, 18)( 24, 28)
( 25, 29)( 26, 27)( 30, 57)( 31, 58)( 32, 59)( 33, 64)( 34, 65)( 35, 63)
( 36, 62)( 37, 60)( 38, 61)( 39, 66)( 40, 67)( 41, 68)( 42, 73)( 43, 74)
( 44, 72)( 45, 71)( 46, 69)( 47, 70)( 48, 75)( 49, 76)( 50, 77)( 51, 82)
( 52, 83)( 53, 81)( 54, 80)( 55, 78)( 56, 79)( 87, 91)( 88, 92)( 89, 90)
( 96,100)( 97,101)( 98, 99)(105,109)(106,110)(107,108)(111,138)(112,139)
(113,140)(114,145)(115,146)(116,144)(117,143)(118,141)(119,142)(120,147)
(121,148)(122,149)(123,154)(124,155)(125,153)(126,152)(127,150)(128,151)
(129,156)(130,157)(131,158)(132,163)(133,164)(134,162)(135,161)(136,159)
(137,160)(168,172)(169,173)(170,171)(177,181)(178,182)(179,180)(186,190)
(187,191)(188,189)(192,219)(193,220)(194,221)(195,226)(196,227)(197,225)
(198,224)(199,222)(200,223)(201,228)(202,229)(203,230)(204,235)(205,236)
(206,234)(207,233)(208,231)(209,232)(210,237)(211,238)(212,239)(213,244)
(214,245)(215,243)(216,242)(217,240)(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, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2 >;
to this polytope