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