Polytope of Type {2,6,54}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,54}*1944a
if this polytope has a name.
Group : SmallGroup(1944,948)
Rank : 4
Schlafli Type : {2,6,54}
Number of vertices, edges, etc : 2, 9, 243, 81
Order of s0s1s2s3 : 54
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-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 : {2,6,18}*648a
9-fold quotients : {2,6,6}*216
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 6, 9)( 7, 10)( 8, 11)( 15, 18)( 16, 19)( 17, 20)( 24, 27)( 25, 28)
( 26, 29)( 30, 57)( 31, 58)( 32, 59)( 33, 63)( 34, 64)( 35, 65)( 36, 60)
( 37, 61)( 38, 62)( 39, 66)( 40, 67)( 41, 68)( 42, 72)( 43, 73)( 44, 74)
( 45, 69)( 46, 70)( 47, 71)( 48, 75)( 49, 76)( 50, 77)( 51, 81)( 52, 82)
( 53, 83)( 54, 78)( 55, 79)( 56, 80)( 87, 90)( 88, 91)( 89, 92)( 96, 99)
( 97,100)( 98,101)(105,108)(106,109)(107,110)(111,138)(112,139)(113,140)
(114,144)(115,145)(116,146)(117,141)(118,142)(119,143)(120,147)(121,148)
(122,149)(123,153)(124,154)(125,155)(126,150)(127,151)(128,152)(129,156)
(130,157)(131,158)(132,162)(133,163)(134,164)(135,159)(136,160)(137,161)
(168,171)(169,172)(170,173)(177,180)(178,181)(179,182)(186,189)(187,190)
(188,191)(192,219)(193,220)(194,221)(195,225)(196,226)(197,227)(198,222)
(199,223)(200,224)(201,228)(202,229)(203,230)(204,234)(205,235)(206,236)
(207,231)(208,232)(209,233)(210,237)(211,238)(212,239)(213,243)(214,244)
(215,245)(216,240)(217,241)(218,242);;
s2 := ( 3, 30)( 4, 32)( 5, 31)( 6, 33)( 7, 35)( 8, 34)( 9, 36)( 10, 38)
( 11, 37)( 12, 50)( 13, 49)( 14, 48)( 15, 53)( 16, 52)( 17, 51)( 18, 56)
( 19, 55)( 20, 54)( 21, 41)( 22, 40)( 23, 39)( 24, 44)( 25, 43)( 26, 42)
( 27, 47)( 28, 46)( 29, 45)( 58, 59)( 61, 62)( 64, 65)( 66, 77)( 67, 76)
( 68, 75)( 69, 80)( 70, 79)( 71, 78)( 72, 83)( 73, 82)( 74, 81)( 84,212)
( 85,211)( 86,210)( 87,215)( 88,214)( 89,213)( 90,218)( 91,217)( 92,216)
( 93,203)( 94,202)( 95,201)( 96,206)( 97,205)( 98,204)( 99,209)(100,208)
(101,207)(102,194)(103,193)(104,192)(105,197)(106,196)(107,195)(108,200)
(109,199)(110,198)(111,185)(112,184)(113,183)(114,188)(115,187)(116,186)
(117,191)(118,190)(119,189)(120,176)(121,175)(122,174)(123,179)(124,178)
(125,177)(126,182)(127,181)(128,180)(129,167)(130,166)(131,165)(132,170)
(133,169)(134,168)(135,173)(136,172)(137,171)(138,239)(139,238)(140,237)
(141,242)(142,241)(143,240)(144,245)(145,244)(146,243)(147,230)(148,229)
(149,228)(150,233)(151,232)(152,231)(153,236)(154,235)(155,234)(156,221)
(157,220)(158,219)(159,224)(160,223)(161,222)(162,227)(163,226)(164,225);;
s3 := ( 3, 84)( 4, 86)( 5, 85)( 6, 90)( 7, 92)( 8, 91)( 9, 87)( 10, 89)
( 11, 88)( 12,104)( 13,103)( 14,102)( 15,110)( 16,109)( 17,108)( 18,107)
( 19,106)( 20,105)( 21, 95)( 22, 94)( 23, 93)( 24,101)( 25,100)( 26, 99)
( 27, 98)( 28, 97)( 29, 96)( 30,114)( 31,116)( 32,115)( 33,111)( 34,113)
( 35,112)( 36,117)( 37,119)( 38,118)( 39,134)( 40,133)( 41,132)( 42,131)
( 43,130)( 44,129)( 45,137)( 46,136)( 47,135)( 48,125)( 49,124)( 50,123)
( 51,122)( 52,121)( 53,120)( 54,128)( 55,127)( 56,126)( 57,144)( 58,146)
( 59,145)( 60,141)( 61,143)( 62,142)( 63,138)( 64,140)( 65,139)( 66,164)
( 67,163)( 68,162)( 69,161)( 70,160)( 71,159)( 72,158)( 73,157)( 74,156)
( 75,155)( 76,154)( 77,153)( 78,152)( 79,151)( 80,150)( 81,149)( 82,148)
( 83,147)(165,185)(166,184)(167,183)(168,191)(169,190)(170,189)(171,188)
(172,187)(173,186)(174,176)(177,182)(178,181)(179,180)(192,215)(193,214)
(194,213)(195,212)(196,211)(197,210)(198,218)(199,217)(200,216)(201,206)
(202,205)(203,204)(207,209)(219,245)(220,244)(221,243)(222,242)(223,241)
(224,240)(225,239)(226,238)(227,237)(228,236)(229,235)(230,234)(231,233);;
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*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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)!( 6, 9)( 7, 10)( 8, 11)( 15, 18)( 16, 19)( 17, 20)( 24, 27)
( 25, 28)( 26, 29)( 30, 57)( 31, 58)( 32, 59)( 33, 63)( 34, 64)( 35, 65)
( 36, 60)( 37, 61)( 38, 62)( 39, 66)( 40, 67)( 41, 68)( 42, 72)( 43, 73)
( 44, 74)( 45, 69)( 46, 70)( 47, 71)( 48, 75)( 49, 76)( 50, 77)( 51, 81)
( 52, 82)( 53, 83)( 54, 78)( 55, 79)( 56, 80)( 87, 90)( 88, 91)( 89, 92)
( 96, 99)( 97,100)( 98,101)(105,108)(106,109)(107,110)(111,138)(112,139)
(113,140)(114,144)(115,145)(116,146)(117,141)(118,142)(119,143)(120,147)
(121,148)(122,149)(123,153)(124,154)(125,155)(126,150)(127,151)(128,152)
(129,156)(130,157)(131,158)(132,162)(133,163)(134,164)(135,159)(136,160)
(137,161)(168,171)(169,172)(170,173)(177,180)(178,181)(179,182)(186,189)
(187,190)(188,191)(192,219)(193,220)(194,221)(195,225)(196,226)(197,227)
(198,222)(199,223)(200,224)(201,228)(202,229)(203,230)(204,234)(205,235)
(206,236)(207,231)(208,232)(209,233)(210,237)(211,238)(212,239)(213,243)
(214,244)(215,245)(216,240)(217,241)(218,242);
s2 := Sym(245)!( 3, 30)( 4, 32)( 5, 31)( 6, 33)( 7, 35)( 8, 34)( 9, 36)
( 10, 38)( 11, 37)( 12, 50)( 13, 49)( 14, 48)( 15, 53)( 16, 52)( 17, 51)
( 18, 56)( 19, 55)( 20, 54)( 21, 41)( 22, 40)( 23, 39)( 24, 44)( 25, 43)
( 26, 42)( 27, 47)( 28, 46)( 29, 45)( 58, 59)( 61, 62)( 64, 65)( 66, 77)
( 67, 76)( 68, 75)( 69, 80)( 70, 79)( 71, 78)( 72, 83)( 73, 82)( 74, 81)
( 84,212)( 85,211)( 86,210)( 87,215)( 88,214)( 89,213)( 90,218)( 91,217)
( 92,216)( 93,203)( 94,202)( 95,201)( 96,206)( 97,205)( 98,204)( 99,209)
(100,208)(101,207)(102,194)(103,193)(104,192)(105,197)(106,196)(107,195)
(108,200)(109,199)(110,198)(111,185)(112,184)(113,183)(114,188)(115,187)
(116,186)(117,191)(118,190)(119,189)(120,176)(121,175)(122,174)(123,179)
(124,178)(125,177)(126,182)(127,181)(128,180)(129,167)(130,166)(131,165)
(132,170)(133,169)(134,168)(135,173)(136,172)(137,171)(138,239)(139,238)
(140,237)(141,242)(142,241)(143,240)(144,245)(145,244)(146,243)(147,230)
(148,229)(149,228)(150,233)(151,232)(152,231)(153,236)(154,235)(155,234)
(156,221)(157,220)(158,219)(159,224)(160,223)(161,222)(162,227)(163,226)
(164,225);
s3 := Sym(245)!( 3, 84)( 4, 86)( 5, 85)( 6, 90)( 7, 92)( 8, 91)( 9, 87)
( 10, 89)( 11, 88)( 12,104)( 13,103)( 14,102)( 15,110)( 16,109)( 17,108)
( 18,107)( 19,106)( 20,105)( 21, 95)( 22, 94)( 23, 93)( 24,101)( 25,100)
( 26, 99)( 27, 98)( 28, 97)( 29, 96)( 30,114)( 31,116)( 32,115)( 33,111)
( 34,113)( 35,112)( 36,117)( 37,119)( 38,118)( 39,134)( 40,133)( 41,132)
( 42,131)( 43,130)( 44,129)( 45,137)( 46,136)( 47,135)( 48,125)( 49,124)
( 50,123)( 51,122)( 52,121)( 53,120)( 54,128)( 55,127)( 56,126)( 57,144)
( 58,146)( 59,145)( 60,141)( 61,143)( 62,142)( 63,138)( 64,140)( 65,139)
( 66,164)( 67,163)( 68,162)( 69,161)( 70,160)( 71,159)( 72,158)( 73,157)
( 74,156)( 75,155)( 76,154)( 77,153)( 78,152)( 79,151)( 80,150)( 81,149)
( 82,148)( 83,147)(165,185)(166,184)(167,183)(168,191)(169,190)(170,189)
(171,188)(172,187)(173,186)(174,176)(177,182)(178,181)(179,180)(192,215)
(193,214)(194,213)(195,212)(196,211)(197,210)(198,218)(199,217)(200,216)
(201,206)(202,205)(203,204)(207,209)(219,245)(220,244)(221,243)(222,242)
(223,241)(224,240)(225,239)(226,238)(227,237)(228,236)(229,235)(230,234)
(231,233);
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*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope