Polytope of Type {2,27,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,27,18}*1944
if this polytope has a name.
Group : SmallGroup(1944,947)
Rank : 4
Schlafli Type : {2,27,18}
Number of vertices, edges, etc : 2, 27, 243, 18
Order of s₀s₁s₂s₃ : 54
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 : {2,9,18}*648, {2,27,6}*648
   9-fold quotients : {2,27,2}*216, {2,9,6}*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,  9)(  7, 11)(  8, 10)( 12, 23)( 13, 22)( 14, 21)( 15, 29)
( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 30, 60)( 31, 62)( 32, 61)
( 33, 57)( 34, 59)( 35, 58)( 36, 63)( 37, 65)( 38, 64)( 39, 80)( 40, 79)
( 41, 78)( 42, 77)( 43, 76)( 44, 75)( 45, 83)( 46, 82)( 47, 81)( 48, 71)
( 49, 70)( 50, 69)( 51, 68)( 52, 67)( 53, 66)( 54, 74)( 55, 73)( 56, 72)
( 84,185)( 85,184)( 86,183)( 87,191)( 88,190)( 89,189)( 90,188)( 91,187)
( 92,186)( 93,176)( 94,175)( 95,174)( 96,182)( 97,181)( 98,180)( 99,179)
(100,178)(101,177)(102,167)(103,166)(104,165)(105,173)(106,172)(107,171)
(108,170)(109,169)(110,168)(111,242)(112,241)(113,240)(114,239)(115,238)
(116,237)(117,245)(118,244)(119,243)(120,233)(121,232)(122,231)(123,230)
(124,229)(125,228)(126,236)(127,235)(128,234)(129,224)(130,223)(131,222)
(132,221)(133,220)(134,219)(135,227)(136,226)(137,225)(138,215)(139,214)
(140,213)(141,212)(142,211)(143,210)(144,218)(145,217)(146,216)(147,206)
(148,205)(149,204)(150,203)(151,202)(152,201)(153,209)(154,208)(155,207)
(156,197)(157,196)(158,195)(159,194)(160,193)(161,192)(162,200)(163,199)
(164,198);;
s2 := (  3,111)(  4,113)(  5,112)(  6,117)(  7,119)(  8,118)(  9,114)( 10,116)
( 11,115)( 12,131)( 13,130)( 14,129)( 15,137)( 16,136)( 17,135)( 18,134)
( 19,133)( 20,132)( 21,122)( 22,121)( 23,120)( 24,128)( 25,127)( 26,126)
( 27,125)( 28,124)( 29,123)( 30, 84)( 31, 86)( 32, 85)( 33, 90)( 34, 92)
( 35, 91)( 36, 87)( 37, 89)( 38, 88)( 39,104)( 40,103)( 41,102)( 42,110)
( 43,109)( 44,108)( 45,107)( 46,106)( 47,105)( 48, 95)( 49, 94)( 50, 93)
( 51,101)( 52,100)( 53, 99)( 54, 98)( 55, 97)( 56, 96)( 57,141)( 58,143)
( 59,142)( 60,138)( 61,140)( 62,139)( 63,144)( 64,146)( 65,145)( 66,161)
( 67,160)( 68,159)( 69,158)( 70,157)( 71,156)( 72,164)( 73,163)( 74,162)
( 75,152)( 76,151)( 77,150)( 78,149)( 79,148)( 80,147)( 81,155)( 82,154)
( 83,153)(165,212)(166,211)(167,210)(168,218)(169,217)(170,216)(171,215)
(172,214)(173,213)(174,203)(175,202)(176,201)(177,209)(178,208)(179,207)
(180,206)(181,205)(182,204)(183,194)(184,193)(185,192)(186,200)(187,199)
(188,198)(189,197)(190,196)(191,195)(219,242)(220,241)(221,240)(222,239)
(223,238)(224,237)(225,245)(226,244)(227,243)(228,233)(229,232)(230,231)
(234,236);;
s3 := (  6,  9)(  7, 10)(  8, 11)( 15, 18)( 16, 19)( 17, 20)( 24, 27)( 25, 28)
( 26, 29)( 30, 60)( 31, 61)( 32, 62)( 33, 57)( 34, 58)( 35, 59)( 36, 63)
( 37, 64)( 38, 65)( 39, 69)( 40, 70)( 41, 71)( 42, 66)( 43, 67)( 44, 68)
( 45, 72)( 46, 73)( 47, 74)( 48, 78)( 49, 79)( 50, 80)( 51, 75)( 52, 76)
( 53, 77)( 54, 81)( 55, 82)( 56, 83)( 87, 90)( 88, 91)( 89, 92)( 96, 99)
( 97,100)( 98,101)(105,108)(106,109)(107,110)(111,141)(112,142)(113,143)
(114,138)(115,139)(116,140)(117,144)(118,145)(119,146)(120,150)(121,151)
(122,152)(123,147)(124,148)(125,149)(126,153)(127,154)(128,155)(129,159)
(130,160)(131,161)(132,156)(133,157)(134,158)(135,162)(136,163)(137,164)
(168,171)(169,172)(170,173)(177,180)(178,181)(179,182)(186,189)(187,190)
(188,191)(192,222)(193,223)(194,224)(195,219)(196,220)(197,221)(198,225)
(199,226)(200,227)(201,231)(202,232)(203,233)(204,228)(205,229)(206,230)
(207,234)(208,235)(209,236)(210,240)(211,241)(212,242)(213,237)(214,238)
(215,239)(216,243)(217,244)(218,245);;
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, 
s3*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*s3*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(245)!(1,2);
s1 := Sym(245)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 12, 23)( 13, 22)( 14, 21)
( 15, 29)( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 30, 60)( 31, 62)
( 32, 61)( 33, 57)( 34, 59)( 35, 58)( 36, 63)( 37, 65)( 38, 64)( 39, 80)
( 40, 79)( 41, 78)( 42, 77)( 43, 76)( 44, 75)( 45, 83)( 46, 82)( 47, 81)
( 48, 71)( 49, 70)( 50, 69)( 51, 68)( 52, 67)( 53, 66)( 54, 74)( 55, 73)
( 56, 72)( 84,185)( 85,184)( 86,183)( 87,191)( 88,190)( 89,189)( 90,188)
( 91,187)( 92,186)( 93,176)( 94,175)( 95,174)( 96,182)( 97,181)( 98,180)
( 99,179)(100,178)(101,177)(102,167)(103,166)(104,165)(105,173)(106,172)
(107,171)(108,170)(109,169)(110,168)(111,242)(112,241)(113,240)(114,239)
(115,238)(116,237)(117,245)(118,244)(119,243)(120,233)(121,232)(122,231)
(123,230)(124,229)(125,228)(126,236)(127,235)(128,234)(129,224)(130,223)
(131,222)(132,221)(133,220)(134,219)(135,227)(136,226)(137,225)(138,215)
(139,214)(140,213)(141,212)(142,211)(143,210)(144,218)(145,217)(146,216)
(147,206)(148,205)(149,204)(150,203)(151,202)(152,201)(153,209)(154,208)
(155,207)(156,197)(157,196)(158,195)(159,194)(160,193)(161,192)(162,200)
(163,199)(164,198);
s2 := Sym(245)!(  3,111)(  4,113)(  5,112)(  6,117)(  7,119)(  8,118)(  9,114)
( 10,116)( 11,115)( 12,131)( 13,130)( 14,129)( 15,137)( 16,136)( 17,135)
( 18,134)( 19,133)( 20,132)( 21,122)( 22,121)( 23,120)( 24,128)( 25,127)
( 26,126)( 27,125)( 28,124)( 29,123)( 30, 84)( 31, 86)( 32, 85)( 33, 90)
( 34, 92)( 35, 91)( 36, 87)( 37, 89)( 38, 88)( 39,104)( 40,103)( 41,102)
( 42,110)( 43,109)( 44,108)( 45,107)( 46,106)( 47,105)( 48, 95)( 49, 94)
( 50, 93)( 51,101)( 52,100)( 53, 99)( 54, 98)( 55, 97)( 56, 96)( 57,141)
( 58,143)( 59,142)( 60,138)( 61,140)( 62,139)( 63,144)( 64,146)( 65,145)
( 66,161)( 67,160)( 68,159)( 69,158)( 70,157)( 71,156)( 72,164)( 73,163)
( 74,162)( 75,152)( 76,151)( 77,150)( 78,149)( 79,148)( 80,147)( 81,155)
( 82,154)( 83,153)(165,212)(166,211)(167,210)(168,218)(169,217)(170,216)
(171,215)(172,214)(173,213)(174,203)(175,202)(176,201)(177,209)(178,208)
(179,207)(180,206)(181,205)(182,204)(183,194)(184,193)(185,192)(186,200)
(187,199)(188,198)(189,197)(190,196)(191,195)(219,242)(220,241)(221,240)
(222,239)(223,238)(224,237)(225,245)(226,244)(227,243)(228,233)(229,232)
(230,231)(234,236);
s3 := Sym(245)!(  6,  9)(  7, 10)(  8, 11)( 15, 18)( 16, 19)( 17, 20)( 24, 27)
( 25, 28)( 26, 29)( 30, 60)( 31, 61)( 32, 62)( 33, 57)( 34, 58)( 35, 59)
( 36, 63)( 37, 64)( 38, 65)( 39, 69)( 40, 70)( 41, 71)( 42, 66)( 43, 67)
( 44, 68)( 45, 72)( 46, 73)( 47, 74)( 48, 78)( 49, 79)( 50, 80)( 51, 75)
( 52, 76)( 53, 77)( 54, 81)( 55, 82)( 56, 83)( 87, 90)( 88, 91)( 89, 92)
( 96, 99)( 97,100)( 98,101)(105,108)(106,109)(107,110)(111,141)(112,142)
(113,143)(114,138)(115,139)(116,140)(117,144)(118,145)(119,146)(120,150)
(121,151)(122,152)(123,147)(124,148)(125,149)(126,153)(127,154)(128,155)
(129,159)(130,160)(131,161)(132,156)(133,157)(134,158)(135,162)(136,163)
(137,164)(168,171)(169,172)(170,173)(177,180)(178,181)(179,182)(186,189)
(187,190)(188,191)(192,222)(193,223)(194,224)(195,219)(196,220)(197,221)
(198,225)(199,226)(200,227)(201,231)(202,232)(203,233)(204,228)(205,229)
(206,230)(207,234)(208,235)(209,236)(210,240)(211,241)(212,242)(213,237)
(214,238)(215,239)(216,243)(217,244)(218,245);
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, s3*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*s3*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 >;

to this polytope