Polytope of Type {27,18,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {27,18,2}*1944
if this polytope has a name.
Group : SmallGroup(1944,947)
Rank : 4
Schlafli Type : {27,18,2}
Number of vertices, edges, etc : 27, 243, 18, 2
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 : {9,18,2}*648, {27,6,2}*648
   9-fold quotients : {27,2,2}*216, {9,6,2}*216
   27-fold quotients : {9,2,2}*72, {3,6,2}*72
   81-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 21)( 11, 20)( 12, 19)( 13, 27)
( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 28, 58)( 29, 60)( 30, 59)
( 31, 55)( 32, 57)( 33, 56)( 34, 61)( 35, 63)( 36, 62)( 37, 78)( 38, 77)
( 39, 76)( 40, 75)( 41, 74)( 42, 73)( 43, 81)( 44, 80)( 45, 79)( 46, 69)
( 47, 68)( 48, 67)( 49, 66)( 50, 65)( 51, 64)( 52, 72)( 53, 71)( 54, 70)
( 82,183)( 83,182)( 84,181)( 85,189)( 86,188)( 87,187)( 88,186)( 89,185)
( 90,184)( 91,174)( 92,173)( 93,172)( 94,180)( 95,179)( 96,178)( 97,177)
( 98,176)( 99,175)(100,165)(101,164)(102,163)(103,171)(104,170)(105,169)
(106,168)(107,167)(108,166)(109,240)(110,239)(111,238)(112,237)(113,236)
(114,235)(115,243)(116,242)(117,241)(118,231)(119,230)(120,229)(121,228)
(122,227)(123,226)(124,234)(125,233)(126,232)(127,222)(128,221)(129,220)
(130,219)(131,218)(132,217)(133,225)(134,224)(135,223)(136,213)(137,212)
(138,211)(139,210)(140,209)(141,208)(142,216)(143,215)(144,214)(145,204)
(146,203)(147,202)(148,201)(149,200)(150,199)(151,207)(152,206)(153,205)
(154,195)(155,194)(156,193)(157,192)(158,191)(159,190)(160,198)(161,197)
(162,196);;
s1 := (  1,109)(  2,111)(  3,110)(  4,115)(  5,117)(  6,116)(  7,112)(  8,114)
(  9,113)( 10,129)( 11,128)( 12,127)( 13,135)( 14,134)( 15,133)( 16,132)
( 17,131)( 18,130)( 19,120)( 20,119)( 21,118)( 22,126)( 23,125)( 24,124)
( 25,123)( 26,122)( 27,121)( 28, 82)( 29, 84)( 30, 83)( 31, 88)( 32, 90)
( 33, 89)( 34, 85)( 35, 87)( 36, 86)( 37,102)( 38,101)( 39,100)( 40,108)
( 41,107)( 42,106)( 43,105)( 44,104)( 45,103)( 46, 93)( 47, 92)( 48, 91)
( 49, 99)( 50, 98)( 51, 97)( 52, 96)( 53, 95)( 54, 94)( 55,139)( 56,141)
( 57,140)( 58,136)( 59,138)( 60,137)( 61,142)( 62,144)( 63,143)( 64,159)
( 65,158)( 66,157)( 67,156)( 68,155)( 69,154)( 70,162)( 71,161)( 72,160)
( 73,150)( 74,149)( 75,148)( 76,147)( 77,146)( 78,145)( 79,153)( 80,152)
( 81,151)(163,210)(164,209)(165,208)(166,216)(167,215)(168,214)(169,213)
(170,212)(171,211)(172,201)(173,200)(174,199)(175,207)(176,206)(177,205)
(178,204)(179,203)(180,202)(181,192)(182,191)(183,190)(184,198)(185,197)
(186,196)(187,195)(188,194)(189,193)(217,240)(218,239)(219,238)(220,237)
(221,236)(222,235)(223,243)(224,242)(225,241)(226,231)(227,230)(228,229)
(232,234);;
s2 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 28, 58)( 29, 59)( 30, 60)( 31, 55)( 32, 56)( 33, 57)( 34, 61)
( 35, 62)( 36, 63)( 37, 67)( 38, 68)( 39, 69)( 40, 64)( 41, 65)( 42, 66)
( 43, 70)( 44, 71)( 45, 72)( 46, 76)( 47, 77)( 48, 78)( 49, 73)( 50, 74)
( 51, 75)( 52, 79)( 53, 80)( 54, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,139)(110,140)(111,141)
(112,136)(113,137)(114,138)(115,142)(116,143)(117,144)(118,148)(119,149)
(120,150)(121,145)(122,146)(123,147)(124,151)(125,152)(126,153)(127,157)
(128,158)(129,159)(130,154)(131,155)(132,156)(133,160)(134,161)(135,162)
(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)
(186,189)(190,220)(191,221)(192,222)(193,217)(194,218)(195,219)(196,223)
(197,224)(198,225)(199,229)(200,230)(201,231)(202,226)(203,227)(204,228)
(205,232)(206,233)(207,234)(208,238)(209,239)(210,240)(211,235)(212,236)
(213,237)(214,241)(215,242)(216,243);;
s3 := (244,245);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(245)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 21)( 11, 20)( 12, 19)
( 13, 27)( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 28, 58)( 29, 60)
( 30, 59)( 31, 55)( 32, 57)( 33, 56)( 34, 61)( 35, 63)( 36, 62)( 37, 78)
( 38, 77)( 39, 76)( 40, 75)( 41, 74)( 42, 73)( 43, 81)( 44, 80)( 45, 79)
( 46, 69)( 47, 68)( 48, 67)( 49, 66)( 50, 65)( 51, 64)( 52, 72)( 53, 71)
( 54, 70)( 82,183)( 83,182)( 84,181)( 85,189)( 86,188)( 87,187)( 88,186)
( 89,185)( 90,184)( 91,174)( 92,173)( 93,172)( 94,180)( 95,179)( 96,178)
( 97,177)( 98,176)( 99,175)(100,165)(101,164)(102,163)(103,171)(104,170)
(105,169)(106,168)(107,167)(108,166)(109,240)(110,239)(111,238)(112,237)
(113,236)(114,235)(115,243)(116,242)(117,241)(118,231)(119,230)(120,229)
(121,228)(122,227)(123,226)(124,234)(125,233)(126,232)(127,222)(128,221)
(129,220)(130,219)(131,218)(132,217)(133,225)(134,224)(135,223)(136,213)
(137,212)(138,211)(139,210)(140,209)(141,208)(142,216)(143,215)(144,214)
(145,204)(146,203)(147,202)(148,201)(149,200)(150,199)(151,207)(152,206)
(153,205)(154,195)(155,194)(156,193)(157,192)(158,191)(159,190)(160,198)
(161,197)(162,196);
s1 := Sym(245)!(  1,109)(  2,111)(  3,110)(  4,115)(  5,117)(  6,116)(  7,112)
(  8,114)(  9,113)( 10,129)( 11,128)( 12,127)( 13,135)( 14,134)( 15,133)
( 16,132)( 17,131)( 18,130)( 19,120)( 20,119)( 21,118)( 22,126)( 23,125)
( 24,124)( 25,123)( 26,122)( 27,121)( 28, 82)( 29, 84)( 30, 83)( 31, 88)
( 32, 90)( 33, 89)( 34, 85)( 35, 87)( 36, 86)( 37,102)( 38,101)( 39,100)
( 40,108)( 41,107)( 42,106)( 43,105)( 44,104)( 45,103)( 46, 93)( 47, 92)
( 48, 91)( 49, 99)( 50, 98)( 51, 97)( 52, 96)( 53, 95)( 54, 94)( 55,139)
( 56,141)( 57,140)( 58,136)( 59,138)( 60,137)( 61,142)( 62,144)( 63,143)
( 64,159)( 65,158)( 66,157)( 67,156)( 68,155)( 69,154)( 70,162)( 71,161)
( 72,160)( 73,150)( 74,149)( 75,148)( 76,147)( 77,146)( 78,145)( 79,153)
( 80,152)( 81,151)(163,210)(164,209)(165,208)(166,216)(167,215)(168,214)
(169,213)(170,212)(171,211)(172,201)(173,200)(174,199)(175,207)(176,206)
(177,205)(178,204)(179,203)(180,202)(181,192)(182,191)(183,190)(184,198)
(185,197)(186,196)(187,195)(188,194)(189,193)(217,240)(218,239)(219,238)
(220,237)(221,236)(222,235)(223,243)(224,242)(225,241)(226,231)(227,230)
(228,229)(232,234);
s2 := Sym(245)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 28, 58)( 29, 59)( 30, 60)( 31, 55)( 32, 56)( 33, 57)
( 34, 61)( 35, 62)( 36, 63)( 37, 67)( 38, 68)( 39, 69)( 40, 64)( 41, 65)
( 42, 66)( 43, 70)( 44, 71)( 45, 72)( 46, 76)( 47, 77)( 48, 78)( 49, 73)
( 50, 74)( 51, 75)( 52, 79)( 53, 80)( 54, 81)( 85, 88)( 86, 89)( 87, 90)
( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,139)(110,140)
(111,141)(112,136)(113,137)(114,138)(115,142)(116,143)(117,144)(118,148)
(119,149)(120,150)(121,145)(122,146)(123,147)(124,151)(125,152)(126,153)
(127,157)(128,158)(129,159)(130,154)(131,155)(132,156)(133,160)(134,161)
(135,162)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)
(185,188)(186,189)(190,220)(191,221)(192,222)(193,217)(194,218)(195,219)
(196,223)(197,224)(198,225)(199,229)(200,230)(201,231)(202,226)(203,227)
(204,228)(205,232)(206,233)(207,234)(208,238)(209,239)(210,240)(211,235)
(212,236)(213,237)(214,241)(215,242)(216,243);
s3 := Sym(245)!(244,245);
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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

to this polytope