Polytope of Type {2,6,27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,27}*1944b
if this polytope has a name.
Group : SmallGroup(1944,953)
Rank : 4
Schlafli Type : {2,6,27}
Number of vertices, edges, etc : 2, 18, 243, 81
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 : {2,6,9}*648a
   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 := (  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);;
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, 89)(  7, 88)(  8, 87)(  9, 91)( 10, 90)
( 11, 92)( 12,104)( 13,103)( 14,102)( 15,106)( 16,105)( 17,107)( 18,108)
( 19,110)( 20,109)( 21, 95)( 22, 94)( 23, 93)( 24, 97)( 25, 96)( 26, 98)
( 27, 99)( 28,101)( 29,100)( 30,145)( 31,144)( 32,146)( 33,138)( 34,140)
( 35,139)( 36,143)( 37,142)( 38,141)( 39,162)( 40,164)( 41,163)( 42,158)
( 43,157)( 44,156)( 45,160)( 46,159)( 47,161)( 48,153)( 49,155)( 50,154)
( 51,149)( 52,148)( 53,147)( 54,151)( 55,150)( 56,152)( 57,114)( 58,116)
( 59,115)( 60,119)( 61,118)( 62,117)( 63,112)( 64,111)( 65,113)( 66,134)
( 67,133)( 68,132)( 69,136)( 70,135)( 71,137)( 72,129)( 73,131)( 74,130)
( 75,125)( 76,124)( 77,123)( 78,127)( 79,126)( 80,128)( 81,120)( 82,122)
( 83,121)(165,185)(166,184)(167,183)(168,187)(169,186)(170,188)(171,189)
(172,191)(173,190)(174,176)(177,178)(181,182)(192,243)(193,245)(194,244)
(195,239)(196,238)(197,237)(198,241)(199,240)(200,242)(201,234)(202,236)
(203,235)(204,230)(205,229)(206,228)(207,232)(208,231)(209,233)(210,225)
(211,227)(212,226)(213,221)(214,220)(215,219)(216,223)(217,222)(218,224);;
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, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s1*s3*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope