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