Polytope of Type {2,6,27}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,27}*1944a
if this polytope has a name.
Group : SmallGroup(1944,948)
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, {2,6,27}*648
9-fold quotients : {2,2,27}*216, {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, 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, 87)( 7, 89)( 8, 88)( 9, 90)( 10, 92)
( 11, 91)( 12,104)( 13,103)( 14,102)( 15,107)( 16,106)( 17,105)( 18,110)
( 19,109)( 20,108)( 21, 95)( 22, 94)( 23, 93)( 24, 98)( 25, 97)( 26, 96)
( 27,101)( 28,100)( 29, 99)( 30,144)( 31,146)( 32,145)( 33,138)( 34,140)
( 35,139)( 36,141)( 37,143)( 38,142)( 39,164)( 40,163)( 41,162)( 42,158)
( 43,157)( 44,156)( 45,161)( 46,160)( 47,159)( 48,155)( 49,154)( 50,153)
( 51,149)( 52,148)( 53,147)( 54,152)( 55,151)( 56,150)( 57,114)( 58,116)
( 59,115)( 60,117)( 61,119)( 62,118)( 63,111)( 64,113)( 65,112)( 66,134)
( 67,133)( 68,132)( 69,137)( 70,136)( 71,135)( 72,131)( 73,130)( 74,129)
( 75,125)( 76,124)( 77,123)( 78,128)( 79,127)( 80,126)( 81,122)( 82,121)
( 83,120)(165,185)(166,184)(167,183)(168,188)(169,187)(170,186)(171,191)
(172,190)(173,189)(174,176)(177,179)(180,182)(192,245)(193,244)(194,243)
(195,239)(196,238)(197,237)(198,242)(199,241)(200,240)(201,236)(202,235)
(203,234)(204,230)(205,229)(206,228)(207,233)(208,232)(209,231)(210,227)
(211,226)(212,225)(213,221)(214,220)(215,219)(216,224)(217,223)(218,222);;
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,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*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, 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, 87)( 7, 89)( 8, 88)( 9, 90)
( 10, 92)( 11, 91)( 12,104)( 13,103)( 14,102)( 15,107)( 16,106)( 17,105)
( 18,110)( 19,109)( 20,108)( 21, 95)( 22, 94)( 23, 93)( 24, 98)( 25, 97)
( 26, 96)( 27,101)( 28,100)( 29, 99)( 30,144)( 31,146)( 32,145)( 33,138)
( 34,140)( 35,139)( 36,141)( 37,143)( 38,142)( 39,164)( 40,163)( 41,162)
( 42,158)( 43,157)( 44,156)( 45,161)( 46,160)( 47,159)( 48,155)( 49,154)
( 50,153)( 51,149)( 52,148)( 53,147)( 54,152)( 55,151)( 56,150)( 57,114)
( 58,116)( 59,115)( 60,117)( 61,119)( 62,118)( 63,111)( 64,113)( 65,112)
( 66,134)( 67,133)( 68,132)( 69,137)( 70,136)( 71,135)( 72,131)( 73,130)
( 74,129)( 75,125)( 76,124)( 77,123)( 78,128)( 79,127)( 80,126)( 81,122)
( 82,121)( 83,120)(165,185)(166,184)(167,183)(168,188)(169,187)(170,186)
(171,191)(172,190)(173,189)(174,176)(177,179)(180,182)(192,245)(193,244)
(194,243)(195,239)(196,238)(197,237)(198,242)(199,241)(200,240)(201,236)
(202,235)(203,234)(204,230)(205,229)(206,228)(207,233)(208,232)(209,231)
(210,227)(211,226)(212,225)(213,221)(214,220)(215,219)(216,224)(217,223)
(218,222);
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,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*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