Polytope of Type {2,27,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,27,6}*1944a
if this polytope has a name.
Group : SmallGroup(1944,948)
Rank : 4
Schlafli Type : {2,27,6}
Number of vertices, edges, etc : 2, 81, 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 : {2,9,6}*648a, {2,27,6}*648
9-fold quotients : {2,27,2}*216, {2,9,6}*216, {2,3,6}*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)( 7, 8)( 10, 11)( 12, 23)( 13, 22)( 14, 21)( 15, 26)( 16, 25)
( 17, 24)( 18, 29)( 19, 28)( 20, 27)( 30, 57)( 31, 59)( 32, 58)( 33, 60)
( 34, 62)( 35, 61)( 36, 63)( 37, 65)( 38, 64)( 39, 77)( 40, 76)( 41, 75)
( 42, 80)( 43, 79)( 44, 78)( 45, 83)( 46, 82)( 47, 81)( 48, 68)( 49, 67)
( 50, 66)( 51, 71)( 52, 70)( 53, 69)( 54, 74)( 55, 73)( 56, 72)( 84,185)
( 85,184)( 86,183)( 87,188)( 88,187)( 89,186)( 90,191)( 91,190)( 92,189)
( 93,176)( 94,175)( 95,174)( 96,179)( 97,178)( 98,177)( 99,182)(100,181)
(101,180)(102,167)(103,166)(104,165)(105,170)(106,169)(107,168)(108,173)
(109,172)(110,171)(111,239)(112,238)(113,237)(114,242)(115,241)(116,240)
(117,245)(118,244)(119,243)(120,230)(121,229)(122,228)(123,233)(124,232)
(125,231)(126,236)(127,235)(128,234)(129,221)(130,220)(131,219)(132,224)
(133,223)(134,222)(135,227)(136,226)(137,225)(138,212)(139,211)(140,210)
(141,215)(142,214)(143,213)(144,218)(145,217)(146,216)(147,203)(148,202)
(149,201)(150,206)(151,205)(152,204)(153,209)(154,208)(155,207)(156,194)
(157,193)(158,192)(159,197)(160,196)(161,195)(162,200)(163,199)(164,198);;
s2 := ( 3,195)( 4,197)( 5,196)( 6,198)( 7,200)( 8,199)( 9,192)( 10,194)
( 11,193)( 12,215)( 13,214)( 14,213)( 15,218)( 16,217)( 17,216)( 18,212)
( 19,211)( 20,210)( 21,206)( 22,205)( 23,204)( 24,209)( 25,208)( 26,207)
( 27,203)( 28,202)( 29,201)( 30,171)( 31,173)( 32,172)( 33,165)( 34,167)
( 35,166)( 36,168)( 37,170)( 38,169)( 39,191)( 40,190)( 41,189)( 42,185)
( 43,184)( 44,183)( 45,188)( 46,187)( 47,186)( 48,182)( 49,181)( 50,180)
( 51,176)( 52,175)( 53,174)( 54,179)( 55,178)( 56,177)( 57,219)( 58,221)
( 59,220)( 60,222)( 61,224)( 62,223)( 63,225)( 64,227)( 65,226)( 66,239)
( 67,238)( 68,237)( 69,242)( 70,241)( 71,240)( 72,245)( 73,244)( 74,243)
( 75,230)( 76,229)( 77,228)( 78,233)( 79,232)( 80,231)( 81,236)( 82,235)
( 83,234)( 84,114)( 85,116)( 86,115)( 87,117)( 88,119)( 89,118)( 90,111)
( 91,113)( 92,112)( 93,134)( 94,133)( 95,132)( 96,137)( 97,136)( 98,135)
( 99,131)(100,130)(101,129)(102,125)(103,124)(104,123)(105,128)(106,127)
(107,126)(108,122)(109,121)(110,120)(139,140)(142,143)(145,146)(147,158)
(148,157)(149,156)(150,161)(151,160)(152,159)(153,164)(154,163)(155,162);;
s3 := ( 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);;
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*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(245)!(1,2);
s1 := Sym(245)!( 4, 5)( 7, 8)( 10, 11)( 12, 23)( 13, 22)( 14, 21)( 15, 26)
( 16, 25)( 17, 24)( 18, 29)( 19, 28)( 20, 27)( 30, 57)( 31, 59)( 32, 58)
( 33, 60)( 34, 62)( 35, 61)( 36, 63)( 37, 65)( 38, 64)( 39, 77)( 40, 76)
( 41, 75)( 42, 80)( 43, 79)( 44, 78)( 45, 83)( 46, 82)( 47, 81)( 48, 68)
( 49, 67)( 50, 66)( 51, 71)( 52, 70)( 53, 69)( 54, 74)( 55, 73)( 56, 72)
( 84,185)( 85,184)( 86,183)( 87,188)( 88,187)( 89,186)( 90,191)( 91,190)
( 92,189)( 93,176)( 94,175)( 95,174)( 96,179)( 97,178)( 98,177)( 99,182)
(100,181)(101,180)(102,167)(103,166)(104,165)(105,170)(106,169)(107,168)
(108,173)(109,172)(110,171)(111,239)(112,238)(113,237)(114,242)(115,241)
(116,240)(117,245)(118,244)(119,243)(120,230)(121,229)(122,228)(123,233)
(124,232)(125,231)(126,236)(127,235)(128,234)(129,221)(130,220)(131,219)
(132,224)(133,223)(134,222)(135,227)(136,226)(137,225)(138,212)(139,211)
(140,210)(141,215)(142,214)(143,213)(144,218)(145,217)(146,216)(147,203)
(148,202)(149,201)(150,206)(151,205)(152,204)(153,209)(154,208)(155,207)
(156,194)(157,193)(158,192)(159,197)(160,196)(161,195)(162,200)(163,199)
(164,198);
s2 := Sym(245)!( 3,195)( 4,197)( 5,196)( 6,198)( 7,200)( 8,199)( 9,192)
( 10,194)( 11,193)( 12,215)( 13,214)( 14,213)( 15,218)( 16,217)( 17,216)
( 18,212)( 19,211)( 20,210)( 21,206)( 22,205)( 23,204)( 24,209)( 25,208)
( 26,207)( 27,203)( 28,202)( 29,201)( 30,171)( 31,173)( 32,172)( 33,165)
( 34,167)( 35,166)( 36,168)( 37,170)( 38,169)( 39,191)( 40,190)( 41,189)
( 42,185)( 43,184)( 44,183)( 45,188)( 46,187)( 47,186)( 48,182)( 49,181)
( 50,180)( 51,176)( 52,175)( 53,174)( 54,179)( 55,178)( 56,177)( 57,219)
( 58,221)( 59,220)( 60,222)( 61,224)( 62,223)( 63,225)( 64,227)( 65,226)
( 66,239)( 67,238)( 68,237)( 69,242)( 70,241)( 71,240)( 72,245)( 73,244)
( 74,243)( 75,230)( 76,229)( 77,228)( 78,233)( 79,232)( 80,231)( 81,236)
( 82,235)( 83,234)( 84,114)( 85,116)( 86,115)( 87,117)( 88,119)( 89,118)
( 90,111)( 91,113)( 92,112)( 93,134)( 94,133)( 95,132)( 96,137)( 97,136)
( 98,135)( 99,131)(100,130)(101,129)(102,125)(103,124)(104,123)(105,128)
(106,127)(107,126)(108,122)(109,121)(110,120)(139,140)(142,143)(145,146)
(147,158)(148,157)(149,156)(150,161)(151,160)(152,159)(153,164)(154,163)
(155,162);
s3 := 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);
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*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*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 >;
to this polytope