Polytope of Type {2,3,6,27}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,6,27}*1944
if this polytope has a name.
Group : SmallGroup(1944,948)
Rank : 5
Schlafli Type : {2,3,6,27}
Number of vertices, edges, etc : 2, 3, 9, 81, 27
Order of s0s1s2s3s4 : 54
Order of s0s1s2s3s4s3s2s1 : 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 : {2,3,6,9}*648, {2,3,2,27}*648
9-fold quotients : {2,3,2,9}*216, {2,3,6,3}*216
27-fold quotients : {2,3,2,3}*72
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, 31)( 5, 32)( 6, 36)( 7, 37)( 8, 38)( 9, 33)( 10, 34)
( 11, 35)( 12, 39)( 13, 40)( 14, 41)( 15, 45)( 16, 46)( 17, 47)( 18, 42)
( 19, 43)( 20, 44)( 21, 48)( 22, 49)( 23, 50)( 24, 54)( 25, 55)( 26, 56)
( 27, 51)( 28, 52)( 29, 53)( 60, 63)( 61, 64)( 62, 65)( 69, 72)( 70, 73)
( 71, 74)( 78, 81)( 79, 82)( 80, 83)( 84,111)( 85,112)( 86,113)( 87,117)
( 88,118)( 89,119)( 90,114)( 91,115)( 92,116)( 93,120)( 94,121)( 95,122)
( 96,126)( 97,127)( 98,128)( 99,123)(100,124)(101,125)(102,129)(103,130)
(104,131)(105,135)(106,136)(107,137)(108,132)(109,133)(110,134)(141,144)
(142,145)(143,146)(150,153)(151,154)(152,155)(159,162)(160,163)(161,164)
(165,192)(166,193)(167,194)(168,198)(169,199)(170,200)(171,195)(172,196)
(173,197)(174,201)(175,202)(176,203)(177,207)(178,208)(179,209)(180,204)
(181,205)(182,206)(183,210)(184,211)(185,212)(186,216)(187,217)(188,218)
(189,213)(190,214)(191,215)(222,225)(223,226)(224,227)(231,234)(232,235)
(233,236)(240,243)(241,244)(242,245);;
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)( 30,114)( 31,116)( 32,115)( 33,111)( 34,113)
( 35,112)( 36,117)( 37,119)( 38,118)( 39,134)( 40,133)( 41,132)( 42,131)
( 43,130)( 44,129)( 45,137)( 46,136)( 47,135)( 48,125)( 49,124)( 50,123)
( 51,122)( 52,121)( 53,120)( 54,128)( 55,127)( 56,126)( 57,144)( 58,146)
( 59,145)( 60,141)( 61,143)( 62,142)( 63,138)( 64,140)( 65,139)( 66,164)
( 67,163)( 68,162)( 69,161)( 70,160)( 71,159)( 72,158)( 73,157)( 74,156)
( 75,155)( 76,154)( 77,153)( 78,152)( 79,151)( 80,150)( 81,149)( 82,148)
( 83,147)(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)(192,215)(193,214)
(194,213)(195,212)(196,211)(197,210)(198,218)(199,217)(200,216)(201,206)
(202,205)(203,204)(207,209)(219,245)(220,244)(221,243)(222,242)(223,241)
(224,240)(225,239)(226,238)(227,237)(228,236)(229,235)(230,234)(231,233);;
s4 := ( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 12, 23)( 13, 22)( 14, 21)( 15, 29)
( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 39, 50)( 40, 49)( 41, 48)( 42, 56)( 43, 55)( 44, 54)( 45, 53)
( 46, 52)( 47, 51)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 66, 77)( 67, 76)
( 68, 75)( 69, 83)( 70, 82)( 71, 81)( 72, 80)( 73, 79)( 74, 78)( 84,185)
( 85,184)( 86,183)( 87,191)( 88,190)( 89,189)( 90,188)( 91,187)( 92,186)
( 93,176)( 94,175)( 95,174)( 96,182)( 97,181)( 98,180)( 99,179)(100,178)
(101,177)(102,167)(103,166)(104,165)(105,173)(106,172)(107,171)(108,170)
(109,169)(110,168)(111,212)(112,211)(113,210)(114,218)(115,217)(116,216)
(117,215)(118,214)(119,213)(120,203)(121,202)(122,201)(123,209)(124,208)
(125,207)(126,206)(127,205)(128,204)(129,194)(130,193)(131,192)(132,200)
(133,199)(134,198)(135,197)(136,196)(137,195)(138,239)(139,238)(140,237)
(141,245)(142,244)(143,243)(144,242)(145,241)(146,240)(147,230)(148,229)
(149,228)(150,236)(151,235)(152,234)(153,233)(154,232)(155,231)(156,221)
(157,220)(158,219)(159,227)(160,226)(161,225)(162,224)(163,223)(164,222);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
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, 31)( 5, 32)( 6, 36)( 7, 37)( 8, 38)( 9, 33)
( 10, 34)( 11, 35)( 12, 39)( 13, 40)( 14, 41)( 15, 45)( 16, 46)( 17, 47)
( 18, 42)( 19, 43)( 20, 44)( 21, 48)( 22, 49)( 23, 50)( 24, 54)( 25, 55)
( 26, 56)( 27, 51)( 28, 52)( 29, 53)( 60, 63)( 61, 64)( 62, 65)( 69, 72)
( 70, 73)( 71, 74)( 78, 81)( 79, 82)( 80, 83)( 84,111)( 85,112)( 86,113)
( 87,117)( 88,118)( 89,119)( 90,114)( 91,115)( 92,116)( 93,120)( 94,121)
( 95,122)( 96,126)( 97,127)( 98,128)( 99,123)(100,124)(101,125)(102,129)
(103,130)(104,131)(105,135)(106,136)(107,137)(108,132)(109,133)(110,134)
(141,144)(142,145)(143,146)(150,153)(151,154)(152,155)(159,162)(160,163)
(161,164)(165,192)(166,193)(167,194)(168,198)(169,199)(170,200)(171,195)
(172,196)(173,197)(174,201)(175,202)(176,203)(177,207)(178,208)(179,209)
(180,204)(181,205)(182,206)(183,210)(184,211)(185,212)(186,216)(187,217)
(188,218)(189,213)(190,214)(191,215)(222,225)(223,226)(224,227)(231,234)
(232,235)(233,236)(240,243)(241,244)(242,245);
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)( 30,114)( 31,116)( 32,115)( 33,111)
( 34,113)( 35,112)( 36,117)( 37,119)( 38,118)( 39,134)( 40,133)( 41,132)
( 42,131)( 43,130)( 44,129)( 45,137)( 46,136)( 47,135)( 48,125)( 49,124)
( 50,123)( 51,122)( 52,121)( 53,120)( 54,128)( 55,127)( 56,126)( 57,144)
( 58,146)( 59,145)( 60,141)( 61,143)( 62,142)( 63,138)( 64,140)( 65,139)
( 66,164)( 67,163)( 68,162)( 69,161)( 70,160)( 71,159)( 72,158)( 73,157)
( 74,156)( 75,155)( 76,154)( 77,153)( 78,152)( 79,151)( 80,150)( 81,149)
( 82,148)( 83,147)(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)(192,215)
(193,214)(194,213)(195,212)(196,211)(197,210)(198,218)(199,217)(200,216)
(201,206)(202,205)(203,204)(207,209)(219,245)(220,244)(221,243)(222,242)
(223,241)(224,240)(225,239)(226,238)(227,237)(228,236)(229,235)(230,234)
(231,233);
s4 := Sym(245)!( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 12, 23)( 13, 22)( 14, 21)
( 15, 29)( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 31, 32)( 33, 36)
( 34, 38)( 35, 37)( 39, 50)( 40, 49)( 41, 48)( 42, 56)( 43, 55)( 44, 54)
( 45, 53)( 46, 52)( 47, 51)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 66, 77)
( 67, 76)( 68, 75)( 69, 83)( 70, 82)( 71, 81)( 72, 80)( 73, 79)( 74, 78)
( 84,185)( 85,184)( 86,183)( 87,191)( 88,190)( 89,189)( 90,188)( 91,187)
( 92,186)( 93,176)( 94,175)( 95,174)( 96,182)( 97,181)( 98,180)( 99,179)
(100,178)(101,177)(102,167)(103,166)(104,165)(105,173)(106,172)(107,171)
(108,170)(109,169)(110,168)(111,212)(112,211)(113,210)(114,218)(115,217)
(116,216)(117,215)(118,214)(119,213)(120,203)(121,202)(122,201)(123,209)
(124,208)(125,207)(126,206)(127,205)(128,204)(129,194)(130,193)(131,192)
(132,200)(133,199)(134,198)(135,197)(136,196)(137,195)(138,239)(139,238)
(140,237)(141,245)(142,244)(143,243)(144,242)(145,241)(146,240)(147,230)
(148,229)(149,228)(150,236)(151,235)(152,234)(153,233)(154,232)(155,231)
(156,221)(157,220)(158,219)(159,227)(160,226)(161,225)(162,224)(163,223)
(164,222);
poly := sub<Sym(245)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope