Polytope of Type {54,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,6,3}*1944b
if this polytope has a name.
Group : SmallGroup(1944,2343)
Rank : 4
Schlafli Type : {54,6,3}
Number of vertices, edges, etc : 54, 162, 9, 3
Order of s0s1s2s3 : 54
Order of s0s1s2s3s2s1 : 2
Special Properties :
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 : {54,2,3}*648, {18,6,3}*648b
6-fold quotients : {27,2,3}*324
9-fold quotients : {18,2,3}*216, {6,6,3}*216b
18-fold quotients : {9,2,3}*108
27-fold quotients : {2,6,3}*72, {6,2,3}*72
54-fold quotients : {3,2,3}*36
81-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 8)( 5, 7)( 6, 9)( 10, 23)( 11, 22)( 12, 24)( 13, 20)
( 14, 19)( 15, 21)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 31, 35)( 32, 34)
( 33, 36)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)( 43, 54)
( 44, 53)( 45, 52)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 64, 77)( 65, 76)
( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 81)( 71, 80)( 72, 79)( 83, 84)
( 85, 89)( 86, 88)( 87, 90)( 91,104)( 92,103)( 93,105)( 94,101)( 95,100)
( 96,102)( 97,108)( 98,107)( 99,106)(110,111)(112,116)(113,115)(114,117)
(118,131)(119,130)(120,132)(121,128)(122,127)(123,129)(124,135)(125,134)
(126,133)(137,138)(139,143)(140,142)(141,144)(145,158)(146,157)(147,159)
(148,155)(149,154)(150,156)(151,162)(152,161)(153,160)(164,165)(166,170)
(167,169)(168,171)(172,185)(173,184)(174,186)(175,182)(176,181)(177,183)
(178,189)(179,188)(180,187)(191,192)(193,197)(194,196)(195,198)(199,212)
(200,211)(201,213)(202,209)(203,208)(204,210)(205,216)(206,215)(207,214)
(218,219)(220,224)(221,223)(222,225)(226,239)(227,238)(228,240)(229,236)
(230,235)(231,237)(232,243)(233,242)(234,241);;
s1 := ( 1, 10)( 2, 12)( 3, 11)( 4, 17)( 5, 16)( 6, 18)( 7, 14)( 8, 13)
( 9, 15)( 19, 23)( 20, 22)( 21, 24)( 25, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 44)( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 46, 50)( 47, 49)
( 48, 51)( 52, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 71)( 59, 70)( 60, 72)
( 61, 68)( 62, 67)( 63, 69)( 73, 77)( 74, 76)( 75, 78)( 79, 81)( 82,172)
( 83,174)( 84,173)( 85,179)( 86,178)( 87,180)( 88,176)( 89,175)( 90,177)
( 91,163)( 92,165)( 93,164)( 94,170)( 95,169)( 96,171)( 97,167)( 98,166)
( 99,168)(100,185)(101,184)(102,186)(103,182)(104,181)(105,183)(106,189)
(107,188)(108,187)(109,199)(110,201)(111,200)(112,206)(113,205)(114,207)
(115,203)(116,202)(117,204)(118,190)(119,192)(120,191)(121,197)(122,196)
(123,198)(124,194)(125,193)(126,195)(127,212)(128,211)(129,213)(130,209)
(131,208)(132,210)(133,216)(134,215)(135,214)(136,226)(137,228)(138,227)
(139,233)(140,232)(141,234)(142,230)(143,229)(144,231)(145,217)(146,219)
(147,218)(148,224)(149,223)(150,225)(151,221)(152,220)(153,222)(154,239)
(155,238)(156,240)(157,236)(158,235)(159,237)(160,243)(161,242)(162,241);;
s2 := ( 1, 82)( 2, 83)( 3, 84)( 4, 85)( 5, 86)( 6, 87)( 7, 88)( 8, 89)
( 9, 90)( 10, 91)( 11, 92)( 12, 93)( 13, 94)( 14, 95)( 15, 96)( 16, 97)
( 17, 98)( 18, 99)( 19,100)( 20,101)( 21,102)( 22,103)( 23,104)( 24,105)
( 25,106)( 26,107)( 27,108)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)
( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)
( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,109)( 56,110)
( 57,111)( 58,112)( 59,113)( 60,114)( 61,115)( 62,116)( 63,117)( 64,118)
( 65,119)( 66,120)( 67,121)( 68,122)( 69,123)( 70,124)( 71,125)( 72,126)
( 73,127)( 74,128)( 75,129)( 76,130)( 77,131)( 78,132)( 79,133)( 80,134)
( 81,135)(190,217)(191,218)(192,219)(193,220)(194,221)(195,222)(196,223)
(197,224)(198,225)(199,226)(200,227)(201,228)(202,229)(203,230)(204,231)
(205,232)(206,233)(207,234)(208,235)(209,236)(210,237)(211,238)(212,239)
(213,240)(214,241)(215,242)(216,243);;
s3 := ( 1, 28)( 2, 29)( 3, 30)( 4, 31)( 5, 32)( 6, 33)( 7, 34)( 8, 35)
( 9, 36)( 10, 37)( 11, 38)( 12, 39)( 13, 40)( 14, 41)( 15, 42)( 16, 43)
( 17, 44)( 18, 45)( 19, 46)( 20, 47)( 21, 48)( 22, 49)( 23, 50)( 24, 51)
( 25, 52)( 26, 53)( 27, 54)( 82,190)( 83,191)( 84,192)( 85,193)( 86,194)
( 87,195)( 88,196)( 89,197)( 90,198)( 91,199)( 92,200)( 93,201)( 94,202)
( 95,203)( 96,204)( 97,205)( 98,206)( 99,207)(100,208)(101,209)(102,210)
(103,211)(104,212)(105,213)(106,214)(107,215)(108,216)(109,163)(110,164)
(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)
(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)
(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)
(135,189)(136,217)(137,218)(138,219)(139,220)(140,221)(141,222)(142,223)
(143,224)(144,225)(145,226)(146,227)(147,228)(148,229)(149,230)(150,231)
(151,232)(152,233)(153,234)(154,235)(155,236)(156,237)(157,238)(158,239)
(159,240)(160,241)(161,242)(162,243);;
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*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(243)!( 2, 3)( 4, 8)( 5, 7)( 6, 9)( 10, 23)( 11, 22)( 12, 24)
( 13, 20)( 14, 19)( 15, 21)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 31, 35)
( 32, 34)( 33, 36)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)
( 43, 54)( 44, 53)( 45, 52)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 64, 77)
( 65, 76)( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 81)( 71, 80)( 72, 79)
( 83, 84)( 85, 89)( 86, 88)( 87, 90)( 91,104)( 92,103)( 93,105)( 94,101)
( 95,100)( 96,102)( 97,108)( 98,107)( 99,106)(110,111)(112,116)(113,115)
(114,117)(118,131)(119,130)(120,132)(121,128)(122,127)(123,129)(124,135)
(125,134)(126,133)(137,138)(139,143)(140,142)(141,144)(145,158)(146,157)
(147,159)(148,155)(149,154)(150,156)(151,162)(152,161)(153,160)(164,165)
(166,170)(167,169)(168,171)(172,185)(173,184)(174,186)(175,182)(176,181)
(177,183)(178,189)(179,188)(180,187)(191,192)(193,197)(194,196)(195,198)
(199,212)(200,211)(201,213)(202,209)(203,208)(204,210)(205,216)(206,215)
(207,214)(218,219)(220,224)(221,223)(222,225)(226,239)(227,238)(228,240)
(229,236)(230,235)(231,237)(232,243)(233,242)(234,241);
s1 := Sym(243)!( 1, 10)( 2, 12)( 3, 11)( 4, 17)( 5, 16)( 6, 18)( 7, 14)
( 8, 13)( 9, 15)( 19, 23)( 20, 22)( 21, 24)( 25, 27)( 28, 37)( 29, 39)
( 30, 38)( 31, 44)( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 46, 50)
( 47, 49)( 48, 51)( 52, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 71)( 59, 70)
( 60, 72)( 61, 68)( 62, 67)( 63, 69)( 73, 77)( 74, 76)( 75, 78)( 79, 81)
( 82,172)( 83,174)( 84,173)( 85,179)( 86,178)( 87,180)( 88,176)( 89,175)
( 90,177)( 91,163)( 92,165)( 93,164)( 94,170)( 95,169)( 96,171)( 97,167)
( 98,166)( 99,168)(100,185)(101,184)(102,186)(103,182)(104,181)(105,183)
(106,189)(107,188)(108,187)(109,199)(110,201)(111,200)(112,206)(113,205)
(114,207)(115,203)(116,202)(117,204)(118,190)(119,192)(120,191)(121,197)
(122,196)(123,198)(124,194)(125,193)(126,195)(127,212)(128,211)(129,213)
(130,209)(131,208)(132,210)(133,216)(134,215)(135,214)(136,226)(137,228)
(138,227)(139,233)(140,232)(141,234)(142,230)(143,229)(144,231)(145,217)
(146,219)(147,218)(148,224)(149,223)(150,225)(151,221)(152,220)(153,222)
(154,239)(155,238)(156,240)(157,236)(158,235)(159,237)(160,243)(161,242)
(162,241);
s2 := Sym(243)!( 1, 82)( 2, 83)( 3, 84)( 4, 85)( 5, 86)( 6, 87)( 7, 88)
( 8, 89)( 9, 90)( 10, 91)( 11, 92)( 12, 93)( 13, 94)( 14, 95)( 15, 96)
( 16, 97)( 17, 98)( 18, 99)( 19,100)( 20,101)( 21,102)( 22,103)( 23,104)
( 24,105)( 25,106)( 26,107)( 27,108)( 28,136)( 29,137)( 30,138)( 31,139)
( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)
( 40,148)( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)
( 48,156)( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,109)
( 56,110)( 57,111)( 58,112)( 59,113)( 60,114)( 61,115)( 62,116)( 63,117)
( 64,118)( 65,119)( 66,120)( 67,121)( 68,122)( 69,123)( 70,124)( 71,125)
( 72,126)( 73,127)( 74,128)( 75,129)( 76,130)( 77,131)( 78,132)( 79,133)
( 80,134)( 81,135)(190,217)(191,218)(192,219)(193,220)(194,221)(195,222)
(196,223)(197,224)(198,225)(199,226)(200,227)(201,228)(202,229)(203,230)
(204,231)(205,232)(206,233)(207,234)(208,235)(209,236)(210,237)(211,238)
(212,239)(213,240)(214,241)(215,242)(216,243);
s3 := Sym(243)!( 1, 28)( 2, 29)( 3, 30)( 4, 31)( 5, 32)( 6, 33)( 7, 34)
( 8, 35)( 9, 36)( 10, 37)( 11, 38)( 12, 39)( 13, 40)( 14, 41)( 15, 42)
( 16, 43)( 17, 44)( 18, 45)( 19, 46)( 20, 47)( 21, 48)( 22, 49)( 23, 50)
( 24, 51)( 25, 52)( 26, 53)( 27, 54)( 82,190)( 83,191)( 84,192)( 85,193)
( 86,194)( 87,195)( 88,196)( 89,197)( 90,198)( 91,199)( 92,200)( 93,201)
( 94,202)( 95,203)( 96,204)( 97,205)( 98,206)( 99,207)(100,208)(101,209)
(102,210)(103,211)(104,212)(105,213)(106,214)(107,215)(108,216)(109,163)
(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)
(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)
(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)
(134,188)(135,189)(136,217)(137,218)(138,219)(139,220)(140,221)(141,222)
(142,223)(143,224)(144,225)(145,226)(146,227)(147,228)(148,229)(149,230)
(150,231)(151,232)(152,233)(153,234)(154,235)(155,236)(156,237)(157,238)
(158,239)(159,240)(160,241)(161,242)(162,243);
poly := sub<Sym(243)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope