Polytope of Type {3,6,54}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,54}*1944b
if this polytope has a name.
Group : SmallGroup(1944,2343)
Rank : 4
Schlafli Type : {3,6,54}
Number of vertices, edges, etc : 3, 9, 162, 54
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 : {3,2,54}*648, {3,6,18}*648b
   6-fold quotients : {3,2,27}*324
   9-fold quotients : {3,2,18}*216, {3,6,6}*216b
   18-fold quotients : {3,2,9}*108
   27-fold quotients : {3,2,6}*72, {3,6,2}*72
   54-fold quotients : {3,2,3}*36
   81-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

References : None.
to this polytope