Polytope of Type {2,54,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,54,6}*1944b
if this polytope has a name.
Group : SmallGroup(1944,953)
Rank : 4
Schlafli Type : {2,54,6}
Number of vertices, edges, etc : 2, 81, 243, 9
Order of s₀s₁s₂s₃ : 54
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-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 : {2,18,6}*648a
   9-fold quotients : {2,6,6}*216
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

to this polytope