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