Polytope of Type {2,54,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,54,4}*864b
if this polytope has a name.
Group : SmallGroup(864,1916)
Rank : 4
Schlafli Type : {2,54,4}
Number of vertices, edges, etc : 2, 54, 108, 4
Order of s₀s₁s₂s₃ : 54
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,54,4,2} of size 1728
Vertex Figure Of :
   {2,2,54,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,27,4}*432
   3-fold quotients : {2,18,4}*288b
   6-fold quotients : {2,9,4}*144
   9-fold quotients : {2,6,4}*96c
   18-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,108,4}*1728b, {2,108,4}*1728c, {4,54,4}*1728b, {2,54,4}*1728
Permutation Representation (GAP) :

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

to this polytope