Polytope of Type {2,4,54}

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

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

to this polytope