Polytope of Type {2,8,54}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,8,54}*1728
if this polytope has a name.
Group : SmallGroup(1728,2555)
Rank : 4
Schlafli Type : {2,8,54}
Number of vertices, edges, etc : 2, 8, 216, 54
Order of s₀s₁s₂s₃ : 216
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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) :
   2-fold quotients : {2,4,54}*864a
   3-fold quotients : {2,8,18}*576
   4-fold quotients : {2,2,54}*432
   6-fold quotients : {2,4,18}*288a
   8-fold quotients : {2,2,27}*216
   9-fold quotients : {2,8,6}*192
   12-fold quotients : {2,2,18}*144
   18-fold quotients : {2,4,6}*96a
   24-fold quotients : {2,2,9}*72
   27-fold quotients : {2,8,2}*64
   36-fold quotients : {2,2,6}*48
   54-fold quotients : {2,4,2}*32
   72-fold quotients : {2,2,3}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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