Polytope of Type {2,54,8}

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

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