Polytope of Type {2,216,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,216,2}*1728
if this polytope has a name.
Group : SmallGroup(1728,2542)
Rank : 4
Schlafli Type : {2,216,2}
Number of vertices, edges, etc : 2, 216, 216, 2
Order of s₀s₁s₂s₃ : 216
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
   Self-Dual
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 : {2,108,2}*864
   3-fold quotients : {2,72,2}*576
   4-fold quotients : {2,54,2}*432
   6-fold quotients : {2,36,2}*288
   8-fold quotients : {2,27,2}*216
   9-fold quotients : {2,24,2}*192
   12-fold quotients : {2,18,2}*144
   18-fold quotients : {2,12,2}*96
   24-fold quotients : {2,9,2}*72
   27-fold quotients : {2,8,2}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {2,4,2}*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, 10)(  7,  9)(  8, 11)( 12, 24)( 13, 26)( 14, 25)( 15, 21)
( 16, 23)( 17, 22)( 18, 28)( 19, 27)( 20, 29)( 31, 32)( 33, 37)( 34, 36)
( 35, 38)( 39, 51)( 40, 53)( 41, 52)( 42, 48)( 43, 50)( 44, 49)( 45, 55)
( 46, 54)( 47, 56)( 57, 84)( 58, 86)( 59, 85)( 60, 91)( 61, 90)( 62, 92)
( 63, 88)( 64, 87)( 65, 89)( 66,105)( 67,107)( 68,106)( 69,102)( 70,104)
( 71,103)( 72,109)( 73,108)( 74,110)( 75, 96)( 76, 98)( 77, 97)( 78, 93)
( 79, 95)( 80, 94)( 81,100)( 82, 99)( 83,101)(111,165)(112,167)(113,166)
(114,172)(115,171)(116,173)(117,169)(118,168)(119,170)(120,186)(121,188)
(122,187)(123,183)(124,185)(125,184)(126,190)(127,189)(128,191)(129,177)
(130,179)(131,178)(132,174)(133,176)(134,175)(135,181)(136,180)(137,182)
(138,192)(139,194)(140,193)(141,199)(142,198)(143,200)(144,196)(145,195)
(146,197)(147,213)(148,215)(149,214)(150,210)(151,212)(152,211)(153,217)
(154,216)(155,218)(156,204)(157,206)(158,205)(159,201)(160,203)(161,202)
(162,208)(163,207)(164,209);;
s2 := (  3,120)(  4,122)(  5,121)(  6,127)(  7,126)(  8,128)(  9,124)( 10,123)
( 11,125)( 12,111)( 13,113)( 14,112)( 15,118)( 16,117)( 17,119)( 18,115)
( 19,114)( 20,116)( 21,132)( 22,134)( 23,133)( 24,129)( 25,131)( 26,130)
( 27,136)( 28,135)( 29,137)( 30,147)( 31,149)( 32,148)( 33,154)( 34,153)
( 35,155)( 36,151)( 37,150)( 38,152)( 39,138)( 40,140)( 41,139)( 42,145)
( 43,144)( 44,146)( 45,142)( 46,141)( 47,143)( 48,159)( 49,161)( 50,160)
( 51,156)( 52,158)( 53,157)( 54,163)( 55,162)( 56,164)( 57,201)( 58,203)
( 59,202)( 60,208)( 61,207)( 62,209)( 63,205)( 64,204)( 65,206)( 66,192)
( 67,194)( 68,193)( 69,199)( 70,198)( 71,200)( 72,196)( 73,195)( 74,197)
( 75,213)( 76,215)( 77,214)( 78,210)( 79,212)( 80,211)( 81,217)( 82,216)
( 83,218)( 84,174)( 85,176)( 86,175)( 87,181)( 88,180)( 89,182)( 90,178)
( 91,177)( 92,179)( 93,165)( 94,167)( 95,166)( 96,172)( 97,171)( 98,173)
( 99,169)(100,168)(101,170)(102,186)(103,188)(104,187)(105,183)(106,185)
(107,184)(108,190)(109,189)(110,191);;
s3 := (219,220);;
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, 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*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*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*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(220)!(1,2);
s1 := Sym(220)!(  4,  5)(  6, 10)(  7,  9)(  8, 11)( 12, 24)( 13, 26)( 14, 25)
( 15, 21)( 16, 23)( 17, 22)( 18, 28)( 19, 27)( 20, 29)( 31, 32)( 33, 37)
( 34, 36)( 35, 38)( 39, 51)( 40, 53)( 41, 52)( 42, 48)( 43, 50)( 44, 49)
( 45, 55)( 46, 54)( 47, 56)( 57, 84)( 58, 86)( 59, 85)( 60, 91)( 61, 90)
( 62, 92)( 63, 88)( 64, 87)( 65, 89)( 66,105)( 67,107)( 68,106)( 69,102)
( 70,104)( 71,103)( 72,109)( 73,108)( 74,110)( 75, 96)( 76, 98)( 77, 97)
( 78, 93)( 79, 95)( 80, 94)( 81,100)( 82, 99)( 83,101)(111,165)(112,167)
(113,166)(114,172)(115,171)(116,173)(117,169)(118,168)(119,170)(120,186)
(121,188)(122,187)(123,183)(124,185)(125,184)(126,190)(127,189)(128,191)
(129,177)(130,179)(131,178)(132,174)(133,176)(134,175)(135,181)(136,180)
(137,182)(138,192)(139,194)(140,193)(141,199)(142,198)(143,200)(144,196)
(145,195)(146,197)(147,213)(148,215)(149,214)(150,210)(151,212)(152,211)
(153,217)(154,216)(155,218)(156,204)(157,206)(158,205)(159,201)(160,203)
(161,202)(162,208)(163,207)(164,209);
s2 := Sym(220)!(  3,120)(  4,122)(  5,121)(  6,127)(  7,126)(  8,128)(  9,124)
( 10,123)( 11,125)( 12,111)( 13,113)( 14,112)( 15,118)( 16,117)( 17,119)
( 18,115)( 19,114)( 20,116)( 21,132)( 22,134)( 23,133)( 24,129)( 25,131)
( 26,130)( 27,136)( 28,135)( 29,137)( 30,147)( 31,149)( 32,148)( 33,154)
( 34,153)( 35,155)( 36,151)( 37,150)( 38,152)( 39,138)( 40,140)( 41,139)
( 42,145)( 43,144)( 44,146)( 45,142)( 46,141)( 47,143)( 48,159)( 49,161)
( 50,160)( 51,156)( 52,158)( 53,157)( 54,163)( 55,162)( 56,164)( 57,201)
( 58,203)( 59,202)( 60,208)( 61,207)( 62,209)( 63,205)( 64,204)( 65,206)
( 66,192)( 67,194)( 68,193)( 69,199)( 70,198)( 71,200)( 72,196)( 73,195)
( 74,197)( 75,213)( 76,215)( 77,214)( 78,210)( 79,212)( 80,211)( 81,217)
( 82,216)( 83,218)( 84,174)( 85,176)( 86,175)( 87,181)( 88,180)( 89,182)
( 90,178)( 91,177)( 92,179)( 93,165)( 94,167)( 95,166)( 96,172)( 97,171)
( 98,173)( 99,169)(100,168)(101,170)(102,186)(103,188)(104,187)(105,183)
(106,185)(107,184)(108,190)(109,189)(110,191);
s3 := Sym(220)!(219,220);
poly := sub<Sym(220)|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, 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*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*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*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