Polytope of Type {2,108,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,108,4}*1728a
if this polytope has a name.
Group : SmallGroup(1728,2289)
Rank : 4
Schlafli Type : {2,108,4}
Number of vertices, edges, etc : 2, 108, 216, 4
Order of s₀s₁s₂s₃ : 108
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 : {2,108,2}*864, {2,54,4}*864a
   3-fold quotients : {2,36,4}*576a
   4-fold quotients : {2,54,2}*432
   6-fold quotients : {2,36,2}*288, {2,18,4}*288a
   8-fold quotients : {2,27,2}*216
   9-fold quotients : {2,12,4}*192a
   12-fold quotients : {2,18,2}*144
   18-fold quotients : {2,12,2}*96, {2,6,4}*96a
   24-fold quotients : {2,9,2}*72
   27-fold quotients : {2,4,4}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {2,2,4}*32, {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, 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)(111,165)(112,167)(113,166)(114,173)
(115,172)(116,171)(117,170)(118,169)(119,168)(120,191)(121,190)(122,189)
(123,188)(124,187)(125,186)(126,185)(127,184)(128,183)(129,182)(130,181)
(131,180)(132,179)(133,178)(134,177)(135,176)(136,175)(137,174)(138,192)
(139,194)(140,193)(141,200)(142,199)(143,198)(144,197)(145,196)(146,195)
(147,218)(148,217)(149,216)(150,215)(151,214)(152,213)(153,212)(154,211)
(155,210)(156,209)(157,208)(158,207)(159,206)(160,205)(161,204)(162,203)
(163,202)(164,201);;
s2 := (  3,120)(  4,122)(  5,121)(  6,128)(  7,127)(  8,126)(  9,125)( 10,124)
( 11,123)( 12,111)( 13,113)( 14,112)( 15,119)( 16,118)( 17,117)( 18,116)
( 19,115)( 20,114)( 21,137)( 22,136)( 23,135)( 24,134)( 25,133)( 26,132)
( 27,131)( 28,130)( 29,129)( 30,147)( 31,149)( 32,148)( 33,155)( 34,154)
( 35,153)( 36,152)( 37,151)( 38,150)( 39,138)( 40,140)( 41,139)( 42,146)
( 43,145)( 44,144)( 45,143)( 46,142)( 47,141)( 48,164)( 49,163)( 50,162)
( 51,161)( 52,160)( 53,159)( 54,158)( 55,157)( 56,156)( 57,174)( 58,176)
( 59,175)( 60,182)( 61,181)( 62,180)( 63,179)( 64,178)( 65,177)( 66,165)
( 67,167)( 68,166)( 69,173)( 70,172)( 71,171)( 72,170)( 73,169)( 74,168)
( 75,191)( 76,190)( 77,189)( 78,188)( 79,187)( 80,186)( 81,185)( 82,184)
( 83,183)( 84,201)( 85,203)( 86,202)( 87,209)( 88,208)( 89,207)( 90,206)
( 91,205)( 92,204)( 93,192)( 94,194)( 95,193)( 96,200)( 97,199)( 98,198)
( 99,197)(100,196)(101,195)(102,218)(103,217)(104,216)(105,215)(106,214)
(107,213)(108,212)(109,211)(110,210);;
s3 := (111,138)(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(118,145)
(119,146)(120,147)(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)
(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)
(135,162)(136,163)(137,164)(165,192)(166,193)(167,194)(168,195)(169,196)
(170,197)(171,198)(172,199)(173,200)(174,201)(175,202)(176,203)(177,204)
(178,205)(179,206)(180,207)(181,208)(182,209)(183,210)(184,211)(185,212)
(186,213)(187,214)(188,215)(189,216)(190,217)(191,218);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)(111,165)(112,167)(113,166)
(114,173)(115,172)(116,171)(117,170)(118,169)(119,168)(120,191)(121,190)
(122,189)(123,188)(124,187)(125,186)(126,185)(127,184)(128,183)(129,182)
(130,181)(131,180)(132,179)(133,178)(134,177)(135,176)(136,175)(137,174)
(138,192)(139,194)(140,193)(141,200)(142,199)(143,198)(144,197)(145,196)
(146,195)(147,218)(148,217)(149,216)(150,215)(151,214)(152,213)(153,212)
(154,211)(155,210)(156,209)(157,208)(158,207)(159,206)(160,205)(161,204)
(162,203)(163,202)(164,201);
s2 := Sym(218)!(  3,120)(  4,122)(  5,121)(  6,128)(  7,127)(  8,126)(  9,125)
( 10,124)( 11,123)( 12,111)( 13,113)( 14,112)( 15,119)( 16,118)( 17,117)
( 18,116)( 19,115)( 20,114)( 21,137)( 22,136)( 23,135)( 24,134)( 25,133)
( 26,132)( 27,131)( 28,130)( 29,129)( 30,147)( 31,149)( 32,148)( 33,155)
( 34,154)( 35,153)( 36,152)( 37,151)( 38,150)( 39,138)( 40,140)( 41,139)
( 42,146)( 43,145)( 44,144)( 45,143)( 46,142)( 47,141)( 48,164)( 49,163)
( 50,162)( 51,161)( 52,160)( 53,159)( 54,158)( 55,157)( 56,156)( 57,174)
( 58,176)( 59,175)( 60,182)( 61,181)( 62,180)( 63,179)( 64,178)( 65,177)
( 66,165)( 67,167)( 68,166)( 69,173)( 70,172)( 71,171)( 72,170)( 73,169)
( 74,168)( 75,191)( 76,190)( 77,189)( 78,188)( 79,187)( 80,186)( 81,185)
( 82,184)( 83,183)( 84,201)( 85,203)( 86,202)( 87,209)( 88,208)( 89,207)
( 90,206)( 91,205)( 92,204)( 93,192)( 94,194)( 95,193)( 96,200)( 97,199)
( 98,198)( 99,197)(100,196)(101,195)(102,218)(103,217)(104,216)(105,215)
(106,214)(107,213)(108,212)(109,211)(110,210);
s3 := Sym(218)!(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)
(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)(124,151)(125,152)
(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)
(134,161)(135,162)(136,163)(137,164)(165,192)(166,193)(167,194)(168,195)
(169,196)(170,197)(171,198)(172,199)(173,200)(174,201)(175,202)(176,203)
(177,204)(178,205)(179,206)(180,207)(181,208)(182,209)(183,210)(184,211)
(185,212)(186,213)(187,214)(188,215)(189,216)(190,217)(191,218);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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