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Polytope of Type {2,18,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,24}*1728a
if this polytope has a name.
Group : SmallGroup(1728,15830)
Rank : 4
Schlafli Type : {2,18,24}
Number of vertices, edges, etc : 2, 18, 216, 24
Order of s0s1s2s3 : 72
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 : {2,18,12}*864a
   3-fold quotients : {2,18,8}*576, {2,6,24}*576a
   4-fold quotients : {2,18,6}*432a
   6-fold quotients : {2,18,4}*288a, {2,6,12}*288a
   9-fold quotients : {2,2,24}*192, {2,6,8}*192
   12-fold quotients : {2,18,2}*144, {2,6,6}*144a
   18-fold quotients : {2,2,12}*96, {2,6,4}*96a
   24-fold quotients : {2,9,2}*72
   27-fold quotients : {2,2,8}*64
   36-fold quotients : {2,2,6}*48, {2,6,2}*48
   54-fold quotients : {2,2,4}*32
   72-fold quotients : {2,2,3}*24, {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)( 13, 14)( 15, 20)( 16, 19)( 17, 18)
( 22, 23)( 24, 29)( 25, 28)( 26, 27)( 31, 32)( 33, 38)( 34, 37)( 35, 36)
( 40, 41)( 42, 47)( 43, 46)( 44, 45)( 49, 50)( 51, 56)( 52, 55)( 53, 54)
( 58, 59)( 60, 65)( 61, 64)( 62, 63)( 67, 68)( 69, 74)( 70, 73)( 71, 72)
( 76, 77)( 78, 83)( 79, 82)( 80, 81)( 85, 86)( 87, 92)( 88, 91)( 89, 90)
( 94, 95)( 96,101)( 97,100)( 98, 99)(103,104)(105,110)(106,109)(107,108)
(112,113)(114,119)(115,118)(116,117)(121,122)(123,128)(124,127)(125,126)
(130,131)(132,137)(133,136)(134,135)(139,140)(141,146)(142,145)(143,144)
(148,149)(150,155)(151,154)(152,153)(157,158)(159,164)(160,163)(161,162)
(166,167)(168,173)(169,172)(170,171)(175,176)(177,182)(178,181)(179,180)
(184,185)(186,191)(187,190)(188,189)(193,194)(195,200)(196,199)(197,198)
(202,203)(204,209)(205,208)(206,207)(211,212)(213,218)(214,217)(215,216);;
s2 := (  3,  6)(  4,  8)(  5,  7)(  9, 11)( 12, 24)( 13, 26)( 14, 25)( 15, 21)
( 16, 23)( 17, 22)( 18, 29)( 19, 28)( 20, 27)( 30, 33)( 31, 35)( 32, 34)
( 36, 38)( 39, 51)( 40, 53)( 41, 52)( 42, 48)( 43, 50)( 44, 49)( 45, 56)
( 46, 55)( 47, 54)( 57, 87)( 58, 89)( 59, 88)( 60, 84)( 61, 86)( 62, 85)
( 63, 92)( 64, 91)( 65, 90)( 66,105)( 67,107)( 68,106)( 69,102)( 70,104)
( 71,103)( 72,110)( 73,109)( 74,108)( 75, 96)( 76, 98)( 77, 97)( 78, 93)
( 79, 95)( 80, 94)( 81,101)( 82,100)( 83, 99)(111,168)(112,170)(113,169)
(114,165)(115,167)(116,166)(117,173)(118,172)(119,171)(120,186)(121,188)
(122,187)(123,183)(124,185)(125,184)(126,191)(127,190)(128,189)(129,177)
(130,179)(131,178)(132,174)(133,176)(134,175)(135,182)(136,181)(137,180)
(138,195)(139,197)(140,196)(141,192)(142,194)(143,193)(144,200)(145,199)
(146,198)(147,213)(148,215)(149,214)(150,210)(151,212)(152,211)(153,218)
(154,217)(155,216)(156,204)(157,206)(158,205)(159,201)(160,203)(161,202)
(162,209)(163,208)(164,207);;
s3 := (  3,120)(  4,121)(  5,122)(  6,123)(  7,124)(  8,125)(  9,126)( 10,127)
( 11,128)( 12,111)( 13,112)( 14,113)( 15,114)( 16,115)( 17,116)( 18,117)
( 19,118)( 20,119)( 21,129)( 22,130)( 23,131)( 24,132)( 25,133)( 26,134)
( 27,135)( 28,136)( 29,137)( 30,147)( 31,148)( 32,149)( 33,150)( 34,151)
( 35,152)( 36,153)( 37,154)( 38,155)( 39,138)( 40,139)( 41,140)( 42,141)
( 43,142)( 44,143)( 45,144)( 46,145)( 47,146)( 48,156)( 49,157)( 50,158)
( 51,159)( 52,160)( 53,161)( 54,162)( 55,163)( 56,164)( 57,201)( 58,202)
( 59,203)( 60,204)( 61,205)( 62,206)( 63,207)( 64,208)( 65,209)( 66,192)
( 67,193)( 68,194)( 69,195)( 70,196)( 71,197)( 72,198)( 73,199)( 74,200)
( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)( 81,216)( 82,217)
( 83,218)( 84,174)( 85,175)( 86,176)( 87,177)( 88,178)( 89,179)( 90,180)
( 91,181)( 92,182)( 93,165)( 94,166)( 95,167)( 96,168)( 97,169)( 98,170)
( 99,171)(100,172)(101,173)(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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!(  4,  5)(  6, 11)(  7, 10)(  8,  9)( 13, 14)( 15, 20)( 16, 19)
( 17, 18)( 22, 23)( 24, 29)( 25, 28)( 26, 27)( 31, 32)( 33, 38)( 34, 37)
( 35, 36)( 40, 41)( 42, 47)( 43, 46)( 44, 45)( 49, 50)( 51, 56)( 52, 55)
( 53, 54)( 58, 59)( 60, 65)( 61, 64)( 62, 63)( 67, 68)( 69, 74)( 70, 73)
( 71, 72)( 76, 77)( 78, 83)( 79, 82)( 80, 81)( 85, 86)( 87, 92)( 88, 91)
( 89, 90)( 94, 95)( 96,101)( 97,100)( 98, 99)(103,104)(105,110)(106,109)
(107,108)(112,113)(114,119)(115,118)(116,117)(121,122)(123,128)(124,127)
(125,126)(130,131)(132,137)(133,136)(134,135)(139,140)(141,146)(142,145)
(143,144)(148,149)(150,155)(151,154)(152,153)(157,158)(159,164)(160,163)
(161,162)(166,167)(168,173)(169,172)(170,171)(175,176)(177,182)(178,181)
(179,180)(184,185)(186,191)(187,190)(188,189)(193,194)(195,200)(196,199)
(197,198)(202,203)(204,209)(205,208)(206,207)(211,212)(213,218)(214,217)
(215,216);
s2 := Sym(218)!(  3,  6)(  4,  8)(  5,  7)(  9, 11)( 12, 24)( 13, 26)( 14, 25)
( 15, 21)( 16, 23)( 17, 22)( 18, 29)( 19, 28)( 20, 27)( 30, 33)( 31, 35)
( 32, 34)( 36, 38)( 39, 51)( 40, 53)( 41, 52)( 42, 48)( 43, 50)( 44, 49)
( 45, 56)( 46, 55)( 47, 54)( 57, 87)( 58, 89)( 59, 88)( 60, 84)( 61, 86)
( 62, 85)( 63, 92)( 64, 91)( 65, 90)( 66,105)( 67,107)( 68,106)( 69,102)
( 70,104)( 71,103)( 72,110)( 73,109)( 74,108)( 75, 96)( 76, 98)( 77, 97)
( 78, 93)( 79, 95)( 80, 94)( 81,101)( 82,100)( 83, 99)(111,168)(112,170)
(113,169)(114,165)(115,167)(116,166)(117,173)(118,172)(119,171)(120,186)
(121,188)(122,187)(123,183)(124,185)(125,184)(126,191)(127,190)(128,189)
(129,177)(130,179)(131,178)(132,174)(133,176)(134,175)(135,182)(136,181)
(137,180)(138,195)(139,197)(140,196)(141,192)(142,194)(143,193)(144,200)
(145,199)(146,198)(147,213)(148,215)(149,214)(150,210)(151,212)(152,211)
(153,218)(154,217)(155,216)(156,204)(157,206)(158,205)(159,201)(160,203)
(161,202)(162,209)(163,208)(164,207);
s3 := Sym(218)!(  3,120)(  4,121)(  5,122)(  6,123)(  7,124)(  8,125)(  9,126)
( 10,127)( 11,128)( 12,111)( 13,112)( 14,113)( 15,114)( 16,115)( 17,116)
( 18,117)( 19,118)( 20,119)( 21,129)( 22,130)( 23,131)( 24,132)( 25,133)
( 26,134)( 27,135)( 28,136)( 29,137)( 30,147)( 31,148)( 32,149)( 33,150)
( 34,151)( 35,152)( 36,153)( 37,154)( 38,155)( 39,138)( 40,139)( 41,140)
( 42,141)( 43,142)( 44,143)( 45,144)( 46,145)( 47,146)( 48,156)( 49,157)
( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,163)( 56,164)( 57,201)
( 58,202)( 59,203)( 60,204)( 61,205)( 62,206)( 63,207)( 64,208)( 65,209)
( 66,192)( 67,193)( 68,194)( 69,195)( 70,196)( 71,197)( 72,198)( 73,199)
( 74,200)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)( 81,216)
( 82,217)( 83,218)( 84,174)( 85,175)( 86,176)( 87,177)( 88,178)( 89,179)
( 90,180)( 91,181)( 92,182)( 93,165)( 94,166)( 95,167)( 96,168)( 97,169)
( 98,170)( 99,171)(100,172)(101,173)(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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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 >; 
 

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