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Polytope of Type {2,6,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,24}*1728a
if this polytope has a name.
Group : SmallGroup(1728,15888)
Rank : 4
Schlafli Type : {2,6,24}
Number of vertices, edges, etc : 2, 18, 216, 72
Order of s0s1s2s3 : 24
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 : {2,6,12}*864a
   3-fold quotients : {2,6,24}*576b
   4-fold quotients : {2,6,6}*432a
   6-fold quotients : {2,6,12}*288b
   8-fold quotients : {2,6,3}*216
   9-fold quotients : {2,2,24}*192
   12-fold quotients : {2,6,6}*144b
   18-fold quotients : {2,2,12}*96
   24-fold quotients : {2,6,3}*72
   27-fold quotients : {2,2,8}*64
   36-fold quotients : {2,2,6}*48
   54-fold quotients : {2,2,4}*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 := (  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)( 35, 37)
( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)( 53, 55)
( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)( 71, 73)
( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)( 89, 91)
( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)(107,109)
(112,113)(114,117)(115,119)(116,118)(121,122)(123,126)(124,128)(125,127)
(130,131)(132,135)(133,137)(134,136)(139,140)(141,144)(142,146)(143,145)
(148,149)(150,153)(151,155)(152,154)(157,158)(159,162)(160,164)(161,163)
(166,167)(168,171)(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)
(184,185)(186,189)(187,191)(188,190)(193,194)(195,198)(196,200)(197,199)
(202,203)(204,207)(205,209)(206,208)(211,212)(213,216)(214,218)(215,217);;
s2 := (  3,  6)(  4,  7)(  5,  8)( 12, 24)( 13, 25)( 14, 26)( 15, 21)( 16, 22)
( 17, 23)( 18, 27)( 19, 28)( 20, 29)( 30, 33)( 31, 34)( 32, 35)( 39, 51)
( 40, 52)( 41, 53)( 42, 48)( 43, 49)( 44, 50)( 45, 54)( 46, 55)( 47, 56)
( 57, 87)( 58, 88)( 59, 89)( 60, 84)( 61, 85)( 62, 86)( 63, 90)( 64, 91)
( 65, 92)( 66,105)( 67,106)( 68,107)( 69,102)( 70,103)( 71,104)( 72,108)
( 73,109)( 74,110)( 75, 96)( 76, 97)( 77, 98)( 78, 93)( 79, 94)( 80, 95)
( 81, 99)( 82,100)( 83,101)(111,168)(112,169)(113,170)(114,165)(115,166)
(116,167)(117,171)(118,172)(119,173)(120,186)(121,187)(122,188)(123,183)
(124,184)(125,185)(126,189)(127,190)(128,191)(129,177)(130,178)(131,179)
(132,174)(133,175)(134,176)(135,180)(136,181)(137,182)(138,195)(139,196)
(140,197)(141,192)(142,193)(143,194)(144,198)(145,199)(146,200)(147,213)
(148,214)(149,215)(150,210)(151,211)(152,212)(153,216)(154,217)(155,218)
(156,204)(157,205)(158,206)(159,201)(160,202)(161,203)(162,207)(163,208)
(164,209);;
s3 := (  3,120)(  4,121)(  5,122)(  6,128)(  7,126)(  8,127)(  9,124)( 10,125)
( 11,123)( 12,111)( 13,112)( 14,113)( 15,119)( 16,117)( 17,118)( 18,115)
( 19,116)( 20,114)( 21,129)( 22,130)( 23,131)( 24,137)( 25,135)( 26,136)
( 27,133)( 28,134)( 29,132)( 30,147)( 31,148)( 32,149)( 33,155)( 34,153)
( 35,154)( 36,151)( 37,152)( 38,150)( 39,138)( 40,139)( 41,140)( 42,146)
( 43,144)( 44,145)( 45,142)( 46,143)( 47,141)( 48,156)( 49,157)( 50,158)
( 51,164)( 52,162)( 53,163)( 54,160)( 55,161)( 56,159)( 57,201)( 58,202)
( 59,203)( 60,209)( 61,207)( 62,208)( 63,205)( 64,206)( 65,204)( 66,192)
( 67,193)( 68,194)( 69,200)( 70,198)( 71,199)( 72,196)( 73,197)( 74,195)
( 75,210)( 76,211)( 77,212)( 78,218)( 79,216)( 80,217)( 81,214)( 82,215)
( 83,213)( 84,174)( 85,175)( 86,176)( 87,182)( 88,180)( 89,181)( 90,178)
( 91,179)( 92,177)( 93,165)( 94,166)( 95,167)( 96,173)( 97,171)( 98,172)
( 99,169)(100,170)(101,168)(102,183)(103,184)(104,185)(105,191)(106,189)
(107,190)(108,187)(109,188)(110,186);;
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*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,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)
( 53, 55)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)
( 71, 73)( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)
( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)
(107,109)(112,113)(114,117)(115,119)(116,118)(121,122)(123,126)(124,128)
(125,127)(130,131)(132,135)(133,137)(134,136)(139,140)(141,144)(142,146)
(143,145)(148,149)(150,153)(151,155)(152,154)(157,158)(159,162)(160,164)
(161,163)(166,167)(168,171)(169,173)(170,172)(175,176)(177,180)(178,182)
(179,181)(184,185)(186,189)(187,191)(188,190)(193,194)(195,198)(196,200)
(197,199)(202,203)(204,207)(205,209)(206,208)(211,212)(213,216)(214,218)
(215,217);
s2 := Sym(218)!(  3,  6)(  4,  7)(  5,  8)( 12, 24)( 13, 25)( 14, 26)( 15, 21)
( 16, 22)( 17, 23)( 18, 27)( 19, 28)( 20, 29)( 30, 33)( 31, 34)( 32, 35)
( 39, 51)( 40, 52)( 41, 53)( 42, 48)( 43, 49)( 44, 50)( 45, 54)( 46, 55)
( 47, 56)( 57, 87)( 58, 88)( 59, 89)( 60, 84)( 61, 85)( 62, 86)( 63, 90)
( 64, 91)( 65, 92)( 66,105)( 67,106)( 68,107)( 69,102)( 70,103)( 71,104)
( 72,108)( 73,109)( 74,110)( 75, 96)( 76, 97)( 77, 98)( 78, 93)( 79, 94)
( 80, 95)( 81, 99)( 82,100)( 83,101)(111,168)(112,169)(113,170)(114,165)
(115,166)(116,167)(117,171)(118,172)(119,173)(120,186)(121,187)(122,188)
(123,183)(124,184)(125,185)(126,189)(127,190)(128,191)(129,177)(130,178)
(131,179)(132,174)(133,175)(134,176)(135,180)(136,181)(137,182)(138,195)
(139,196)(140,197)(141,192)(142,193)(143,194)(144,198)(145,199)(146,200)
(147,213)(148,214)(149,215)(150,210)(151,211)(152,212)(153,216)(154,217)
(155,218)(156,204)(157,205)(158,206)(159,201)(160,202)(161,203)(162,207)
(163,208)(164,209);
s3 := Sym(218)!(  3,120)(  4,121)(  5,122)(  6,128)(  7,126)(  8,127)(  9,124)
( 10,125)( 11,123)( 12,111)( 13,112)( 14,113)( 15,119)( 16,117)( 17,118)
( 18,115)( 19,116)( 20,114)( 21,129)( 22,130)( 23,131)( 24,137)( 25,135)
( 26,136)( 27,133)( 28,134)( 29,132)( 30,147)( 31,148)( 32,149)( 33,155)
( 34,153)( 35,154)( 36,151)( 37,152)( 38,150)( 39,138)( 40,139)( 41,140)
( 42,146)( 43,144)( 44,145)( 45,142)( 46,143)( 47,141)( 48,156)( 49,157)
( 50,158)( 51,164)( 52,162)( 53,163)( 54,160)( 55,161)( 56,159)( 57,201)
( 58,202)( 59,203)( 60,209)( 61,207)( 62,208)( 63,205)( 64,206)( 65,204)
( 66,192)( 67,193)( 68,194)( 69,200)( 70,198)( 71,199)( 72,196)( 73,197)
( 74,195)( 75,210)( 76,211)( 77,212)( 78,218)( 79,216)( 80,217)( 81,214)
( 82,215)( 83,213)( 84,174)( 85,175)( 86,176)( 87,182)( 88,180)( 89,181)
( 90,178)( 91,179)( 92,177)( 93,165)( 94,166)( 95,167)( 96,173)( 97,171)
( 98,172)( 99,169)(100,170)(101,168)(102,183)(103,184)(104,185)(105,191)
(106,189)(107,190)(108,187)(109,188)(110,186);
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*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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