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Polytope of Type {2,24,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,24,18}*1728b
if this polytope has a name.
Group : SmallGroup(1728,17171)
Rank : 4
Schlafli Type : {2,24,18}
Number of vertices, edges, etc : 2, 24, 216, 18
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 : {2,12,18}*864b
   3-fold quotients : {2,8,18}*576, {2,24,6}*576c
   4-fold quotients : {2,6,18}*432b
   6-fold quotients : {2,4,18}*288a, {2,12,6}*288c
   8-fold quotients : {2,6,9}*216
   9-fold quotients : {2,8,6}*192
   12-fold quotients : {2,2,18}*144, {2,6,6}*144b
   18-fold quotients : {2,4,6}*96a
   24-fold quotients : {2,2,9}*72, {2,6,3}*72
   27-fold quotients : {2,8,2}*64
   36-fold quotients : {2,2,6}*48
   54-fold quotients : {2,4,2}*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 := (  6,  9)(  7, 10)(  8, 11)( 15, 18)( 16, 19)( 17, 20)( 24, 27)( 25, 28)
( 26, 29)( 33, 36)( 34, 37)( 35, 38)( 42, 45)( 43, 46)( 44, 47)( 51, 54)
( 52, 55)( 53, 56)( 57, 84)( 58, 85)( 59, 86)( 60, 90)( 61, 91)( 62, 92)
( 63, 87)( 64, 88)( 65, 89)( 66, 93)( 67, 94)( 68, 95)( 69, 99)( 70,100)
( 71,101)( 72, 96)( 73, 97)( 74, 98)( 75,102)( 76,103)( 77,104)( 78,108)
( 79,109)( 80,110)( 81,105)( 82,106)( 83,107)(111,165)(112,166)(113,167)
(114,171)(115,172)(116,173)(117,168)(118,169)(119,170)(120,174)(121,175)
(122,176)(123,180)(124,181)(125,182)(126,177)(127,178)(128,179)(129,183)
(130,184)(131,185)(132,189)(133,190)(134,191)(135,186)(136,187)(137,188)
(138,192)(139,193)(140,194)(141,198)(142,199)(143,200)(144,195)(145,196)
(146,197)(147,201)(148,202)(149,203)(150,207)(151,208)(152,209)(153,204)
(154,205)(155,206)(156,210)(157,211)(158,212)(159,216)(160,217)(161,218)
(162,213)(163,214)(164,215);;
s2 := (  3,114)(  4,116)(  5,115)(  6,111)(  7,113)(  8,112)(  9,117)( 10,119)
( 11,118)( 12,134)( 13,133)( 14,132)( 15,131)( 16,130)( 17,129)( 18,137)
( 19,136)( 20,135)( 21,125)( 22,124)( 23,123)( 24,122)( 25,121)( 26,120)
( 27,128)( 28,127)( 29,126)( 30,141)( 31,143)( 32,142)( 33,138)( 34,140)
( 35,139)( 36,144)( 37,146)( 38,145)( 39,161)( 40,160)( 41,159)( 42,158)
( 43,157)( 44,156)( 45,164)( 46,163)( 47,162)( 48,152)( 49,151)( 50,150)
( 51,149)( 52,148)( 53,147)( 54,155)( 55,154)( 56,153)( 57,195)( 58,197)
( 59,196)( 60,192)( 61,194)( 62,193)( 63,198)( 64,200)( 65,199)( 66,215)
( 67,214)( 68,213)( 69,212)( 70,211)( 71,210)( 72,218)( 73,217)( 74,216)
( 75,206)( 76,205)( 77,204)( 78,203)( 79,202)( 80,201)( 81,209)( 82,208)
( 83,207)( 84,168)( 85,170)( 86,169)( 87,165)( 88,167)( 89,166)( 90,171)
( 91,173)( 92,172)( 93,188)( 94,187)( 95,186)( 96,185)( 97,184)( 98,183)
( 99,191)(100,190)(101,189)(102,179)(103,178)(104,177)(105,176)(106,175)
(107,174)(108,182)(109,181)(110,180);;
s3 := (  3, 12)(  4, 14)(  5, 13)(  6, 18)(  7, 20)(  8, 19)(  9, 15)( 10, 17)
( 11, 16)( 21, 23)( 24, 29)( 25, 28)( 26, 27)( 30, 39)( 31, 41)( 32, 40)
( 33, 45)( 34, 47)( 35, 46)( 36, 42)( 37, 44)( 38, 43)( 48, 50)( 51, 56)
( 52, 55)( 53, 54)( 57, 66)( 58, 68)( 59, 67)( 60, 72)( 61, 74)( 62, 73)
( 63, 69)( 64, 71)( 65, 70)( 75, 77)( 78, 83)( 79, 82)( 80, 81)( 84, 93)
( 85, 95)( 86, 94)( 87, 99)( 88,101)( 89,100)( 90, 96)( 91, 98)( 92, 97)
(102,104)(105,110)(106,109)(107,108)(111,120)(112,122)(113,121)(114,126)
(115,128)(116,127)(117,123)(118,125)(119,124)(129,131)(132,137)(133,136)
(134,135)(138,147)(139,149)(140,148)(141,153)(142,155)(143,154)(144,150)
(145,152)(146,151)(156,158)(159,164)(160,163)(161,162)(165,174)(166,176)
(167,175)(168,180)(169,182)(170,181)(171,177)(172,179)(173,178)(183,185)
(186,191)(187,190)(188,189)(192,201)(193,203)(194,202)(195,207)(196,209)
(197,208)(198,204)(199,206)(200,205)(210,212)(213,218)(214,217)(215,216);;
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, 
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2, 
s3*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!(  6,  9)(  7, 10)(  8, 11)( 15, 18)( 16, 19)( 17, 20)( 24, 27)
( 25, 28)( 26, 29)( 33, 36)( 34, 37)( 35, 38)( 42, 45)( 43, 46)( 44, 47)
( 51, 54)( 52, 55)( 53, 56)( 57, 84)( 58, 85)( 59, 86)( 60, 90)( 61, 91)
( 62, 92)( 63, 87)( 64, 88)( 65, 89)( 66, 93)( 67, 94)( 68, 95)( 69, 99)
( 70,100)( 71,101)( 72, 96)( 73, 97)( 74, 98)( 75,102)( 76,103)( 77,104)
( 78,108)( 79,109)( 80,110)( 81,105)( 82,106)( 83,107)(111,165)(112,166)
(113,167)(114,171)(115,172)(116,173)(117,168)(118,169)(119,170)(120,174)
(121,175)(122,176)(123,180)(124,181)(125,182)(126,177)(127,178)(128,179)
(129,183)(130,184)(131,185)(132,189)(133,190)(134,191)(135,186)(136,187)
(137,188)(138,192)(139,193)(140,194)(141,198)(142,199)(143,200)(144,195)
(145,196)(146,197)(147,201)(148,202)(149,203)(150,207)(151,208)(152,209)
(153,204)(154,205)(155,206)(156,210)(157,211)(158,212)(159,216)(160,217)
(161,218)(162,213)(163,214)(164,215);
s2 := Sym(218)!(  3,114)(  4,116)(  5,115)(  6,111)(  7,113)(  8,112)(  9,117)
( 10,119)( 11,118)( 12,134)( 13,133)( 14,132)( 15,131)( 16,130)( 17,129)
( 18,137)( 19,136)( 20,135)( 21,125)( 22,124)( 23,123)( 24,122)( 25,121)
( 26,120)( 27,128)( 28,127)( 29,126)( 30,141)( 31,143)( 32,142)( 33,138)
( 34,140)( 35,139)( 36,144)( 37,146)( 38,145)( 39,161)( 40,160)( 41,159)
( 42,158)( 43,157)( 44,156)( 45,164)( 46,163)( 47,162)( 48,152)( 49,151)
( 50,150)( 51,149)( 52,148)( 53,147)( 54,155)( 55,154)( 56,153)( 57,195)
( 58,197)( 59,196)( 60,192)( 61,194)( 62,193)( 63,198)( 64,200)( 65,199)
( 66,215)( 67,214)( 68,213)( 69,212)( 70,211)( 71,210)( 72,218)( 73,217)
( 74,216)( 75,206)( 76,205)( 77,204)( 78,203)( 79,202)( 80,201)( 81,209)
( 82,208)( 83,207)( 84,168)( 85,170)( 86,169)( 87,165)( 88,167)( 89,166)
( 90,171)( 91,173)( 92,172)( 93,188)( 94,187)( 95,186)( 96,185)( 97,184)
( 98,183)( 99,191)(100,190)(101,189)(102,179)(103,178)(104,177)(105,176)
(106,175)(107,174)(108,182)(109,181)(110,180);
s3 := Sym(218)!(  3, 12)(  4, 14)(  5, 13)(  6, 18)(  7, 20)(  8, 19)(  9, 15)
( 10, 17)( 11, 16)( 21, 23)( 24, 29)( 25, 28)( 26, 27)( 30, 39)( 31, 41)
( 32, 40)( 33, 45)( 34, 47)( 35, 46)( 36, 42)( 37, 44)( 38, 43)( 48, 50)
( 51, 56)( 52, 55)( 53, 54)( 57, 66)( 58, 68)( 59, 67)( 60, 72)( 61, 74)
( 62, 73)( 63, 69)( 64, 71)( 65, 70)( 75, 77)( 78, 83)( 79, 82)( 80, 81)
( 84, 93)( 85, 95)( 86, 94)( 87, 99)( 88,101)( 89,100)( 90, 96)( 91, 98)
( 92, 97)(102,104)(105,110)(106,109)(107,108)(111,120)(112,122)(113,121)
(114,126)(115,128)(116,127)(117,123)(118,125)(119,124)(129,131)(132,137)
(133,136)(134,135)(138,147)(139,149)(140,148)(141,153)(142,155)(143,154)
(144,150)(145,152)(146,151)(156,158)(159,164)(160,163)(161,162)(165,174)
(166,176)(167,175)(168,180)(169,182)(170,181)(171,177)(172,179)(173,178)
(183,185)(186,191)(187,190)(188,189)(192,201)(193,203)(194,202)(195,207)
(196,209)(197,208)(198,204)(199,206)(200,205)(210,212)(213,218)(214,217)
(215,216);
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, s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2, 
s3*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*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 >; 
 

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