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Polytope of Type {2,18,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,12}*1728b
if this polytope has a name.
Group : SmallGroup(1728,46115)
Rank : 4
Schlafli Type : {2,18,12}
Number of vertices, edges, etc : 2, 36, 216, 24
Order of s0s1s2s3 : 18
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 : {2,9,12}*864
   3-fold quotients : {2,18,4}*576, {2,6,12}*576b
   4-fold quotients : {2,18,6}*432b
   6-fold quotients : {2,9,4}*288, {2,18,4}*288b, {2,18,4}*288c, {2,3,12}*288
   8-fold quotients : {2,9,6}*216
   9-fold quotients : {2,6,4}*192
   12-fold quotients : {2,9,4}*144, {2,18,2}*144, {2,6,6}*144c
   18-fold quotients : {2,3,4}*96, {2,6,4}*96b, {2,6,4}*96c
   24-fold quotients : {2,9,2}*72, {2,3,6}*72
   36-fold quotients : {2,3,4}*48, {2,6,2}*48
   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)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 27)( 16, 29)( 17, 28)
( 18, 30)( 19, 35)( 20, 37)( 21, 36)( 22, 38)( 23, 31)( 24, 33)( 25, 32)
( 26, 34)( 39, 79)( 40, 81)( 41, 80)( 42, 82)( 43, 75)( 44, 77)( 45, 76)
( 46, 78)( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51,103)( 52,105)( 53,104)
( 54,106)( 55, 99)( 56,101)( 57,100)( 58,102)( 59,107)( 60,109)( 61,108)
( 62,110)( 63, 91)( 64, 93)( 65, 92)( 66, 94)( 67, 87)( 68, 89)( 69, 88)
( 70, 90)( 71, 95)( 72, 97)( 73, 96)( 74, 98)(112,113)(115,119)(116,121)
(117,120)(118,122)(123,135)(124,137)(125,136)(126,138)(127,143)(128,145)
(129,144)(130,146)(131,139)(132,141)(133,140)(134,142)(147,187)(148,189)
(149,188)(150,190)(151,183)(152,185)(153,184)(154,186)(155,191)(156,193)
(157,192)(158,194)(159,211)(160,213)(161,212)(162,214)(163,207)(164,209)
(165,208)(166,210)(167,215)(168,217)(169,216)(170,218)(171,199)(172,201)
(173,200)(174,202)(175,195)(176,197)(177,196)(178,198)(179,203)(180,205)
(181,204)(182,206);;
s2 := (  3,159)(  4,160)(  5,162)(  6,161)(  7,167)(  8,168)(  9,170)( 10,169)
( 11,163)( 12,164)( 13,166)( 14,165)( 15,147)( 16,148)( 17,150)( 18,149)
( 19,155)( 20,156)( 21,158)( 22,157)( 23,151)( 24,152)( 25,154)( 26,153)
( 27,171)( 28,172)( 29,174)( 30,173)( 31,179)( 32,180)( 33,182)( 34,181)
( 35,175)( 36,176)( 37,178)( 38,177)( 39,123)( 40,124)( 41,126)( 42,125)
( 43,131)( 44,132)( 45,134)( 46,133)( 47,127)( 48,128)( 49,130)( 50,129)
( 51,111)( 52,112)( 53,114)( 54,113)( 55,119)( 56,120)( 57,122)( 58,121)
( 59,115)( 60,116)( 61,118)( 62,117)( 63,135)( 64,136)( 65,138)( 66,137)
( 67,143)( 68,144)( 69,146)( 70,145)( 71,139)( 72,140)( 73,142)( 74,141)
( 75,199)( 76,200)( 77,202)( 78,201)( 79,195)( 80,196)( 81,198)( 82,197)
( 83,203)( 84,204)( 85,206)( 86,205)( 87,187)( 88,188)( 89,190)( 90,189)
( 91,183)( 92,184)( 93,186)( 94,185)( 95,191)( 96,192)( 97,194)( 98,193)
( 99,211)(100,212)(101,214)(102,213)(103,207)(104,208)(105,210)(106,209)
(107,215)(108,216)(109,218)(110,217);;
s3 := (  3,  6)(  4,  5)(  7, 10)(  8,  9)( 11, 14)( 12, 13)( 15, 30)( 16, 29)
( 17, 28)( 18, 27)( 19, 34)( 20, 33)( 21, 32)( 22, 31)( 23, 38)( 24, 37)
( 25, 36)( 26, 35)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)( 48, 49)
( 51, 66)( 52, 65)( 53, 64)( 54, 63)( 55, 70)( 56, 69)( 57, 68)( 58, 67)
( 59, 74)( 60, 73)( 61, 72)( 62, 71)( 75, 78)( 76, 77)( 79, 82)( 80, 81)
( 83, 86)( 84, 85)( 87,102)( 88,101)( 89,100)( 90, 99)( 91,106)( 92,105)
( 93,104)( 94,103)( 95,110)( 96,109)( 97,108)( 98,107)(111,114)(112,113)
(115,118)(116,117)(119,122)(120,121)(123,138)(124,137)(125,136)(126,135)
(127,142)(128,141)(129,140)(130,139)(131,146)(132,145)(133,144)(134,143)
(147,150)(148,149)(151,154)(152,153)(155,158)(156,157)(159,174)(160,173)
(161,172)(162,171)(163,178)(164,177)(165,176)(166,175)(167,182)(168,181)
(169,180)(170,179)(183,186)(184,185)(187,190)(188,189)(191,194)(192,193)
(195,210)(196,209)(197,208)(198,207)(199,214)(200,213)(201,212)(202,211)
(203,218)(204,217)(205,216)(206,215);;
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*s1*s2*s3*s2*s1*s2, 
s1*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s3*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!(  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 27)( 16, 29)
( 17, 28)( 18, 30)( 19, 35)( 20, 37)( 21, 36)( 22, 38)( 23, 31)( 24, 33)
( 25, 32)( 26, 34)( 39, 79)( 40, 81)( 41, 80)( 42, 82)( 43, 75)( 44, 77)
( 45, 76)( 46, 78)( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51,103)( 52,105)
( 53,104)( 54,106)( 55, 99)( 56,101)( 57,100)( 58,102)( 59,107)( 60,109)
( 61,108)( 62,110)( 63, 91)( 64, 93)( 65, 92)( 66, 94)( 67, 87)( 68, 89)
( 69, 88)( 70, 90)( 71, 95)( 72, 97)( 73, 96)( 74, 98)(112,113)(115,119)
(116,121)(117,120)(118,122)(123,135)(124,137)(125,136)(126,138)(127,143)
(128,145)(129,144)(130,146)(131,139)(132,141)(133,140)(134,142)(147,187)
(148,189)(149,188)(150,190)(151,183)(152,185)(153,184)(154,186)(155,191)
(156,193)(157,192)(158,194)(159,211)(160,213)(161,212)(162,214)(163,207)
(164,209)(165,208)(166,210)(167,215)(168,217)(169,216)(170,218)(171,199)
(172,201)(173,200)(174,202)(175,195)(176,197)(177,196)(178,198)(179,203)
(180,205)(181,204)(182,206);
s2 := Sym(218)!(  3,159)(  4,160)(  5,162)(  6,161)(  7,167)(  8,168)(  9,170)
( 10,169)( 11,163)( 12,164)( 13,166)( 14,165)( 15,147)( 16,148)( 17,150)
( 18,149)( 19,155)( 20,156)( 21,158)( 22,157)( 23,151)( 24,152)( 25,154)
( 26,153)( 27,171)( 28,172)( 29,174)( 30,173)( 31,179)( 32,180)( 33,182)
( 34,181)( 35,175)( 36,176)( 37,178)( 38,177)( 39,123)( 40,124)( 41,126)
( 42,125)( 43,131)( 44,132)( 45,134)( 46,133)( 47,127)( 48,128)( 49,130)
( 50,129)( 51,111)( 52,112)( 53,114)( 54,113)( 55,119)( 56,120)( 57,122)
( 58,121)( 59,115)( 60,116)( 61,118)( 62,117)( 63,135)( 64,136)( 65,138)
( 66,137)( 67,143)( 68,144)( 69,146)( 70,145)( 71,139)( 72,140)( 73,142)
( 74,141)( 75,199)( 76,200)( 77,202)( 78,201)( 79,195)( 80,196)( 81,198)
( 82,197)( 83,203)( 84,204)( 85,206)( 86,205)( 87,187)( 88,188)( 89,190)
( 90,189)( 91,183)( 92,184)( 93,186)( 94,185)( 95,191)( 96,192)( 97,194)
( 98,193)( 99,211)(100,212)(101,214)(102,213)(103,207)(104,208)(105,210)
(106,209)(107,215)(108,216)(109,218)(110,217);
s3 := Sym(218)!(  3,  6)(  4,  5)(  7, 10)(  8,  9)( 11, 14)( 12, 13)( 15, 30)
( 16, 29)( 17, 28)( 18, 27)( 19, 34)( 20, 33)( 21, 32)( 22, 31)( 23, 38)
( 24, 37)( 25, 36)( 26, 35)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)
( 48, 49)( 51, 66)( 52, 65)( 53, 64)( 54, 63)( 55, 70)( 56, 69)( 57, 68)
( 58, 67)( 59, 74)( 60, 73)( 61, 72)( 62, 71)( 75, 78)( 76, 77)( 79, 82)
( 80, 81)( 83, 86)( 84, 85)( 87,102)( 88,101)( 89,100)( 90, 99)( 91,106)
( 92,105)( 93,104)( 94,103)( 95,110)( 96,109)( 97,108)( 98,107)(111,114)
(112,113)(115,118)(116,117)(119,122)(120,121)(123,138)(124,137)(125,136)
(126,135)(127,142)(128,141)(129,140)(130,139)(131,146)(132,145)(133,144)
(134,143)(147,150)(148,149)(151,154)(152,153)(155,158)(156,157)(159,174)
(160,173)(161,172)(162,171)(163,178)(164,177)(165,176)(166,175)(167,182)
(168,181)(169,180)(170,179)(183,186)(184,185)(187,190)(188,189)(191,194)
(192,193)(195,210)(196,209)(197,208)(198,207)(199,214)(200,213)(201,212)
(202,211)(203,218)(204,217)(205,216)(206,215);
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*s1*s2*s3*s2*s1*s2, 
s1*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s3*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*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 >; 
 

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