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Polytope of Type {18,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6,3}*864
Also Known As : {{18,6|2},{6,3}4}. if this polytope has another name.
Group : SmallGroup(864,3998)
Rank : 4
Schlafli Type : {18,6,3}
Number of vertices, edges, etc : 18, 72, 12, 4
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {18,6,3,2} of size 1728
Vertex Figure Of :
   {2,18,6,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,6,3}*288
   9-fold quotients : {2,6,3}*96
   18-fold quotients : {2,3,3}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,6,3}*1728, {18,12,3}*1728, {18,6,6}*1728
Permutation Representation (GAP) :
s0 := (  5,  9)(  6, 10)(  7, 11)(  8, 12)( 13, 29)( 14, 30)( 15, 31)( 16, 32)
( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 33)( 22, 34)( 23, 35)( 24, 36)
( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 65)( 50, 66)( 51, 67)( 52, 68)
( 53, 61)( 54, 62)( 55, 63)( 56, 64)( 57, 69)( 58, 70)( 59, 71)( 60, 72)
( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 85,101)( 86,102)( 87,103)( 88,104)
( 89, 97)( 90, 98)( 91, 99)( 92,100)( 93,105)( 94,106)( 95,107)( 96,108)
(113,117)(114,118)(115,119)(116,120)(121,137)(122,138)(123,139)(124,140)
(125,133)(126,134)(127,135)(128,136)(129,141)(130,142)(131,143)(132,144)
(149,153)(150,154)(151,155)(152,156)(157,173)(158,174)(159,175)(160,176)
(161,169)(162,170)(163,171)(164,172)(165,177)(166,178)(167,179)(168,180)
(185,189)(186,190)(187,191)(188,192)(193,209)(194,210)(195,211)(196,212)
(197,205)(198,206)(199,207)(200,208)(201,213)(202,214)(203,215)(204,216);;
s1 := (  1, 13)(  2, 15)(  3, 14)(  4, 16)(  5, 21)(  6, 23)(  7, 22)(  8, 24)
(  9, 17)( 10, 19)( 11, 18)( 12, 20)( 25, 29)( 26, 31)( 27, 30)( 28, 32)
( 34, 35)( 37, 85)( 38, 87)( 39, 86)( 40, 88)( 41, 93)( 42, 95)( 43, 94)
( 44, 96)( 45, 89)( 46, 91)( 47, 90)( 48, 92)( 49, 73)( 50, 75)( 51, 74)
( 52, 76)( 53, 81)( 54, 83)( 55, 82)( 56, 84)( 57, 77)( 58, 79)( 59, 78)
( 60, 80)( 61,101)( 62,103)( 63,102)( 64,104)( 65, 97)( 66, 99)( 67, 98)
( 68,100)( 69,105)( 70,107)( 71,106)( 72,108)(109,121)(110,123)(111,122)
(112,124)(113,129)(114,131)(115,130)(116,132)(117,125)(118,127)(119,126)
(120,128)(133,137)(134,139)(135,138)(136,140)(142,143)(145,193)(146,195)
(147,194)(148,196)(149,201)(150,203)(151,202)(152,204)(153,197)(154,199)
(155,198)(156,200)(157,181)(158,183)(159,182)(160,184)(161,189)(162,191)
(163,190)(164,192)(165,185)(166,187)(167,186)(168,188)(169,209)(170,211)
(171,210)(172,212)(173,205)(174,207)(175,206)(176,208)(177,213)(178,215)
(179,214)(180,216);;
s2 := (  1,145)(  2,146)(  3,148)(  4,147)(  5,149)(  6,150)(  7,152)(  8,151)
(  9,153)( 10,154)( 11,156)( 12,155)( 13,157)( 14,158)( 15,160)( 16,159)
( 17,161)( 18,162)( 19,164)( 20,163)( 21,165)( 22,166)( 23,168)( 24,167)
( 25,169)( 26,170)( 27,172)( 28,171)( 29,173)( 30,174)( 31,176)( 32,175)
( 33,177)( 34,178)( 35,180)( 36,179)( 37,109)( 38,110)( 39,112)( 40,111)
( 41,113)( 42,114)( 43,116)( 44,115)( 45,117)( 46,118)( 47,120)( 48,119)
( 49,121)( 50,122)( 51,124)( 52,123)( 53,125)( 54,126)( 55,128)( 56,127)
( 57,129)( 58,130)( 59,132)( 60,131)( 61,133)( 62,134)( 63,136)( 64,135)
( 65,137)( 66,138)( 67,140)( 68,139)( 69,141)( 70,142)( 71,144)( 72,143)
( 73,181)( 74,182)( 75,184)( 76,183)( 77,185)( 78,186)( 79,188)( 80,187)
( 81,189)( 82,190)( 83,192)( 84,191)( 85,193)( 86,194)( 87,196)( 88,195)
( 89,197)( 90,198)( 91,200)( 92,199)( 93,201)( 94,202)( 95,204)( 96,203)
( 97,205)( 98,206)( 99,208)(100,207)(101,209)(102,210)(103,212)(104,211)
(105,213)(106,214)(107,216)(108,215);;
s3 := (  1,112)(  2,110)(  3,111)(  4,109)(  5,116)(  6,114)(  7,115)(  8,113)
(  9,120)( 10,118)( 11,119)( 12,117)( 13,124)( 14,122)( 15,123)( 16,121)
( 17,128)( 18,126)( 19,127)( 20,125)( 21,132)( 22,130)( 23,131)( 24,129)
( 25,136)( 26,134)( 27,135)( 28,133)( 29,140)( 30,138)( 31,139)( 32,137)
( 33,144)( 34,142)( 35,143)( 36,141)( 37,184)( 38,182)( 39,183)( 40,181)
( 41,188)( 42,186)( 43,187)( 44,185)( 45,192)( 46,190)( 47,191)( 48,189)
( 49,196)( 50,194)( 51,195)( 52,193)( 53,200)( 54,198)( 55,199)( 56,197)
( 57,204)( 58,202)( 59,203)( 60,201)( 61,208)( 62,206)( 63,207)( 64,205)
( 65,212)( 66,210)( 67,211)( 68,209)( 69,216)( 70,214)( 71,215)( 72,213)
( 73,148)( 74,146)( 75,147)( 76,145)( 77,152)( 78,150)( 79,151)( 80,149)
( 81,156)( 82,154)( 83,155)( 84,153)( 85,160)( 86,158)( 87,159)( 88,157)
( 89,164)( 90,162)( 91,163)( 92,161)( 93,168)( 94,166)( 95,167)( 96,165)
( 97,172)( 98,170)( 99,171)(100,169)(101,176)(102,174)(103,175)(104,173)
(105,180)(106,178)(107,179)(108,177);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(216)!(  5,  9)(  6, 10)(  7, 11)(  8, 12)( 13, 29)( 14, 30)( 15, 31)
( 16, 32)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 33)( 22, 34)( 23, 35)
( 24, 36)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 65)( 50, 66)( 51, 67)
( 52, 68)( 53, 61)( 54, 62)( 55, 63)( 56, 64)( 57, 69)( 58, 70)( 59, 71)
( 60, 72)( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 85,101)( 86,102)( 87,103)
( 88,104)( 89, 97)( 90, 98)( 91, 99)( 92,100)( 93,105)( 94,106)( 95,107)
( 96,108)(113,117)(114,118)(115,119)(116,120)(121,137)(122,138)(123,139)
(124,140)(125,133)(126,134)(127,135)(128,136)(129,141)(130,142)(131,143)
(132,144)(149,153)(150,154)(151,155)(152,156)(157,173)(158,174)(159,175)
(160,176)(161,169)(162,170)(163,171)(164,172)(165,177)(166,178)(167,179)
(168,180)(185,189)(186,190)(187,191)(188,192)(193,209)(194,210)(195,211)
(196,212)(197,205)(198,206)(199,207)(200,208)(201,213)(202,214)(203,215)
(204,216);
s1 := Sym(216)!(  1, 13)(  2, 15)(  3, 14)(  4, 16)(  5, 21)(  6, 23)(  7, 22)
(  8, 24)(  9, 17)( 10, 19)( 11, 18)( 12, 20)( 25, 29)( 26, 31)( 27, 30)
( 28, 32)( 34, 35)( 37, 85)( 38, 87)( 39, 86)( 40, 88)( 41, 93)( 42, 95)
( 43, 94)( 44, 96)( 45, 89)( 46, 91)( 47, 90)( 48, 92)( 49, 73)( 50, 75)
( 51, 74)( 52, 76)( 53, 81)( 54, 83)( 55, 82)( 56, 84)( 57, 77)( 58, 79)
( 59, 78)( 60, 80)( 61,101)( 62,103)( 63,102)( 64,104)( 65, 97)( 66, 99)
( 67, 98)( 68,100)( 69,105)( 70,107)( 71,106)( 72,108)(109,121)(110,123)
(111,122)(112,124)(113,129)(114,131)(115,130)(116,132)(117,125)(118,127)
(119,126)(120,128)(133,137)(134,139)(135,138)(136,140)(142,143)(145,193)
(146,195)(147,194)(148,196)(149,201)(150,203)(151,202)(152,204)(153,197)
(154,199)(155,198)(156,200)(157,181)(158,183)(159,182)(160,184)(161,189)
(162,191)(163,190)(164,192)(165,185)(166,187)(167,186)(168,188)(169,209)
(170,211)(171,210)(172,212)(173,205)(174,207)(175,206)(176,208)(177,213)
(178,215)(179,214)(180,216);
s2 := Sym(216)!(  1,145)(  2,146)(  3,148)(  4,147)(  5,149)(  6,150)(  7,152)
(  8,151)(  9,153)( 10,154)( 11,156)( 12,155)( 13,157)( 14,158)( 15,160)
( 16,159)( 17,161)( 18,162)( 19,164)( 20,163)( 21,165)( 22,166)( 23,168)
( 24,167)( 25,169)( 26,170)( 27,172)( 28,171)( 29,173)( 30,174)( 31,176)
( 32,175)( 33,177)( 34,178)( 35,180)( 36,179)( 37,109)( 38,110)( 39,112)
( 40,111)( 41,113)( 42,114)( 43,116)( 44,115)( 45,117)( 46,118)( 47,120)
( 48,119)( 49,121)( 50,122)( 51,124)( 52,123)( 53,125)( 54,126)( 55,128)
( 56,127)( 57,129)( 58,130)( 59,132)( 60,131)( 61,133)( 62,134)( 63,136)
( 64,135)( 65,137)( 66,138)( 67,140)( 68,139)( 69,141)( 70,142)( 71,144)
( 72,143)( 73,181)( 74,182)( 75,184)( 76,183)( 77,185)( 78,186)( 79,188)
( 80,187)( 81,189)( 82,190)( 83,192)( 84,191)( 85,193)( 86,194)( 87,196)
( 88,195)( 89,197)( 90,198)( 91,200)( 92,199)( 93,201)( 94,202)( 95,204)
( 96,203)( 97,205)( 98,206)( 99,208)(100,207)(101,209)(102,210)(103,212)
(104,211)(105,213)(106,214)(107,216)(108,215);
s3 := Sym(216)!(  1,112)(  2,110)(  3,111)(  4,109)(  5,116)(  6,114)(  7,115)
(  8,113)(  9,120)( 10,118)( 11,119)( 12,117)( 13,124)( 14,122)( 15,123)
( 16,121)( 17,128)( 18,126)( 19,127)( 20,125)( 21,132)( 22,130)( 23,131)
( 24,129)( 25,136)( 26,134)( 27,135)( 28,133)( 29,140)( 30,138)( 31,139)
( 32,137)( 33,144)( 34,142)( 35,143)( 36,141)( 37,184)( 38,182)( 39,183)
( 40,181)( 41,188)( 42,186)( 43,187)( 44,185)( 45,192)( 46,190)( 47,191)
( 48,189)( 49,196)( 50,194)( 51,195)( 52,193)( 53,200)( 54,198)( 55,199)
( 56,197)( 57,204)( 58,202)( 59,203)( 60,201)( 61,208)( 62,206)( 63,207)
( 64,205)( 65,212)( 66,210)( 67,211)( 68,209)( 69,216)( 70,214)( 71,215)
( 72,213)( 73,148)( 74,146)( 75,147)( 76,145)( 77,152)( 78,150)( 79,151)
( 80,149)( 81,156)( 82,154)( 83,155)( 84,153)( 85,160)( 86,158)( 87,159)
( 88,157)( 89,164)( 90,162)( 91,163)( 92,161)( 93,168)( 94,166)( 95,167)
( 96,165)( 97,172)( 98,170)( 99,171)(100,169)(101,176)(102,174)(103,175)
(104,173)(105,180)(106,178)(107,179)(108,177);
poly := sub<Sym(216)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope