Polytope of Type {18,4,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,4,3}*864
Also Known As : {{18,4|2},{4,3}}. if this polytope has another name.
Group : SmallGroup(864,3998)
Rank : 4
Schlafli Type : {18,4,3}
Number of vertices, edges, etc : 18, 72, 12, 6
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {18,4,3,2} of size 1728
Vertex Figure Of :
   {2,18,4,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,4,3}*288
   4-fold quotients : {18,2,3}*216
   8-fold quotients : {9,2,3}*108
   9-fold quotients : {2,4,3}*96
   12-fold quotients : {6,2,3}*72
   18-fold quotients : {2,4,3}*48
   24-fold quotients : {3,2,3}*36
   36-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,4,3}*1728, {18,8,3}*1728, {18,4,6}*1728a
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 := (  1,123)(  2,124)(  3,121)(  4,122)(  5,131)(  6,132)(  7,129)(  8,130)
(  9,127)( 10,128)( 11,125)( 12,126)( 13,111)( 14,112)( 15,109)( 16,110)
( 17,119)( 18,120)( 19,117)( 20,118)( 21,115)( 22,116)( 23,113)( 24,114)
( 25,139)( 26,140)( 27,137)( 28,138)( 29,135)( 30,136)( 31,133)( 32,134)
( 33,143)( 34,144)( 35,141)( 36,142)( 37,159)( 38,160)( 39,157)( 40,158)
( 41,167)( 42,168)( 43,165)( 44,166)( 45,163)( 46,164)( 47,161)( 48,162)
( 49,147)( 50,148)( 51,145)( 52,146)( 53,155)( 54,156)( 55,153)( 56,154)
( 57,151)( 58,152)( 59,149)( 60,150)( 61,175)( 62,176)( 63,173)( 64,174)
( 65,171)( 66,172)( 67,169)( 68,170)( 69,179)( 70,180)( 71,177)( 72,178)
( 73,195)( 74,196)( 75,193)( 76,194)( 77,203)( 78,204)( 79,201)( 80,202)
( 81,199)( 82,200)( 83,197)( 84,198)( 85,183)( 86,184)( 87,181)( 88,182)
( 89,191)( 90,192)( 91,189)( 92,190)( 93,187)( 94,188)( 95,185)( 96,186)
( 97,211)( 98,212)( 99,209)(100,210)(101,207)(102,208)(103,205)(104,206)
(105,215)(106,216)(107,213)(108,214);;
s2 := (  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 22, 23)( 26, 27)( 30, 31)
( 34, 35)( 37, 73)( 38, 75)( 39, 74)( 40, 76)( 41, 77)( 42, 79)( 43, 78)
( 44, 80)( 45, 81)( 46, 83)( 47, 82)( 48, 84)( 49, 85)( 50, 87)( 51, 86)
( 52, 88)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 93)( 58, 95)( 59, 94)
( 60, 96)( 61, 97)( 62, 99)( 63, 98)( 64,100)( 65,101)( 66,103)( 67,102)
( 68,104)( 69,105)( 70,107)( 71,106)( 72,108)(110,111)(114,115)(118,119)
(122,123)(126,127)(130,131)(134,135)(138,139)(142,143)(145,181)(146,183)
(147,182)(148,184)(149,185)(150,187)(151,186)(152,188)(153,189)(154,191)
(155,190)(156,192)(157,193)(158,195)(159,194)(160,196)(161,197)(162,199)
(163,198)(164,200)(165,201)(166,203)(167,202)(168,204)(169,205)(170,207)
(171,206)(172,208)(173,209)(174,211)(175,210)(176,212)(177,213)(178,215)
(179,214)(180,216);;
s3 := (  1, 73)(  2, 76)(  3, 75)(  4, 74)(  5, 77)(  6, 80)(  7, 79)(  8, 78)
(  9, 81)( 10, 84)( 11, 83)( 12, 82)( 13, 85)( 14, 88)( 15, 87)( 16, 86)
( 17, 89)( 18, 92)( 19, 91)( 20, 90)( 21, 93)( 22, 96)( 23, 95)( 24, 94)
( 25, 97)( 26,100)( 27, 99)( 28, 98)( 29,101)( 30,104)( 31,103)( 32,102)
( 33,105)( 34,108)( 35,107)( 36,106)( 38, 40)( 42, 44)( 46, 48)( 50, 52)
( 54, 56)( 58, 60)( 62, 64)( 66, 68)( 70, 72)(109,181)(110,184)(111,183)
(112,182)(113,185)(114,188)(115,187)(116,186)(117,189)(118,192)(119,191)
(120,190)(121,193)(122,196)(123,195)(124,194)(125,197)(126,200)(127,199)
(128,198)(129,201)(130,204)(131,203)(132,202)(133,205)(134,208)(135,207)
(136,206)(137,209)(138,212)(139,211)(140,210)(141,213)(142,216)(143,215)
(144,214)(146,148)(150,152)(154,156)(158,160)(162,164)(166,168)(170,172)
(174,176)(178,180);;
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*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)!(  1,123)(  2,124)(  3,121)(  4,122)(  5,131)(  6,132)(  7,129)
(  8,130)(  9,127)( 10,128)( 11,125)( 12,126)( 13,111)( 14,112)( 15,109)
( 16,110)( 17,119)( 18,120)( 19,117)( 20,118)( 21,115)( 22,116)( 23,113)
( 24,114)( 25,139)( 26,140)( 27,137)( 28,138)( 29,135)( 30,136)( 31,133)
( 32,134)( 33,143)( 34,144)( 35,141)( 36,142)( 37,159)( 38,160)( 39,157)
( 40,158)( 41,167)( 42,168)( 43,165)( 44,166)( 45,163)( 46,164)( 47,161)
( 48,162)( 49,147)( 50,148)( 51,145)( 52,146)( 53,155)( 54,156)( 55,153)
( 56,154)( 57,151)( 58,152)( 59,149)( 60,150)( 61,175)( 62,176)( 63,173)
( 64,174)( 65,171)( 66,172)( 67,169)( 68,170)( 69,179)( 70,180)( 71,177)
( 72,178)( 73,195)( 74,196)( 75,193)( 76,194)( 77,203)( 78,204)( 79,201)
( 80,202)( 81,199)( 82,200)( 83,197)( 84,198)( 85,183)( 86,184)( 87,181)
( 88,182)( 89,191)( 90,192)( 91,189)( 92,190)( 93,187)( 94,188)( 95,185)
( 96,186)( 97,211)( 98,212)( 99,209)(100,210)(101,207)(102,208)(103,205)
(104,206)(105,215)(106,216)(107,213)(108,214);
s2 := Sym(216)!(  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 22, 23)( 26, 27)
( 30, 31)( 34, 35)( 37, 73)( 38, 75)( 39, 74)( 40, 76)( 41, 77)( 42, 79)
( 43, 78)( 44, 80)( 45, 81)( 46, 83)( 47, 82)( 48, 84)( 49, 85)( 50, 87)
( 51, 86)( 52, 88)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 93)( 58, 95)
( 59, 94)( 60, 96)( 61, 97)( 62, 99)( 63, 98)( 64,100)( 65,101)( 66,103)
( 67,102)( 68,104)( 69,105)( 70,107)( 71,106)( 72,108)(110,111)(114,115)
(118,119)(122,123)(126,127)(130,131)(134,135)(138,139)(142,143)(145,181)
(146,183)(147,182)(148,184)(149,185)(150,187)(151,186)(152,188)(153,189)
(154,191)(155,190)(156,192)(157,193)(158,195)(159,194)(160,196)(161,197)
(162,199)(163,198)(164,200)(165,201)(166,203)(167,202)(168,204)(169,205)
(170,207)(171,206)(172,208)(173,209)(174,211)(175,210)(176,212)(177,213)
(178,215)(179,214)(180,216);
s3 := Sym(216)!(  1, 73)(  2, 76)(  3, 75)(  4, 74)(  5, 77)(  6, 80)(  7, 79)
(  8, 78)(  9, 81)( 10, 84)( 11, 83)( 12, 82)( 13, 85)( 14, 88)( 15, 87)
( 16, 86)( 17, 89)( 18, 92)( 19, 91)( 20, 90)( 21, 93)( 22, 96)( 23, 95)
( 24, 94)( 25, 97)( 26,100)( 27, 99)( 28, 98)( 29,101)( 30,104)( 31,103)
( 32,102)( 33,105)( 34,108)( 35,107)( 36,106)( 38, 40)( 42, 44)( 46, 48)
( 50, 52)( 54, 56)( 58, 60)( 62, 64)( 66, 68)( 70, 72)(109,181)(110,184)
(111,183)(112,182)(113,185)(114,188)(115,187)(116,186)(117,189)(118,192)
(119,191)(120,190)(121,193)(122,196)(123,195)(124,194)(125,197)(126,200)
(127,199)(128,198)(129,201)(130,204)(131,203)(132,202)(133,205)(134,208)
(135,207)(136,206)(137,209)(138,212)(139,211)(140,210)(141,213)(142,216)
(143,215)(144,214)(146,148)(150,152)(154,156)(158,160)(162,164)(166,168)
(170,172)(174,176)(178,180);
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*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