Polytope of Type {3,4,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4,18}*864
Also Known As : {{3,4},{4,18|2}}. if this polytope has another name.
Group : SmallGroup(864,3998)
Rank : 4
Schlafli Type : {3,4,18}
Number of vertices, edges, etc : 6, 12, 72, 18
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,4,18,2} of size 1728
Vertex Figure Of :
   {2,3,4,18} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,4,6}*288
   4-fold quotients : {3,2,18}*216
   8-fold quotients : {3,2,9}*108
   9-fold quotients : {3,4,2}*96
   12-fold quotients : {3,2,6}*72
   18-fold quotients : {3,4,2}*48
   24-fold quotients : {3,2,3}*36
   36-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,4,36}*1728, {3,8,18}*1728, {6,4,18}*1728a
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope