Polytope of Type {4,6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,18}*864b
if this polytope has a name.
Group : SmallGroup(864,2501)
Rank : 4
Schlafli Type : {4,6,18}
Number of vertices, edges, etc : 4, 12, 54, 18
Order of s₀s₁s₂s₃ : 36
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,6,18,2} of size 1728
Vertex Figure Of :
   {2,4,6,18} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6,9}*432, {2,6,18}*432b
   3-fold quotients : {4,2,18}*288, {4,6,6}*288c
   4-fold quotients : {2,6,9}*216
   6-fold quotients : {4,2,9}*144, {2,2,18}*144, {4,6,3}*144, {2,6,6}*144b
   9-fold quotients : {4,2,6}*96
   12-fold quotients : {2,2,9}*72, {2,6,3}*72
   18-fold quotients : {4,2,3}*48, {2,2,6}*48
   27-fold quotients : {4,2,2}*32
   36-fold quotients : {2,2,3}*24
   54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,6,36}*1728b, {8,6,18}*1728b, {4,12,18}*1728b
Permutation Representation (GAP) :

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

References : None.
to this polytope