Polytope of Type {18,8,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,8,6}*1728
Also Known As : {{18,8|2},{8,6|2}}. if this polytope has another name.
Group : SmallGroup(1728,15957)
Rank : 4
Schlafli Type : {18,8,6}
Number of vertices, edges, etc : 18, 72, 24, 6
Order of s₀s₁s₂s₃ : 72
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   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 : {18,4,6}*864
   3-fold quotients : {18,8,2}*576, {6,8,6}*576
   4-fold quotients : {18,2,6}*432
   6-fold quotients : {18,4,2}*288a, {6,4,6}*288
   8-fold quotients : {9,2,6}*216, {18,2,3}*216
   9-fold quotients : {2,8,6}*192, {6,8,2}*192
   12-fold quotients : {18,2,2}*144, {6,2,6}*144
   16-fold quotients : {9,2,3}*108
   18-fold quotients : {2,4,6}*96a, {6,4,2}*96a
   24-fold quotients : {9,2,2}*72, {3,2,6}*72, {6,2,3}*72
   27-fold quotients : {2,8,2}*64
   36-fold quotients : {2,2,6}*48, {6,2,2}*48
   48-fold quotients : {3,2,3}*36
   54-fold quotients : {2,4,2}*32
   72-fold quotients : {2,2,3}*24, {3,2,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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