Questions?
See the FAQ
or other info.

Polytope of Type {18,6,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6,8}*1728a
Also Known As : {{18,6|2},{6,8|2}}. if this polytope has another name.
Group : SmallGroup(1728,15957)
Rank : 4
Schlafli Type : {18,6,8}
Number of vertices, edges, etc : 18, 54, 24, 8
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 : {18,6,4}*864a
   3-fold quotients : {18,2,8}*576, {6,6,8}*576a
   4-fold quotients : {18,6,2}*432a
   6-fold quotients : {9,2,8}*288, {18,2,4}*288, {6,6,4}*288a
   9-fold quotients : {2,6,8}*192, {6,2,8}*192
   12-fold quotients : {9,2,4}*144, {18,2,2}*144, {6,6,2}*144a
   18-fold quotients : {3,2,8}*96, {2,6,4}*96a, {6,2,4}*96
   24-fold quotients : {9,2,2}*72
   27-fold quotients : {2,2,8}*64
   36-fold quotients : {3,2,4}*48, {2,6,2}*48, {6,2,2}*48
   54-fold quotients : {2,2,4}*32
   72-fold quotients : {2,3,2}*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, 16)(  5, 18)(  6, 17)(  7, 13)(  8, 15)
(  9, 14)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 37)( 29, 39)( 30, 38)
( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 48)( 49, 54)
( 50, 53)( 51, 52)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)
( 61, 67)( 62, 69)( 63, 68)( 73, 75)( 76, 81)( 77, 80)( 78, 79)( 82, 91)
( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)( 90, 95)
(100,102)(103,108)(104,107)(105,106)(109,118)(110,120)(111,119)(112,124)
(113,126)(114,125)(115,121)(116,123)(117,122)(127,129)(130,135)(131,134)
(132,133)(136,145)(137,147)(138,146)(139,151)(140,153)(141,152)(142,148)
(143,150)(144,149)(154,156)(157,162)(158,161)(159,160)(163,172)(164,174)
(165,173)(166,178)(167,180)(168,179)(169,175)(170,177)(171,176)(181,183)
(184,189)(185,188)(186,187)(190,199)(191,201)(192,200)(193,205)(194,207)
(195,206)(196,202)(197,204)(198,203)(208,210)(211,216)(212,215)(213,214);;
s2 := (  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, 85)( 56, 86)( 57, 87)( 58, 82)( 59, 83)( 60, 84)
( 61, 88)( 62, 89)( 63, 90)( 64, 94)( 65, 95)( 66, 96)( 67, 91)( 68, 92)
( 69, 93)( 70, 97)( 71, 98)( 72, 99)( 73,103)( 74,104)( 75,105)( 76,100)
( 77,101)( 78,102)( 79,106)( 80,107)( 81,108)(109,166)(110,167)(111,168)
(112,163)(113,164)(114,165)(115,169)(116,170)(117,171)(118,175)(119,176)
(120,177)(121,172)(122,173)(123,174)(124,178)(125,179)(126,180)(127,184)
(128,185)(129,186)(130,181)(131,182)(132,183)(133,187)(134,188)(135,189)
(136,193)(137,194)(138,195)(139,190)(140,191)(141,192)(142,196)(143,197)
(144,198)(145,202)(146,203)(147,204)(148,199)(149,200)(150,201)(151,205)
(152,206)(153,207)(154,211)(155,212)(156,213)(157,208)(158,209)(159,210)
(160,214)(161,215)(162,216);;
s3 := (  1,109)(  2,110)(  3,111)(  4,112)(  5,113)(  6,114)(  7,115)(  8,116)
(  9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)
( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)
( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)
( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)
( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)( 56,191)
( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)
( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)
( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)
( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)
( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)( 96,177)
( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)(104,185)
(105,186)(106,187)(107,188)(108,189);;
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, 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, 
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, 16)(  5, 18)(  6, 17)(  7, 13)
(  8, 15)(  9, 14)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 37)( 29, 39)
( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 48)
( 49, 54)( 50, 53)( 51, 52)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)
( 60, 71)( 61, 67)( 62, 69)( 63, 68)( 73, 75)( 76, 81)( 77, 80)( 78, 79)
( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)
( 90, 95)(100,102)(103,108)(104,107)(105,106)(109,118)(110,120)(111,119)
(112,124)(113,126)(114,125)(115,121)(116,123)(117,122)(127,129)(130,135)
(131,134)(132,133)(136,145)(137,147)(138,146)(139,151)(140,153)(141,152)
(142,148)(143,150)(144,149)(154,156)(157,162)(158,161)(159,160)(163,172)
(164,174)(165,173)(166,178)(167,180)(168,179)(169,175)(170,177)(171,176)
(181,183)(184,189)(185,188)(186,187)(190,199)(191,201)(192,200)(193,205)
(194,207)(195,206)(196,202)(197,204)(198,203)(208,210)(211,216)(212,215)
(213,214);
s2 := 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, 85)( 56, 86)( 57, 87)( 58, 82)( 59, 83)
( 60, 84)( 61, 88)( 62, 89)( 63, 90)( 64, 94)( 65, 95)( 66, 96)( 67, 91)
( 68, 92)( 69, 93)( 70, 97)( 71, 98)( 72, 99)( 73,103)( 74,104)( 75,105)
( 76,100)( 77,101)( 78,102)( 79,106)( 80,107)( 81,108)(109,166)(110,167)
(111,168)(112,163)(113,164)(114,165)(115,169)(116,170)(117,171)(118,175)
(119,176)(120,177)(121,172)(122,173)(123,174)(124,178)(125,179)(126,180)
(127,184)(128,185)(129,186)(130,181)(131,182)(132,183)(133,187)(134,188)
(135,189)(136,193)(137,194)(138,195)(139,190)(140,191)(141,192)(142,196)
(143,197)(144,198)(145,202)(146,203)(147,204)(148,199)(149,200)(150,201)
(151,205)(152,206)(153,207)(154,211)(155,212)(156,213)(157,208)(158,209)
(159,210)(160,214)(161,215)(162,216);
s3 := Sym(216)!(  1,109)(  2,110)(  3,111)(  4,112)(  5,113)(  6,114)(  7,115)
(  8,116)(  9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)
( 16,124)( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)
( 24,132)( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)
( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)
( 40,148)( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)
( 48,156)( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)
( 56,191)( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)
( 64,199)( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)
( 72,207)( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)
( 80,215)( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)
( 88,169)( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)
( 96,177)( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)
(104,185)(105,186)(106,187)(107,188)(108,189);
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, 
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, 
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