Polytope of Type {6,8,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,8,18}*1728
Also Known As : {{6,8|2},{8,18|2}}. if this polytope has another name.
Group : SmallGroup(1728,15957)
Rank : 4
Schlafli Type : {6,8,18}
Number of vertices, edges, etc : 6, 24, 72, 18
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 : {6,4,18}*864
   3-fold quotients : {2,8,18}*576, {6,8,6}*576
   4-fold quotients : {6,2,18}*432
   6-fold quotients : {2,4,18}*288a, {6,4,6}*288
   8-fold quotients : {3,2,18}*216, {6,2,9}*216
   9-fold quotients : {2,8,6}*192, {6,8,2}*192
   12-fold quotients : {2,2,18}*144, {6,2,6}*144
   16-fold quotients : {3,2,9}*108
   18-fold quotients : {2,4,6}*96a, {6,4,2}*96a
   24-fold quotients : {2,2,9}*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 := (  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)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)
( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)
( 96, 99)(103,106)(104,107)(105,108)(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)(166,169)(167,170)
(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)(193,196)
(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)(213,216);;
s1 := (  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);;
s2 := (  1,109)(  2,111)(  3,110)(  4,112)(  5,114)(  6,113)(  7,115)(  8,117)
(  9,116)( 10,129)( 11,128)( 12,127)( 13,132)( 14,131)( 15,130)( 16,135)
( 17,134)( 18,133)( 19,120)( 20,119)( 21,118)( 22,123)( 23,122)( 24,121)
( 25,126)( 26,125)( 27,124)( 28,136)( 29,138)( 30,137)( 31,139)( 32,141)
( 33,140)( 34,142)( 35,144)( 36,143)( 37,156)( 38,155)( 39,154)( 40,159)
( 41,158)( 42,157)( 43,162)( 44,161)( 45,160)( 46,147)( 47,146)( 48,145)
( 49,150)( 50,149)( 51,148)( 52,153)( 53,152)( 54,151)( 55,190)( 56,192)
( 57,191)( 58,193)( 59,195)( 60,194)( 61,196)( 62,198)( 63,197)( 64,210)
( 65,209)( 66,208)( 67,213)( 68,212)( 69,211)( 70,216)( 71,215)( 72,214)
( 73,201)( 74,200)( 75,199)( 76,204)( 77,203)( 78,202)( 79,207)( 80,206)
( 81,205)( 82,163)( 83,165)( 84,164)( 85,166)( 86,168)( 87,167)( 88,169)
( 89,171)( 90,170)( 91,183)( 92,182)( 93,181)( 94,186)( 95,185)( 96,184)
( 97,189)( 98,188)( 99,187)(100,174)(101,173)(102,172)(103,177)(104,176)
(105,175)(106,180)(107,179)(108,178);;
s3 := (  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, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)( 62, 72)
( 63, 71)( 73, 75)( 76, 78)( 79, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 94)
( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(100,102)(103,105)(106,108)
(109,118)(110,120)(111,119)(112,121)(113,123)(114,122)(115,124)(116,126)
(117,125)(127,129)(130,132)(133,135)(136,145)(137,147)(138,146)(139,148)
(140,150)(141,149)(142,151)(143,153)(144,152)(154,156)(157,159)(160,162)
(163,172)(164,174)(165,173)(166,175)(167,177)(168,176)(169,178)(170,180)
(171,179)(181,183)(184,186)(187,189)(190,199)(191,201)(192,200)(193,202)
(194,204)(195,203)(196,205)(197,207)(198,206)(208,210)(211,213)(214,216);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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)!(  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)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)
( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(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)(166,169)
(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)
(193,196)(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)
(213,216);
s1 := 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);
s2 := Sym(216)!(  1,109)(  2,111)(  3,110)(  4,112)(  5,114)(  6,113)(  7,115)
(  8,117)(  9,116)( 10,129)( 11,128)( 12,127)( 13,132)( 14,131)( 15,130)
( 16,135)( 17,134)( 18,133)( 19,120)( 20,119)( 21,118)( 22,123)( 23,122)
( 24,121)( 25,126)( 26,125)( 27,124)( 28,136)( 29,138)( 30,137)( 31,139)
( 32,141)( 33,140)( 34,142)( 35,144)( 36,143)( 37,156)( 38,155)( 39,154)
( 40,159)( 41,158)( 42,157)( 43,162)( 44,161)( 45,160)( 46,147)( 47,146)
( 48,145)( 49,150)( 50,149)( 51,148)( 52,153)( 53,152)( 54,151)( 55,190)
( 56,192)( 57,191)( 58,193)( 59,195)( 60,194)( 61,196)( 62,198)( 63,197)
( 64,210)( 65,209)( 66,208)( 67,213)( 68,212)( 69,211)( 70,216)( 71,215)
( 72,214)( 73,201)( 74,200)( 75,199)( 76,204)( 77,203)( 78,202)( 79,207)
( 80,206)( 81,205)( 82,163)( 83,165)( 84,164)( 85,166)( 86,168)( 87,167)
( 88,169)( 89,171)( 90,170)( 91,183)( 92,182)( 93,181)( 94,186)( 95,185)
( 96,184)( 97,189)( 98,188)( 99,187)(100,174)(101,173)(102,172)(103,177)
(104,176)(105,175)(106,180)(107,179)(108,178);
s3 := 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, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)
( 62, 72)( 63, 71)( 73, 75)( 76, 78)( 79, 81)( 82, 91)( 83, 93)( 84, 92)
( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(100,102)(103,105)
(106,108)(109,118)(110,120)(111,119)(112,121)(113,123)(114,122)(115,124)
(116,126)(117,125)(127,129)(130,132)(133,135)(136,145)(137,147)(138,146)
(139,148)(140,150)(141,149)(142,151)(143,153)(144,152)(154,156)(157,159)
(160,162)(163,172)(164,174)(165,173)(166,175)(167,177)(168,176)(169,178)
(170,180)(171,179)(181,183)(184,186)(187,189)(190,199)(191,201)(192,200)
(193,202)(194,204)(195,203)(196,205)(197,207)(198,206)(208,210)(211,213)
(214,216);
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*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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