Polytope of Type {2,6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,18}*1728
if this polytope has a name.
Group : SmallGroup(1728,46114)
Rank : 4
Schlafli Type : {2,6,18}
Number of vertices, edges, etc : 2, 24, 216, 72
Order of s₀s₁s₂s₃ : 36
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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 : {2,6,9}*864
   3-fold quotients : {2,6,6}*576a
   4-fold quotients : {2,6,18}*432b
   6-fold quotients : {2,6,3}*288
   8-fold quotients : {2,6,9}*216
   9-fold quotients : {2,6,6}*192
   12-fold quotients : {2,2,18}*144, {2,6,6}*144b
   18-fold quotients : {2,3,6}*96, {2,6,3}*96
   24-fold quotients : {2,2,9}*72, {2,6,3}*72
   36-fold quotients : {2,3,3}*48, {2,2,6}*48
   72-fold quotients : {2,2,3}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  8,  9)( 12, 13)( 16, 17)( 20, 21)( 24, 25)( 28, 29)( 32, 33)
( 36, 37)( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)( 45, 80)
( 46, 82)( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51, 87)( 52, 89)( 53, 88)
( 54, 90)( 55, 91)( 56, 93)( 57, 92)( 58, 94)( 59, 95)( 60, 97)( 61, 96)
( 62, 98)( 63, 99)( 64,101)( 65,100)( 66,102)( 67,103)( 68,105)( 69,104)
( 70,106)( 71,107)( 72,109)( 73,108)( 74,110)(112,113)(116,117)(120,121)
(124,125)(128,129)(132,133)(136,137)(140,141)(144,145)(147,183)(148,185)
(149,184)(150,186)(151,187)(152,189)(153,188)(154,190)(155,191)(156,193)
(157,192)(158,194)(159,195)(160,197)(161,196)(162,198)(163,199)(164,201)
(165,200)(166,202)(167,203)(168,205)(169,204)(170,206)(171,207)(172,209)
(173,208)(174,210)(175,211)(176,213)(177,212)(178,214)(179,215)(180,217)
(181,216)(182,218);;
s2 := (  3, 39)(  4, 40)(  5, 42)(  6, 41)(  7, 47)(  8, 48)(  9, 50)( 10, 49)
( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 67)( 16, 68)( 17, 70)( 18, 69)
( 19, 63)( 20, 64)( 21, 66)( 22, 65)( 23, 71)( 24, 72)( 25, 74)( 26, 73)
( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 31, 51)( 32, 52)( 33, 54)( 34, 53)
( 35, 59)( 36, 60)( 37, 62)( 38, 61)( 77, 78)( 79, 83)( 80, 84)( 81, 86)
( 82, 85)( 87,103)( 88,104)( 89,106)( 90,105)( 91, 99)( 92,100)( 93,102)
( 94,101)( 95,107)( 96,108)( 97,110)( 98,109)(111,147)(112,148)(113,150)
(114,149)(115,155)(116,156)(117,158)(118,157)(119,151)(120,152)(121,154)
(122,153)(123,175)(124,176)(125,178)(126,177)(127,171)(128,172)(129,174)
(130,173)(131,179)(132,180)(133,182)(134,181)(135,163)(136,164)(137,166)
(138,165)(139,159)(140,160)(141,162)(142,161)(143,167)(144,168)(145,170)
(146,169)(185,186)(187,191)(188,192)(189,194)(190,193)(195,211)(196,212)
(197,214)(198,213)(199,207)(200,208)(201,210)(202,209)(203,215)(204,216)
(205,218)(206,217);;
s3 := (  3,126)(  4,124)(  5,125)(  6,123)(  7,134)(  8,132)(  9,133)( 10,131)
( 11,130)( 12,128)( 13,129)( 14,127)( 15,114)( 16,112)( 17,113)( 18,111)
( 19,122)( 20,120)( 21,121)( 22,119)( 23,118)( 24,116)( 25,117)( 26,115)
( 27,142)( 28,140)( 29,141)( 30,139)( 31,138)( 32,136)( 33,137)( 34,135)
( 35,146)( 36,144)( 37,145)( 38,143)( 39,198)( 40,196)( 41,197)( 42,195)
( 43,206)( 44,204)( 45,205)( 46,203)( 47,202)( 48,200)( 49,201)( 50,199)
( 51,186)( 52,184)( 53,185)( 54,183)( 55,194)( 56,192)( 57,193)( 58,191)
( 59,190)( 60,188)( 61,189)( 62,187)( 63,214)( 64,212)( 65,213)( 66,211)
( 67,210)( 68,208)( 69,209)( 70,207)( 71,218)( 72,216)( 73,217)( 74,215)
( 75,162)( 76,160)( 77,161)( 78,159)( 79,170)( 80,168)( 81,169)( 82,167)
( 83,166)( 84,164)( 85,165)( 86,163)( 87,150)( 88,148)( 89,149)( 90,147)
( 91,158)( 92,156)( 93,157)( 94,155)( 95,154)( 96,152)( 97,153)( 98,151)
( 99,178)(100,176)(101,177)(102,175)(103,174)(104,172)(105,173)(106,171)
(107,182)(108,180)(109,181)(110,179);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*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(218)!(1,2);
s1 := Sym(218)!(  4,  5)(  8,  9)( 12, 13)( 16, 17)( 20, 21)( 24, 25)( 28, 29)
( 32, 33)( 36, 37)( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)
( 45, 80)( 46, 82)( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51, 87)( 52, 89)
( 53, 88)( 54, 90)( 55, 91)( 56, 93)( 57, 92)( 58, 94)( 59, 95)( 60, 97)
( 61, 96)( 62, 98)( 63, 99)( 64,101)( 65,100)( 66,102)( 67,103)( 68,105)
( 69,104)( 70,106)( 71,107)( 72,109)( 73,108)( 74,110)(112,113)(116,117)
(120,121)(124,125)(128,129)(132,133)(136,137)(140,141)(144,145)(147,183)
(148,185)(149,184)(150,186)(151,187)(152,189)(153,188)(154,190)(155,191)
(156,193)(157,192)(158,194)(159,195)(160,197)(161,196)(162,198)(163,199)
(164,201)(165,200)(166,202)(167,203)(168,205)(169,204)(170,206)(171,207)
(172,209)(173,208)(174,210)(175,211)(176,213)(177,212)(178,214)(179,215)
(180,217)(181,216)(182,218);
s2 := Sym(218)!(  3, 39)(  4, 40)(  5, 42)(  6, 41)(  7, 47)(  8, 48)(  9, 50)
( 10, 49)( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 67)( 16, 68)( 17, 70)
( 18, 69)( 19, 63)( 20, 64)( 21, 66)( 22, 65)( 23, 71)( 24, 72)( 25, 74)
( 26, 73)( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 31, 51)( 32, 52)( 33, 54)
( 34, 53)( 35, 59)( 36, 60)( 37, 62)( 38, 61)( 77, 78)( 79, 83)( 80, 84)
( 81, 86)( 82, 85)( 87,103)( 88,104)( 89,106)( 90,105)( 91, 99)( 92,100)
( 93,102)( 94,101)( 95,107)( 96,108)( 97,110)( 98,109)(111,147)(112,148)
(113,150)(114,149)(115,155)(116,156)(117,158)(118,157)(119,151)(120,152)
(121,154)(122,153)(123,175)(124,176)(125,178)(126,177)(127,171)(128,172)
(129,174)(130,173)(131,179)(132,180)(133,182)(134,181)(135,163)(136,164)
(137,166)(138,165)(139,159)(140,160)(141,162)(142,161)(143,167)(144,168)
(145,170)(146,169)(185,186)(187,191)(188,192)(189,194)(190,193)(195,211)
(196,212)(197,214)(198,213)(199,207)(200,208)(201,210)(202,209)(203,215)
(204,216)(205,218)(206,217);
s3 := Sym(218)!(  3,126)(  4,124)(  5,125)(  6,123)(  7,134)(  8,132)(  9,133)
( 10,131)( 11,130)( 12,128)( 13,129)( 14,127)( 15,114)( 16,112)( 17,113)
( 18,111)( 19,122)( 20,120)( 21,121)( 22,119)( 23,118)( 24,116)( 25,117)
( 26,115)( 27,142)( 28,140)( 29,141)( 30,139)( 31,138)( 32,136)( 33,137)
( 34,135)( 35,146)( 36,144)( 37,145)( 38,143)( 39,198)( 40,196)( 41,197)
( 42,195)( 43,206)( 44,204)( 45,205)( 46,203)( 47,202)( 48,200)( 49,201)
( 50,199)( 51,186)( 52,184)( 53,185)( 54,183)( 55,194)( 56,192)( 57,193)
( 58,191)( 59,190)( 60,188)( 61,189)( 62,187)( 63,214)( 64,212)( 65,213)
( 66,211)( 67,210)( 68,208)( 69,209)( 70,207)( 71,218)( 72,216)( 73,217)
( 74,215)( 75,162)( 76,160)( 77,161)( 78,159)( 79,170)( 80,168)( 81,169)
( 82,167)( 83,166)( 84,164)( 85,165)( 86,163)( 87,150)( 88,148)( 89,149)
( 90,147)( 91,158)( 92,156)( 93,157)( 94,155)( 95,154)( 96,152)( 97,153)
( 98,151)( 99,178)(100,176)(101,177)(102,175)(103,174)(104,172)(105,173)
(106,171)(107,182)(108,180)(109,181)(110,179);
poly := sub<Sym(218)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*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 >;

to this polytope