Polytope of Type {2,18,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,12}*1728a
if this polytope has a name.
Group : SmallGroup(1728,46115)
Rank : 4
Schlafli Type : {2,18,12}
Number of vertices, edges, etc : 2, 36, 216, 24
Order of s₀s₁s₂s₃ : 18
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 : {2,18,12}*864c
   3-fold quotients : {2,18,4}*576, {2,6,12}*576a
   4-fold quotients : {2,18,6}*432a
   6-fold quotients : {2,9,4}*288, {2,18,4}*288b, {2,18,4}*288c, {2,6,12}*288d
   9-fold quotients : {2,6,4}*192
   12-fold quotients : {2,9,4}*144, {2,18,2}*144, {2,6,6}*144a
   18-fold quotients : {2,3,4}*96, {2,6,4}*96b, {2,6,4}*96c
   24-fold quotients : {2,9,2}*72
   36-fold quotients : {2,3,4}*48, {2,2,6}*48, {2,6,2}*48
   72-fold quotients : {2,2,3}*24, {2,3,2}*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)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 16, 17)( 19, 23)( 20, 25)
( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)( 39, 79)
( 40, 81)( 41, 80)( 42, 82)( 43, 75)( 44, 77)( 45, 76)( 46, 78)( 47, 83)
( 48, 85)( 49, 84)( 50, 86)( 51, 91)( 52, 93)( 53, 92)( 54, 94)( 55, 87)
( 56, 89)( 57, 88)( 58, 90)( 59, 95)( 60, 97)( 61, 96)( 62, 98)( 63,103)
( 64,105)( 65,104)( 66,106)( 67, 99)( 68,101)( 69,100)( 70,102)( 71,107)
( 72,109)( 73,108)( 74,110)(112,113)(115,119)(116,121)(117,120)(118,122)
(124,125)(127,131)(128,133)(129,132)(130,134)(136,137)(139,143)(140,145)
(141,144)(142,146)(147,187)(148,189)(149,188)(150,190)(151,183)(152,185)
(153,184)(154,186)(155,191)(156,193)(157,192)(158,194)(159,199)(160,201)
(161,200)(162,202)(163,195)(164,197)(165,196)(166,198)(167,203)(168,205)
(169,204)(170,206)(171,211)(172,213)(173,212)(174,214)(175,207)(176,209)
(177,208)(178,210)(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, 63)( 16, 64)( 17, 66)( 18, 65)
( 19, 71)( 20, 72)( 21, 74)( 22, 73)( 23, 67)( 24, 68)( 25, 70)( 26, 69)
( 27, 51)( 28, 52)( 29, 54)( 30, 53)( 31, 59)( 32, 60)( 33, 62)( 34, 61)
( 35, 55)( 36, 56)( 37, 58)( 38, 57)( 75, 79)( 76, 80)( 77, 82)( 78, 81)
( 85, 86)( 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)(123,171)(124,172)(125,174)(126,173)(127,179)(128,180)(129,182)
(130,181)(131,175)(132,176)(133,178)(134,177)(135,159)(136,160)(137,162)
(138,161)(139,167)(140,168)(141,170)(142,169)(143,163)(144,164)(145,166)
(146,165)(183,187)(184,188)(185,190)(186,189)(193,194)(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)(  4,125)(  5,124)(  6,123)(  7,130)(  8,129)(  9,128)( 10,127)
( 11,134)( 12,133)( 13,132)( 14,131)( 15,114)( 16,113)( 17,112)( 18,111)
( 19,118)( 20,117)( 21,116)( 22,115)( 23,122)( 24,121)( 25,120)( 26,119)
( 27,138)( 28,137)( 29,136)( 30,135)( 31,142)( 32,141)( 33,140)( 34,139)
( 35,146)( 36,145)( 37,144)( 38,143)( 39,162)( 40,161)( 41,160)( 42,159)
( 43,166)( 44,165)( 45,164)( 46,163)( 47,170)( 48,169)( 49,168)( 50,167)
( 51,150)( 52,149)( 53,148)( 54,147)( 55,154)( 56,153)( 57,152)( 58,151)
( 59,158)( 60,157)( 61,156)( 62,155)( 63,174)( 64,173)( 65,172)( 66,171)
( 67,178)( 68,177)( 69,176)( 70,175)( 71,182)( 72,181)( 73,180)( 74,179)
( 75,198)( 76,197)( 77,196)( 78,195)( 79,202)( 80,201)( 81,200)( 82,199)
( 83,206)( 84,205)( 85,204)( 86,203)( 87,186)( 88,185)( 89,184)( 90,183)
( 91,190)( 92,189)( 93,188)( 94,187)( 95,194)( 96,193)( 97,192)( 98,191)
( 99,210)(100,209)(101,208)(102,207)(103,214)(104,213)(105,212)(106,211)
(107,218)(108,217)(109,216)(110,215);;
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, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope