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