Polytope of Type {2,4,18,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,18,6}*1728d
if this polytope has a name.
Group : SmallGroup(1728,46115)
Rank : 5
Schlafli Type : {2,4,18,6}
Number of vertices, edges, etc : 2, 4, 36, 54, 6
Order of s0s1s2s3s4 : 18
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-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 : {2,4,9,6}*864
3-fold quotients : {2,4,18,2}*576b, {2,4,6,6}*576e
6-fold quotients : {2,4,9,2}*288, {2,4,3,6}*288
9-fold quotients : {2,4,6,2}*192c
18-fold quotients : {2,4,3,2}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3, 5)( 4, 6)( 7, 9)( 8, 10)( 11, 13)( 12, 14)( 15, 17)( 16, 18)
( 19, 21)( 20, 22)( 23, 25)( 24, 26)( 27, 29)( 28, 30)( 31, 33)( 32, 34)
( 35, 37)( 36, 38)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)( 48, 50)
( 51, 53)( 52, 54)( 55, 57)( 56, 58)( 59, 61)( 60, 62)( 63, 65)( 64, 66)
( 67, 69)( 68, 70)( 71, 73)( 72, 74)( 75, 77)( 76, 78)( 79, 81)( 80, 82)
( 83, 85)( 84, 86)( 87, 89)( 88, 90)( 91, 93)( 92, 94)( 95, 97)( 96, 98)
( 99,101)(100,102)(103,105)(104,106)(107,109)(108,110)(111,113)(112,114)
(115,117)(116,118)(119,121)(120,122)(123,125)(124,126)(127,129)(128,130)
(131,133)(132,134)(135,137)(136,138)(139,141)(140,142)(143,145)(144,146)
(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,161)(160,162)
(163,165)(164,166)(167,169)(168,170)(171,173)(172,174)(175,177)(176,178)
(179,181)(180,182)(183,185)(184,186)(187,189)(188,190)(191,193)(192,194)
(195,197)(196,198)(199,201)(200,202)(203,205)(204,206)(207,209)(208,210)
(211,213)(212,214)(215,217)(216,218);;
s2 := ( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 15, 27)( 16, 29)( 17, 28)
( 18, 30)( 19, 35)( 20, 37)( 21, 36)( 22, 38)( 23, 31)( 24, 33)( 25, 32)
( 26, 34)( 39, 79)( 40, 81)( 41, 80)( 42, 82)( 43, 75)( 44, 77)( 45, 76)
( 46, 78)( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51,103)( 52,105)( 53,104)
( 54,106)( 55, 99)( 56,101)( 57,100)( 58,102)( 59,107)( 60,109)( 61,108)
( 62,110)( 63, 91)( 64, 93)( 65, 92)( 66, 94)( 67, 87)( 68, 89)( 69, 88)
( 70, 90)( 71, 95)( 72, 97)( 73, 96)( 74, 98)(112,113)(115,119)(116,121)
(117,120)(118,122)(123,135)(124,137)(125,136)(126,138)(127,143)(128,145)
(129,144)(130,146)(131,139)(132,141)(133,140)(134,142)(147,187)(148,189)
(149,188)(150,190)(151,183)(152,185)(153,184)(154,186)(155,191)(156,193)
(157,192)(158,194)(159,211)(160,213)(161,212)(162,214)(163,207)(164,209)
(165,208)(166,210)(167,215)(168,217)(169,216)(170,218)(171,199)(172,201)
(173,200)(174,202)(175,195)(176,197)(177,196)(178,198)(179,203)(180,205)
(181,204)(182,206);;
s3 := ( 3,195)( 4,198)( 5,197)( 6,196)( 7,203)( 8,206)( 9,205)( 10,204)
( 11,199)( 12,202)( 13,201)( 14,200)( 15,183)( 16,186)( 17,185)( 18,184)
( 19,191)( 20,194)( 21,193)( 22,192)( 23,187)( 24,190)( 25,189)( 26,188)
( 27,207)( 28,210)( 29,209)( 30,208)( 31,215)( 32,218)( 33,217)( 34,216)
( 35,211)( 36,214)( 37,213)( 38,212)( 39,159)( 40,162)( 41,161)( 42,160)
( 43,167)( 44,170)( 45,169)( 46,168)( 47,163)( 48,166)( 49,165)( 50,164)
( 51,147)( 52,150)( 53,149)( 54,148)( 55,155)( 56,158)( 57,157)( 58,156)
( 59,151)( 60,154)( 61,153)( 62,152)( 63,171)( 64,174)( 65,173)( 66,172)
( 67,179)( 68,182)( 69,181)( 70,180)( 71,175)( 72,178)( 73,177)( 74,176)
( 75,123)( 76,126)( 77,125)( 78,124)( 79,131)( 80,134)( 81,133)( 82,132)
( 83,127)( 84,130)( 85,129)( 86,128)( 87,111)( 88,114)( 89,113)( 90,112)
( 91,119)( 92,122)( 93,121)( 94,120)( 95,115)( 96,118)( 97,117)( 98,116)
( 99,135)(100,138)(101,137)(102,136)(103,143)(104,146)(105,145)(106,144)
(107,139)(108,142)(109,141)(110,140);;
s4 := ( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 33)( 22, 34)
( 23, 35)( 24, 36)( 25, 37)( 26, 38)( 51, 63)( 52, 64)( 53, 65)( 54, 66)
( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 61, 73)( 62, 74)
( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)( 93,105)( 94,106)
( 95,107)( 96,108)( 97,109)( 98,110)(123,135)(124,136)(125,137)(126,138)
(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)(133,145)(134,146)
(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)(165,177)(166,178)
(167,179)(168,180)(169,181)(170,182)(195,207)(196,208)(197,209)(198,210)
(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)(205,217)(206,218);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s1*s2,
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s2*s3*s4*s3*s2*s3*s2*s3*s4*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*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 3, 5)( 4, 6)( 7, 9)( 8, 10)( 11, 13)( 12, 14)( 15, 17)
( 16, 18)( 19, 21)( 20, 22)( 23, 25)( 24, 26)( 27, 29)( 28, 30)( 31, 33)
( 32, 34)( 35, 37)( 36, 38)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)
( 48, 50)( 51, 53)( 52, 54)( 55, 57)( 56, 58)( 59, 61)( 60, 62)( 63, 65)
( 64, 66)( 67, 69)( 68, 70)( 71, 73)( 72, 74)( 75, 77)( 76, 78)( 79, 81)
( 80, 82)( 83, 85)( 84, 86)( 87, 89)( 88, 90)( 91, 93)( 92, 94)( 95, 97)
( 96, 98)( 99,101)(100,102)(103,105)(104,106)(107,109)(108,110)(111,113)
(112,114)(115,117)(116,118)(119,121)(120,122)(123,125)(124,126)(127,129)
(128,130)(131,133)(132,134)(135,137)(136,138)(139,141)(140,142)(143,145)
(144,146)(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,161)
(160,162)(163,165)(164,166)(167,169)(168,170)(171,173)(172,174)(175,177)
(176,178)(179,181)(180,182)(183,185)(184,186)(187,189)(188,190)(191,193)
(192,194)(195,197)(196,198)(199,201)(200,202)(203,205)(204,206)(207,209)
(208,210)(211,213)(212,214)(215,217)(216,218);
s2 := Sym(218)!( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 15, 27)( 16, 29)
( 17, 28)( 18, 30)( 19, 35)( 20, 37)( 21, 36)( 22, 38)( 23, 31)( 24, 33)
( 25, 32)( 26, 34)( 39, 79)( 40, 81)( 41, 80)( 42, 82)( 43, 75)( 44, 77)
( 45, 76)( 46, 78)( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51,103)( 52,105)
( 53,104)( 54,106)( 55, 99)( 56,101)( 57,100)( 58,102)( 59,107)( 60,109)
( 61,108)( 62,110)( 63, 91)( 64, 93)( 65, 92)( 66, 94)( 67, 87)( 68, 89)
( 69, 88)( 70, 90)( 71, 95)( 72, 97)( 73, 96)( 74, 98)(112,113)(115,119)
(116,121)(117,120)(118,122)(123,135)(124,137)(125,136)(126,138)(127,143)
(128,145)(129,144)(130,146)(131,139)(132,141)(133,140)(134,142)(147,187)
(148,189)(149,188)(150,190)(151,183)(152,185)(153,184)(154,186)(155,191)
(156,193)(157,192)(158,194)(159,211)(160,213)(161,212)(162,214)(163,207)
(164,209)(165,208)(166,210)(167,215)(168,217)(169,216)(170,218)(171,199)
(172,201)(173,200)(174,202)(175,195)(176,197)(177,196)(178,198)(179,203)
(180,205)(181,204)(182,206);
s3 := Sym(218)!( 3,195)( 4,198)( 5,197)( 6,196)( 7,203)( 8,206)( 9,205)
( 10,204)( 11,199)( 12,202)( 13,201)( 14,200)( 15,183)( 16,186)( 17,185)
( 18,184)( 19,191)( 20,194)( 21,193)( 22,192)( 23,187)( 24,190)( 25,189)
( 26,188)( 27,207)( 28,210)( 29,209)( 30,208)( 31,215)( 32,218)( 33,217)
( 34,216)( 35,211)( 36,214)( 37,213)( 38,212)( 39,159)( 40,162)( 41,161)
( 42,160)( 43,167)( 44,170)( 45,169)( 46,168)( 47,163)( 48,166)( 49,165)
( 50,164)( 51,147)( 52,150)( 53,149)( 54,148)( 55,155)( 56,158)( 57,157)
( 58,156)( 59,151)( 60,154)( 61,153)( 62,152)( 63,171)( 64,174)( 65,173)
( 66,172)( 67,179)( 68,182)( 69,181)( 70,180)( 71,175)( 72,178)( 73,177)
( 74,176)( 75,123)( 76,126)( 77,125)( 78,124)( 79,131)( 80,134)( 81,133)
( 82,132)( 83,127)( 84,130)( 85,129)( 86,128)( 87,111)( 88,114)( 89,113)
( 90,112)( 91,119)( 92,122)( 93,121)( 94,120)( 95,115)( 96,118)( 97,117)
( 98,116)( 99,135)(100,138)(101,137)(102,136)(103,143)(104,146)(105,145)
(106,144)(107,139)(108,142)(109,141)(110,140);
s4 := Sym(218)!( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 33)
( 22, 34)( 23, 35)( 24, 36)( 25, 37)( 26, 38)( 51, 63)( 52, 64)( 53, 65)
( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 61, 73)
( 62, 74)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)( 93,105)
( 94,106)( 95,107)( 96,108)( 97,109)( 98,110)(123,135)(124,136)(125,137)
(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)(133,145)
(134,146)(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)(165,177)
(166,178)(167,179)(168,180)(169,181)(170,182)(195,207)(196,208)(197,209)
(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)(205,217)
(206,218);
poly := sub<Sym(218)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s1*s2, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3,
s2*s3*s4*s3*s2*s3*s2*s3*s4*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*s2*s3*s2*s3 >;
to this polytope