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