Polytope of Type {2,18,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,6,4}*1728b
if this polytope has a name.
Group : SmallGroup(1728,30872)
Rank : 5
Schlafli Type : {2,18,6,4}
Number of vertices, edges, etc : 2, 18, 54, 12, 4
Order of s0s1s2s3s4 : 36
Order of s0s1s2s3s4s3s2s1 : 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 : {2,9,6,4}*864, {2,18,6,2}*864b
3-fold quotients : {2,18,2,4}*576, {2,6,6,4}*576c
4-fold quotients : {2,9,6,2}*432
6-fold quotients : {2,9,2,4}*288, {2,18,2,2}*288, {2,3,6,4}*288, {2,6,6,2}*288c
9-fold quotients : {2,6,2,4}*192
12-fold quotients : {2,9,2,2}*144, {2,3,6,2}*144
18-fold quotients : {2,3,2,4}*96, {2,6,2,2}*96
27-fold quotients : {2,2,2,4}*64
36-fold quotients : {2,3,2,2}*48
54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 12, 23)( 13, 22)( 14, 21)( 15, 29)
( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 39, 50)( 40, 49)( 41, 48)( 42, 56)( 43, 55)( 44, 54)( 45, 53)
( 46, 52)( 47, 51)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 66, 77)( 67, 76)
( 68, 75)( 69, 83)( 70, 82)( 71, 81)( 72, 80)( 73, 79)( 74, 78)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 93,104)( 94,103)( 95,102)( 96,110)( 97,109)
( 98,108)( 99,107)(100,106)(101,105)(112,113)(114,117)(115,119)(116,118)
(120,131)(121,130)(122,129)(123,137)(124,136)(125,135)(126,134)(127,133)
(128,132)(139,140)(141,144)(142,146)(143,145)(147,158)(148,157)(149,156)
(150,164)(151,163)(152,162)(153,161)(154,160)(155,159)(166,167)(168,171)
(169,173)(170,172)(174,185)(175,184)(176,183)(177,191)(178,190)(179,189)
(180,188)(181,187)(182,186)(193,194)(195,198)(196,200)(197,199)(201,212)
(202,211)(203,210)(204,218)(205,217)(206,216)(207,215)(208,214)(209,213);;
s2 := ( 3,123)( 4,125)( 5,124)( 6,120)( 7,122)( 8,121)( 9,126)( 10,128)
( 11,127)( 12,114)( 13,116)( 14,115)( 15,111)( 16,113)( 17,112)( 18,117)
( 19,119)( 20,118)( 21,134)( 22,133)( 23,132)( 24,131)( 25,130)( 26,129)
( 27,137)( 28,136)( 29,135)( 30,150)( 31,152)( 32,151)( 33,147)( 34,149)
( 35,148)( 36,153)( 37,155)( 38,154)( 39,141)( 40,143)( 41,142)( 42,138)
( 43,140)( 44,139)( 45,144)( 46,146)( 47,145)( 48,161)( 49,160)( 50,159)
( 51,158)( 52,157)( 53,156)( 54,164)( 55,163)( 56,162)( 57,177)( 58,179)
( 59,178)( 60,174)( 61,176)( 62,175)( 63,180)( 64,182)( 65,181)( 66,168)
( 67,170)( 68,169)( 69,165)( 70,167)( 71,166)( 72,171)( 73,173)( 74,172)
( 75,188)( 76,187)( 77,186)( 78,185)( 79,184)( 80,183)( 81,191)( 82,190)
( 83,189)( 84,204)( 85,206)( 86,205)( 87,201)( 88,203)( 89,202)( 90,207)
( 91,209)( 92,208)( 93,195)( 94,197)( 95,196)( 96,192)( 97,194)( 98,193)
( 99,198)(100,200)(101,199)(102,215)(103,214)(104,213)(105,212)(106,211)
(107,210)(108,218)(109,217)(110,216);;
s3 := ( 6, 9)( 7, 10)( 8, 11)( 15, 18)( 16, 19)( 17, 20)( 24, 27)( 25, 28)
( 26, 29)( 33, 36)( 34, 37)( 35, 38)( 42, 45)( 43, 46)( 44, 47)( 51, 54)
( 52, 55)( 53, 56)( 57, 84)( 58, 85)( 59, 86)( 60, 90)( 61, 91)( 62, 92)
( 63, 87)( 64, 88)( 65, 89)( 66, 93)( 67, 94)( 68, 95)( 69, 99)( 70,100)
( 71,101)( 72, 96)( 73, 97)( 74, 98)( 75,102)( 76,103)( 77,104)( 78,108)
( 79,109)( 80,110)( 81,105)( 82,106)( 83,107)(114,117)(115,118)(116,119)
(123,126)(124,127)(125,128)(132,135)(133,136)(134,137)(141,144)(142,145)
(143,146)(150,153)(151,154)(152,155)(159,162)(160,163)(161,164)(165,192)
(166,193)(167,194)(168,198)(169,199)(170,200)(171,195)(172,196)(173,197)
(174,201)(175,202)(176,203)(177,207)(178,208)(179,209)(180,204)(181,205)
(182,206)(183,210)(184,211)(185,212)(186,216)(187,217)(188,218)(189,213)
(190,214)(191,215);;
s4 := ( 3, 57)( 4, 58)( 5, 59)( 6, 60)( 7, 61)( 8, 62)( 9, 63)( 10, 64)
( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)( 18, 72)
( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)
( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)( 34, 88)
( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)( 42, 96)
( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)( 50,104)
( 51,105)( 52,106)( 53,107)( 54,108)( 55,109)( 56,110)(111,165)(112,166)
(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)
(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)
(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)
(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)
(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)
(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)
(161,215)(162,216)(163,217)(164,218);;
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,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*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*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 12, 23)( 13, 22)( 14, 21)
( 15, 29)( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 31, 32)( 33, 36)
( 34, 38)( 35, 37)( 39, 50)( 40, 49)( 41, 48)( 42, 56)( 43, 55)( 44, 54)
( 45, 53)( 46, 52)( 47, 51)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 66, 77)
( 67, 76)( 68, 75)( 69, 83)( 70, 82)( 71, 81)( 72, 80)( 73, 79)( 74, 78)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 93,104)( 94,103)( 95,102)( 96,110)
( 97,109)( 98,108)( 99,107)(100,106)(101,105)(112,113)(114,117)(115,119)
(116,118)(120,131)(121,130)(122,129)(123,137)(124,136)(125,135)(126,134)
(127,133)(128,132)(139,140)(141,144)(142,146)(143,145)(147,158)(148,157)
(149,156)(150,164)(151,163)(152,162)(153,161)(154,160)(155,159)(166,167)
(168,171)(169,173)(170,172)(174,185)(175,184)(176,183)(177,191)(178,190)
(179,189)(180,188)(181,187)(182,186)(193,194)(195,198)(196,200)(197,199)
(201,212)(202,211)(203,210)(204,218)(205,217)(206,216)(207,215)(208,214)
(209,213);
s2 := Sym(218)!( 3,123)( 4,125)( 5,124)( 6,120)( 7,122)( 8,121)( 9,126)
( 10,128)( 11,127)( 12,114)( 13,116)( 14,115)( 15,111)( 16,113)( 17,112)
( 18,117)( 19,119)( 20,118)( 21,134)( 22,133)( 23,132)( 24,131)( 25,130)
( 26,129)( 27,137)( 28,136)( 29,135)( 30,150)( 31,152)( 32,151)( 33,147)
( 34,149)( 35,148)( 36,153)( 37,155)( 38,154)( 39,141)( 40,143)( 41,142)
( 42,138)( 43,140)( 44,139)( 45,144)( 46,146)( 47,145)( 48,161)( 49,160)
( 50,159)( 51,158)( 52,157)( 53,156)( 54,164)( 55,163)( 56,162)( 57,177)
( 58,179)( 59,178)( 60,174)( 61,176)( 62,175)( 63,180)( 64,182)( 65,181)
( 66,168)( 67,170)( 68,169)( 69,165)( 70,167)( 71,166)( 72,171)( 73,173)
( 74,172)( 75,188)( 76,187)( 77,186)( 78,185)( 79,184)( 80,183)( 81,191)
( 82,190)( 83,189)( 84,204)( 85,206)( 86,205)( 87,201)( 88,203)( 89,202)
( 90,207)( 91,209)( 92,208)( 93,195)( 94,197)( 95,196)( 96,192)( 97,194)
( 98,193)( 99,198)(100,200)(101,199)(102,215)(103,214)(104,213)(105,212)
(106,211)(107,210)(108,218)(109,217)(110,216);
s3 := Sym(218)!( 6, 9)( 7, 10)( 8, 11)( 15, 18)( 16, 19)( 17, 20)( 24, 27)
( 25, 28)( 26, 29)( 33, 36)( 34, 37)( 35, 38)( 42, 45)( 43, 46)( 44, 47)
( 51, 54)( 52, 55)( 53, 56)( 57, 84)( 58, 85)( 59, 86)( 60, 90)( 61, 91)
( 62, 92)( 63, 87)( 64, 88)( 65, 89)( 66, 93)( 67, 94)( 68, 95)( 69, 99)
( 70,100)( 71,101)( 72, 96)( 73, 97)( 74, 98)( 75,102)( 76,103)( 77,104)
( 78,108)( 79,109)( 80,110)( 81,105)( 82,106)( 83,107)(114,117)(115,118)
(116,119)(123,126)(124,127)(125,128)(132,135)(133,136)(134,137)(141,144)
(142,145)(143,146)(150,153)(151,154)(152,155)(159,162)(160,163)(161,164)
(165,192)(166,193)(167,194)(168,198)(169,199)(170,200)(171,195)(172,196)
(173,197)(174,201)(175,202)(176,203)(177,207)(178,208)(179,209)(180,204)
(181,205)(182,206)(183,210)(184,211)(185,212)(186,216)(187,217)(188,218)
(189,213)(190,214)(191,215);
s4 := Sym(218)!( 3, 57)( 4, 58)( 5, 59)( 6, 60)( 7, 61)( 8, 62)( 9, 63)
( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)
( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)
( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)
( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)
( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)
( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)( 55,109)( 56,110)(111,165)
(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)
(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)
(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)
(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)
(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)
(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)
(160,214)(161,215)(162,216)(163,217)(164,218);
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, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s1*s2*s3*s2*s1*s2*s1*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*s1*s2 >;
to this polytope