Polytope of Type {2,4,6,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,6,18}*1728b
if this polytope has a name.
Group : SmallGroup(1728,30872)
Rank : 5
Schlafli Type : {2,4,6,18}
Number of vertices, edges, etc : 2, 4, 12, 54, 18
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 : {2,4,6,9}*864, {2,2,6,18}*864b
3-fold quotients : {2,4,2,18}*576, {2,4,6,6}*576c
4-fold quotients : {2,2,6,9}*432
6-fold quotients : {2,4,2,9}*288, {2,2,2,18}*288, {2,4,6,3}*288, {2,2,6,6}*288b
9-fold quotients : {2,4,2,6}*192
12-fold quotients : {2,2,2,9}*144, {2,2,6,3}*144
18-fold quotients : {2,4,2,3}*96, {2,2,2,6}*96
27-fold quotients : {2,4,2,2}*64
36-fold quotients : {2,2,2,3}*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 := ( 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);;
s2 := ( 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);;
s3 := ( 3, 6)( 4, 8)( 5, 7)( 10, 11)( 12, 26)( 13, 25)( 14, 24)( 15, 23)
( 16, 22)( 17, 21)( 18, 29)( 19, 28)( 20, 27)( 30, 33)( 31, 35)( 32, 34)
( 37, 38)( 39, 53)( 40, 52)( 41, 51)( 42, 50)( 43, 49)( 44, 48)( 45, 56)
( 46, 55)( 47, 54)( 57, 60)( 58, 62)( 59, 61)( 64, 65)( 66, 80)( 67, 79)
( 68, 78)( 69, 77)( 70, 76)( 71, 75)( 72, 83)( 73, 82)( 74, 81)( 84, 87)
( 85, 89)( 86, 88)( 91, 92)( 93,107)( 94,106)( 95,105)( 96,104)( 97,103)
( 98,102)( 99,110)(100,109)(101,108)(111,114)(112,116)(113,115)(118,119)
(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,137)(127,136)
(128,135)(138,141)(139,143)(140,142)(145,146)(147,161)(148,160)(149,159)
(150,158)(151,157)(152,156)(153,164)(154,163)(155,162)(165,168)(166,170)
(167,169)(172,173)(174,188)(175,187)(176,186)(177,185)(178,184)(179,183)
(180,191)(181,190)(182,189)(192,195)(193,197)(194,196)(199,200)(201,215)
(202,214)(203,213)(204,212)(205,211)(206,210)(207,218)(208,217)(209,216);;
s4 := ( 3,120)( 4,122)( 5,121)( 6,126)( 7,128)( 8,127)( 9,123)( 10,125)
( 11,124)( 12,111)( 13,113)( 14,112)( 15,117)( 16,119)( 17,118)( 18,114)
( 19,116)( 20,115)( 21,131)( 22,130)( 23,129)( 24,137)( 25,136)( 26,135)
( 27,134)( 28,133)( 29,132)( 30,147)( 31,149)( 32,148)( 33,153)( 34,155)
( 35,154)( 36,150)( 37,152)( 38,151)( 39,138)( 40,140)( 41,139)( 42,144)
( 43,146)( 44,145)( 45,141)( 46,143)( 47,142)( 48,158)( 49,157)( 50,156)
( 51,164)( 52,163)( 53,162)( 54,161)( 55,160)( 56,159)( 57,174)( 58,176)
( 59,175)( 60,180)( 61,182)( 62,181)( 63,177)( 64,179)( 65,178)( 66,165)
( 67,167)( 68,166)( 69,171)( 70,173)( 71,172)( 72,168)( 73,170)( 74,169)
( 75,185)( 76,184)( 77,183)( 78,191)( 79,190)( 80,189)( 81,188)( 82,187)
( 83,186)( 84,201)( 85,203)( 86,202)( 87,207)( 88,209)( 89,208)( 90,204)
( 91,206)( 92,205)( 93,192)( 94,194)( 95,193)( 96,198)( 97,200)( 98,199)
( 99,195)(100,197)(101,196)(102,212)(103,211)(104,210)(105,218)(106,217)
(107,216)(108,215)(109,214)(110,213);;
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*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := 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);
s2 := 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);
s3 := Sym(218)!( 3, 6)( 4, 8)( 5, 7)( 10, 11)( 12, 26)( 13, 25)( 14, 24)
( 15, 23)( 16, 22)( 17, 21)( 18, 29)( 19, 28)( 20, 27)( 30, 33)( 31, 35)
( 32, 34)( 37, 38)( 39, 53)( 40, 52)( 41, 51)( 42, 50)( 43, 49)( 44, 48)
( 45, 56)( 46, 55)( 47, 54)( 57, 60)( 58, 62)( 59, 61)( 64, 65)( 66, 80)
( 67, 79)( 68, 78)( 69, 77)( 70, 76)( 71, 75)( 72, 83)( 73, 82)( 74, 81)
( 84, 87)( 85, 89)( 86, 88)( 91, 92)( 93,107)( 94,106)( 95,105)( 96,104)
( 97,103)( 98,102)( 99,110)(100,109)(101,108)(111,114)(112,116)(113,115)
(118,119)(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,137)
(127,136)(128,135)(138,141)(139,143)(140,142)(145,146)(147,161)(148,160)
(149,159)(150,158)(151,157)(152,156)(153,164)(154,163)(155,162)(165,168)
(166,170)(167,169)(172,173)(174,188)(175,187)(176,186)(177,185)(178,184)
(179,183)(180,191)(181,190)(182,189)(192,195)(193,197)(194,196)(199,200)
(201,215)(202,214)(203,213)(204,212)(205,211)(206,210)(207,218)(208,217)
(209,216);
s4 := Sym(218)!( 3,120)( 4,122)( 5,121)( 6,126)( 7,128)( 8,127)( 9,123)
( 10,125)( 11,124)( 12,111)( 13,113)( 14,112)( 15,117)( 16,119)( 17,118)
( 18,114)( 19,116)( 20,115)( 21,131)( 22,130)( 23,129)( 24,137)( 25,136)
( 26,135)( 27,134)( 28,133)( 29,132)( 30,147)( 31,149)( 32,148)( 33,153)
( 34,155)( 35,154)( 36,150)( 37,152)( 38,151)( 39,138)( 40,140)( 41,139)
( 42,144)( 43,146)( 44,145)( 45,141)( 46,143)( 47,142)( 48,158)( 49,157)
( 50,156)( 51,164)( 52,163)( 53,162)( 54,161)( 55,160)( 56,159)( 57,174)
( 58,176)( 59,175)( 60,180)( 61,182)( 62,181)( 63,177)( 64,179)( 65,178)
( 66,165)( 67,167)( 68,166)( 69,171)( 70,173)( 71,172)( 72,168)( 73,170)
( 74,169)( 75,185)( 76,184)( 77,183)( 78,191)( 79,190)( 80,189)( 81,188)
( 82,187)( 83,186)( 84,201)( 85,203)( 86,202)( 87,207)( 88,209)( 89,208)
( 90,204)( 91,206)( 92,205)( 93,192)( 94,194)( 95,193)( 96,198)( 97,200)
( 98,199)( 99,195)(100,197)(101,196)(102,212)(103,211)(104,210)(105,218)
(106,217)(107,216)(108,215)(109,214)(110,213);
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*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope