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