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}*1728c
if this polytope has a name.
Group : SmallGroup(1728,46114)
Rank : 5
Schlafli Type : {2,4,6,18}
Number of vertices, edges, etc : 2, 4, 12, 54, 18
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) :
3-fold quotients : {2,4,6,6}*576d
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)( 8, 9)( 12, 13)( 16, 17)( 20, 21)( 24, 25)( 28, 29)( 32, 33)
( 36, 37)( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)( 45, 80)
( 46, 82)( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51, 87)( 52, 89)( 53, 88)
( 54, 90)( 55, 91)( 56, 93)( 57, 92)( 58, 94)( 59, 95)( 60, 97)( 61, 96)
( 62, 98)( 63, 99)( 64,101)( 65,100)( 66,102)( 67,103)( 68,105)( 69,104)
( 70,106)( 71,107)( 72,109)( 73,108)( 74,110)(112,113)(116,117)(120,121)
(124,125)(128,129)(132,133)(136,137)(140,141)(144,145)(147,183)(148,185)
(149,184)(150,186)(151,187)(152,189)(153,188)(154,190)(155,191)(156,193)
(157,192)(158,194)(159,195)(160,197)(161,196)(162,198)(163,199)(164,201)
(165,200)(166,202)(167,203)(168,205)(169,204)(170,206)(171,207)(172,209)
(173,208)(174,210)(175,211)(176,213)(177,212)(178,214)(179,215)(180,217)
(181,216)(182,218);;
s3 := ( 3, 75)( 4, 78)( 5, 77)( 6, 76)( 7, 83)( 8, 86)( 9, 85)( 10, 84)
( 11, 79)( 12, 82)( 13, 81)( 14, 80)( 15,103)( 16,106)( 17,105)( 18,104)
( 19, 99)( 20,102)( 21,101)( 22,100)( 23,107)( 24,110)( 25,109)( 26,108)
( 27, 91)( 28, 94)( 29, 93)( 30, 92)( 31, 87)( 32, 90)( 33, 89)( 34, 88)
( 35, 95)( 36, 98)( 37, 97)( 38, 96)( 40, 42)( 43, 47)( 44, 50)( 45, 49)
( 46, 48)( 51, 67)( 52, 70)( 53, 69)( 54, 68)( 55, 63)( 56, 66)( 57, 65)
( 58, 64)( 59, 71)( 60, 74)( 61, 73)( 62, 72)(111,183)(112,186)(113,185)
(114,184)(115,191)(116,194)(117,193)(118,192)(119,187)(120,190)(121,189)
(122,188)(123,211)(124,214)(125,213)(126,212)(127,207)(128,210)(129,209)
(130,208)(131,215)(132,218)(133,217)(134,216)(135,199)(136,202)(137,201)
(138,200)(139,195)(140,198)(141,197)(142,196)(143,203)(144,206)(145,205)
(146,204)(148,150)(151,155)(152,158)(153,157)(154,156)(159,175)(160,178)
(161,177)(162,176)(163,171)(164,174)(165,173)(166,172)(167,179)(168,182)
(169,181)(170,180);;
s4 := ( 3,123)( 4,124)( 5,125)( 6,126)( 7,131)( 8,132)( 9,133)( 10,134)
( 11,127)( 12,128)( 13,129)( 14,130)( 15,111)( 16,112)( 17,113)( 18,114)
( 19,119)( 20,120)( 21,121)( 22,122)( 23,115)( 24,116)( 25,117)( 26,118)
( 27,139)( 28,140)( 29,141)( 30,142)( 31,135)( 32,136)( 33,137)( 34,138)
( 35,143)( 36,144)( 37,145)( 38,146)( 39,159)( 40,160)( 41,161)( 42,162)
( 43,167)( 44,168)( 45,169)( 46,170)( 47,163)( 48,164)( 49,165)( 50,166)
( 51,147)( 52,148)( 53,149)( 54,150)( 55,155)( 56,156)( 57,157)( 58,158)
( 59,151)( 60,152)( 61,153)( 62,154)( 63,175)( 64,176)( 65,177)( 66,178)
( 67,171)( 68,172)( 69,173)( 70,174)( 71,179)( 72,180)( 73,181)( 74,182)
( 75,195)( 76,196)( 77,197)( 78,198)( 79,203)( 80,204)( 81,205)( 82,206)
( 83,199)( 84,200)( 85,201)( 86,202)( 87,183)( 88,184)( 89,185)( 90,186)
( 91,191)( 92,192)( 93,193)( 94,194)( 95,187)( 96,188)( 97,189)( 98,190)
( 99,211)(100,212)(101,213)(102,214)(103,207)(104,208)(105,209)(106,210)
(107,215)(108,216)(109,217)(110,218);;
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, s2*s3*s4*s3*s2*s3*s4*s3,
s1*s2*s3*s2*s1*s2*s3*s1*s2, 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, 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)( 8, 9)( 12, 13)( 16, 17)( 20, 21)( 24, 25)( 28, 29)
( 32, 33)( 36, 37)( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)
( 45, 80)( 46, 82)( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51, 87)( 52, 89)
( 53, 88)( 54, 90)( 55, 91)( 56, 93)( 57, 92)( 58, 94)( 59, 95)( 60, 97)
( 61, 96)( 62, 98)( 63, 99)( 64,101)( 65,100)( 66,102)( 67,103)( 68,105)
( 69,104)( 70,106)( 71,107)( 72,109)( 73,108)( 74,110)(112,113)(116,117)
(120,121)(124,125)(128,129)(132,133)(136,137)(140,141)(144,145)(147,183)
(148,185)(149,184)(150,186)(151,187)(152,189)(153,188)(154,190)(155,191)
(156,193)(157,192)(158,194)(159,195)(160,197)(161,196)(162,198)(163,199)
(164,201)(165,200)(166,202)(167,203)(168,205)(169,204)(170,206)(171,207)
(172,209)(173,208)(174,210)(175,211)(176,213)(177,212)(178,214)(179,215)
(180,217)(181,216)(182,218);
s3 := Sym(218)!( 3, 75)( 4, 78)( 5, 77)( 6, 76)( 7, 83)( 8, 86)( 9, 85)
( 10, 84)( 11, 79)( 12, 82)( 13, 81)( 14, 80)( 15,103)( 16,106)( 17,105)
( 18,104)( 19, 99)( 20,102)( 21,101)( 22,100)( 23,107)( 24,110)( 25,109)
( 26,108)( 27, 91)( 28, 94)( 29, 93)( 30, 92)( 31, 87)( 32, 90)( 33, 89)
( 34, 88)( 35, 95)( 36, 98)( 37, 97)( 38, 96)( 40, 42)( 43, 47)( 44, 50)
( 45, 49)( 46, 48)( 51, 67)( 52, 70)( 53, 69)( 54, 68)( 55, 63)( 56, 66)
( 57, 65)( 58, 64)( 59, 71)( 60, 74)( 61, 73)( 62, 72)(111,183)(112,186)
(113,185)(114,184)(115,191)(116,194)(117,193)(118,192)(119,187)(120,190)
(121,189)(122,188)(123,211)(124,214)(125,213)(126,212)(127,207)(128,210)
(129,209)(130,208)(131,215)(132,218)(133,217)(134,216)(135,199)(136,202)
(137,201)(138,200)(139,195)(140,198)(141,197)(142,196)(143,203)(144,206)
(145,205)(146,204)(148,150)(151,155)(152,158)(153,157)(154,156)(159,175)
(160,178)(161,177)(162,176)(163,171)(164,174)(165,173)(166,172)(167,179)
(168,182)(169,181)(170,180);
s4 := Sym(218)!( 3,123)( 4,124)( 5,125)( 6,126)( 7,131)( 8,132)( 9,133)
( 10,134)( 11,127)( 12,128)( 13,129)( 14,130)( 15,111)( 16,112)( 17,113)
( 18,114)( 19,119)( 20,120)( 21,121)( 22,122)( 23,115)( 24,116)( 25,117)
( 26,118)( 27,139)( 28,140)( 29,141)( 30,142)( 31,135)( 32,136)( 33,137)
( 34,138)( 35,143)( 36,144)( 37,145)( 38,146)( 39,159)( 40,160)( 41,161)
( 42,162)( 43,167)( 44,168)( 45,169)( 46,170)( 47,163)( 48,164)( 49,165)
( 50,166)( 51,147)( 52,148)( 53,149)( 54,150)( 55,155)( 56,156)( 57,157)
( 58,158)( 59,151)( 60,152)( 61,153)( 62,154)( 63,175)( 64,176)( 65,177)
( 66,178)( 67,171)( 68,172)( 69,173)( 70,174)( 71,179)( 72,180)( 73,181)
( 74,182)( 75,195)( 76,196)( 77,197)( 78,198)( 79,203)( 80,204)( 81,205)
( 82,206)( 83,199)( 84,200)( 85,201)( 86,202)( 87,183)( 88,184)( 89,185)
( 90,186)( 91,191)( 92,192)( 93,193)( 94,194)( 95,187)( 96,188)( 97,189)
( 98,190)( 99,211)(100,212)(101,213)(102,214)(103,207)(104,208)(105,209)
(106,210)(107,215)(108,216)(109,217)(110,218);
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,
s2*s3*s4*s3*s2*s3*s4*s3, s1*s2*s3*s2*s1*s2*s3*s1*s2,
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