Polytope of Type {2,3,6,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,6,18}*1728
if this polytope has a name.
Group : SmallGroup(1728,46114)
Rank : 5
Schlafli Type : {2,3,6,18}
Number of vertices, edges, etc : 2, 4, 12, 72, 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) :
3-fold quotients : {2,3,6,6}*576
9-fold quotients : {2,3,6,2}*192
18-fold quotients : {2,3,3,2}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 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);;
s2 := ( 3, 39)( 4, 40)( 5, 42)( 6, 41)( 7, 43)( 8, 44)( 9, 46)( 10, 45)
( 11, 47)( 12, 48)( 13, 50)( 14, 49)( 15, 51)( 16, 52)( 17, 54)( 18, 53)
( 19, 55)( 20, 56)( 21, 58)( 22, 57)( 23, 59)( 24, 60)( 25, 62)( 26, 61)
( 27, 63)( 28, 64)( 29, 66)( 30, 65)( 31, 67)( 32, 68)( 33, 70)( 34, 69)
( 35, 71)( 36, 72)( 37, 74)( 38, 73)( 77, 78)( 81, 82)( 85, 86)( 89, 90)
( 93, 94)( 97, 98)(101,102)(105,106)(109,110)(111,147)(112,148)(113,150)
(114,149)(115,151)(116,152)(117,154)(118,153)(119,155)(120,156)(121,158)
(122,157)(123,159)(124,160)(125,162)(126,161)(127,163)(128,164)(129,166)
(130,165)(131,167)(132,168)(133,170)(134,169)(135,171)(136,172)(137,174)
(138,173)(139,175)(140,176)(141,178)(142,177)(143,179)(144,180)(145,182)
(146,181)(185,186)(189,190)(193,194)(197,198)(201,202)(205,206)(209,210)
(213,214)(217,218);;
s3 := ( 3, 6)( 7, 14)( 8, 12)( 9, 13)( 10, 11)( 15, 34)( 16, 32)( 17, 33)
( 18, 31)( 19, 30)( 20, 28)( 21, 29)( 22, 27)( 23, 38)( 24, 36)( 25, 37)
( 26, 35)( 39, 78)( 40, 76)( 41, 77)( 42, 75)( 43, 86)( 44, 84)( 45, 85)
( 46, 83)( 47, 82)( 48, 80)( 49, 81)( 50, 79)( 51,106)( 52,104)( 53,105)
( 54,103)( 55,102)( 56,100)( 57,101)( 58, 99)( 59,110)( 60,108)( 61,109)
( 62,107)( 63, 94)( 64, 92)( 65, 93)( 66, 91)( 67, 90)( 68, 88)( 69, 89)
( 70, 87)( 71, 98)( 72, 96)( 73, 97)( 74, 95)(111,114)(115,122)(116,120)
(117,121)(118,119)(123,142)(124,140)(125,141)(126,139)(127,138)(128,136)
(129,137)(130,135)(131,146)(132,144)(133,145)(134,143)(147,186)(148,184)
(149,185)(150,183)(151,194)(152,192)(153,193)(154,191)(155,190)(156,188)
(157,189)(158,187)(159,214)(160,212)(161,213)(162,211)(163,210)(164,208)
(165,209)(166,207)(167,218)(168,216)(169,217)(170,215)(171,202)(172,200)
(173,201)(174,199)(175,198)(176,196)(177,197)(178,195)(179,206)(180,204)
(181,205)(182,203);;
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, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
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)!( 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);
s2 := Sym(218)!( 3, 39)( 4, 40)( 5, 42)( 6, 41)( 7, 43)( 8, 44)( 9, 46)
( 10, 45)( 11, 47)( 12, 48)( 13, 50)( 14, 49)( 15, 51)( 16, 52)( 17, 54)
( 18, 53)( 19, 55)( 20, 56)( 21, 58)( 22, 57)( 23, 59)( 24, 60)( 25, 62)
( 26, 61)( 27, 63)( 28, 64)( 29, 66)( 30, 65)( 31, 67)( 32, 68)( 33, 70)
( 34, 69)( 35, 71)( 36, 72)( 37, 74)( 38, 73)( 77, 78)( 81, 82)( 85, 86)
( 89, 90)( 93, 94)( 97, 98)(101,102)(105,106)(109,110)(111,147)(112,148)
(113,150)(114,149)(115,151)(116,152)(117,154)(118,153)(119,155)(120,156)
(121,158)(122,157)(123,159)(124,160)(125,162)(126,161)(127,163)(128,164)
(129,166)(130,165)(131,167)(132,168)(133,170)(134,169)(135,171)(136,172)
(137,174)(138,173)(139,175)(140,176)(141,178)(142,177)(143,179)(144,180)
(145,182)(146,181)(185,186)(189,190)(193,194)(197,198)(201,202)(205,206)
(209,210)(213,214)(217,218);
s3 := Sym(218)!( 3, 6)( 7, 14)( 8, 12)( 9, 13)( 10, 11)( 15, 34)( 16, 32)
( 17, 33)( 18, 31)( 19, 30)( 20, 28)( 21, 29)( 22, 27)( 23, 38)( 24, 36)
( 25, 37)( 26, 35)( 39, 78)( 40, 76)( 41, 77)( 42, 75)( 43, 86)( 44, 84)
( 45, 85)( 46, 83)( 47, 82)( 48, 80)( 49, 81)( 50, 79)( 51,106)( 52,104)
( 53,105)( 54,103)( 55,102)( 56,100)( 57,101)( 58, 99)( 59,110)( 60,108)
( 61,109)( 62,107)( 63, 94)( 64, 92)( 65, 93)( 66, 91)( 67, 90)( 68, 88)
( 69, 89)( 70, 87)( 71, 98)( 72, 96)( 73, 97)( 74, 95)(111,114)(115,122)
(116,120)(117,121)(118,119)(123,142)(124,140)(125,141)(126,139)(127,138)
(128,136)(129,137)(130,135)(131,146)(132,144)(133,145)(134,143)(147,186)
(148,184)(149,185)(150,183)(151,194)(152,192)(153,193)(154,191)(155,190)
(156,188)(157,189)(158,187)(159,214)(160,212)(161,213)(162,211)(163,210)
(164,208)(165,209)(166,207)(167,218)(168,216)(169,217)(170,215)(171,202)
(172,200)(173,201)(174,199)(175,198)(176,196)(177,197)(178,195)(179,206)
(180,204)(181,205)(182,203);
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,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
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