Polytope of Type {2,18,4,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,4,3}*1728
if this polytope has a name.
Group : SmallGroup(1728,46114)
Rank : 5
Schlafli Type : {2,18,4,3}
Number of vertices, edges, etc : 2, 18, 72, 12, 6
Order of s0s1s2s3s4 : 18
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 : {2,6,4,3}*576
4-fold quotients : {2,18,2,3}*432
8-fold quotients : {2,9,2,3}*216
9-fold quotients : {2,2,4,3}*192
12-fold quotients : {2,6,2,3}*144
18-fold quotients : {2,2,4,3}*96
24-fold quotients : {2,3,2,3}*72
36-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 7, 11)( 8, 12)( 9, 13)( 10, 14)( 15, 31)( 16, 32)( 17, 33)( 18, 34)
( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 35)( 24, 36)( 25, 37)( 26, 38)
( 43, 47)( 44, 48)( 45, 49)( 46, 50)( 51, 67)( 52, 68)( 53, 69)( 54, 70)
( 55, 63)( 56, 64)( 57, 65)( 58, 66)( 59, 71)( 60, 72)( 61, 73)( 62, 74)
( 79, 83)( 80, 84)( 81, 85)( 82, 86)( 87,103)( 88,104)( 89,105)( 90,106)
( 91, 99)( 92,100)( 93,101)( 94,102)( 95,107)( 96,108)( 97,109)( 98,110)
(115,119)(116,120)(117,121)(118,122)(123,139)(124,140)(125,141)(126,142)
(127,135)(128,136)(129,137)(130,138)(131,143)(132,144)(133,145)(134,146)
(151,155)(152,156)(153,157)(154,158)(159,175)(160,176)(161,177)(162,178)
(163,171)(164,172)(165,173)(166,174)(167,179)(168,180)(169,181)(170,182)
(187,191)(188,192)(189,193)(190,194)(195,211)(196,212)(197,213)(198,214)
(199,207)(200,208)(201,209)(202,210)(203,215)(204,216)(205,217)(206,218);;
s2 := ( 3,125)( 4,126)( 5,123)( 6,124)( 7,133)( 8,134)( 9,131)( 10,132)
( 11,129)( 12,130)( 13,127)( 14,128)( 15,113)( 16,114)( 17,111)( 18,112)
( 19,121)( 20,122)( 21,119)( 22,120)( 23,117)( 24,118)( 25,115)( 26,116)
( 27,141)( 28,142)( 29,139)( 30,140)( 31,137)( 32,138)( 33,135)( 34,136)
( 35,145)( 36,146)( 37,143)( 38,144)( 39,161)( 40,162)( 41,159)( 42,160)
( 43,169)( 44,170)( 45,167)( 46,168)( 47,165)( 48,166)( 49,163)( 50,164)
( 51,149)( 52,150)( 53,147)( 54,148)( 55,157)( 56,158)( 57,155)( 58,156)
( 59,153)( 60,154)( 61,151)( 62,152)( 63,177)( 64,178)( 65,175)( 66,176)
( 67,173)( 68,174)( 69,171)( 70,172)( 71,181)( 72,182)( 73,179)( 74,180)
( 75,197)( 76,198)( 77,195)( 78,196)( 79,205)( 80,206)( 81,203)( 82,204)
( 83,201)( 84,202)( 85,199)( 86,200)( 87,185)( 88,186)( 89,183)( 90,184)
( 91,193)( 92,194)( 93,191)( 94,192)( 95,189)( 96,190)( 97,187)( 98,188)
( 99,213)(100,214)(101,211)(102,212)(103,209)(104,210)(105,207)(106,208)
(107,217)(108,218)(109,215)(110,216);;
s3 := ( 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);;
s4 := ( 3, 75)( 4, 78)( 5, 77)( 6, 76)( 7, 79)( 8, 82)( 9, 81)( 10, 80)
( 11, 83)( 12, 86)( 13, 85)( 14, 84)( 15, 87)( 16, 90)( 17, 89)( 18, 88)
( 19, 91)( 20, 94)( 21, 93)( 22, 92)( 23, 95)( 24, 98)( 25, 97)( 26, 96)
( 27, 99)( 28,102)( 29,101)( 30,100)( 31,103)( 32,106)( 33,105)( 34,104)
( 35,107)( 36,110)( 37,109)( 38,108)( 40, 42)( 44, 46)( 48, 50)( 52, 54)
( 56, 58)( 60, 62)( 64, 66)( 68, 70)( 72, 74)(111,183)(112,186)(113,185)
(114,184)(115,187)(116,190)(117,189)(118,188)(119,191)(120,194)(121,193)
(122,192)(123,195)(124,198)(125,197)(126,196)(127,199)(128,202)(129,201)
(130,200)(131,203)(132,206)(133,205)(134,204)(135,207)(136,210)(137,209)
(138,208)(139,211)(140,214)(141,213)(142,212)(143,215)(144,218)(145,217)
(146,216)(148,150)(152,154)(156,158)(160,162)(164,166)(168,170)(172,174)
(176,178)(180,182);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, 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)!( 7, 11)( 8, 12)( 9, 13)( 10, 14)( 15, 31)( 16, 32)( 17, 33)
( 18, 34)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 35)( 24, 36)( 25, 37)
( 26, 38)( 43, 47)( 44, 48)( 45, 49)( 46, 50)( 51, 67)( 52, 68)( 53, 69)
( 54, 70)( 55, 63)( 56, 64)( 57, 65)( 58, 66)( 59, 71)( 60, 72)( 61, 73)
( 62, 74)( 79, 83)( 80, 84)( 81, 85)( 82, 86)( 87,103)( 88,104)( 89,105)
( 90,106)( 91, 99)( 92,100)( 93,101)( 94,102)( 95,107)( 96,108)( 97,109)
( 98,110)(115,119)(116,120)(117,121)(118,122)(123,139)(124,140)(125,141)
(126,142)(127,135)(128,136)(129,137)(130,138)(131,143)(132,144)(133,145)
(134,146)(151,155)(152,156)(153,157)(154,158)(159,175)(160,176)(161,177)
(162,178)(163,171)(164,172)(165,173)(166,174)(167,179)(168,180)(169,181)
(170,182)(187,191)(188,192)(189,193)(190,194)(195,211)(196,212)(197,213)
(198,214)(199,207)(200,208)(201,209)(202,210)(203,215)(204,216)(205,217)
(206,218);
s2 := Sym(218)!( 3,125)( 4,126)( 5,123)( 6,124)( 7,133)( 8,134)( 9,131)
( 10,132)( 11,129)( 12,130)( 13,127)( 14,128)( 15,113)( 16,114)( 17,111)
( 18,112)( 19,121)( 20,122)( 21,119)( 22,120)( 23,117)( 24,118)( 25,115)
( 26,116)( 27,141)( 28,142)( 29,139)( 30,140)( 31,137)( 32,138)( 33,135)
( 34,136)( 35,145)( 36,146)( 37,143)( 38,144)( 39,161)( 40,162)( 41,159)
( 42,160)( 43,169)( 44,170)( 45,167)( 46,168)( 47,165)( 48,166)( 49,163)
( 50,164)( 51,149)( 52,150)( 53,147)( 54,148)( 55,157)( 56,158)( 57,155)
( 58,156)( 59,153)( 60,154)( 61,151)( 62,152)( 63,177)( 64,178)( 65,175)
( 66,176)( 67,173)( 68,174)( 69,171)( 70,172)( 71,181)( 72,182)( 73,179)
( 74,180)( 75,197)( 76,198)( 77,195)( 78,196)( 79,205)( 80,206)( 81,203)
( 82,204)( 83,201)( 84,202)( 85,199)( 86,200)( 87,185)( 88,186)( 89,183)
( 90,184)( 91,193)( 92,194)( 93,191)( 94,192)( 95,189)( 96,190)( 97,187)
( 98,188)( 99,213)(100,214)(101,211)(102,212)(103,209)(104,210)(105,207)
(106,208)(107,217)(108,218)(109,215)(110,216);
s3 := 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);
s4 := Sym(218)!( 3, 75)( 4, 78)( 5, 77)( 6, 76)( 7, 79)( 8, 82)( 9, 81)
( 10, 80)( 11, 83)( 12, 86)( 13, 85)( 14, 84)( 15, 87)( 16, 90)( 17, 89)
( 18, 88)( 19, 91)( 20, 94)( 21, 93)( 22, 92)( 23, 95)( 24, 98)( 25, 97)
( 26, 96)( 27, 99)( 28,102)( 29,101)( 30,100)( 31,103)( 32,106)( 33,105)
( 34,104)( 35,107)( 36,110)( 37,109)( 38,108)( 40, 42)( 44, 46)( 48, 50)
( 52, 54)( 56, 58)( 60, 62)( 64, 66)( 68, 70)( 72, 74)(111,183)(112,186)
(113,185)(114,184)(115,187)(116,190)(117,189)(118,188)(119,191)(120,194)
(121,193)(122,192)(123,195)(124,198)(125,197)(126,196)(127,199)(128,202)
(129,201)(130,200)(131,203)(132,206)(133,205)(134,204)(135,207)(136,210)
(137,209)(138,208)(139,211)(140,214)(141,213)(142,212)(143,215)(144,218)
(145,217)(146,216)(148,150)(152,154)(156,158)(160,162)(164,166)(168,170)
(172,174)(176,178)(180,182);
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
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