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