Polytope of Type {2,9,4,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,4,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,46115)
Rank : 5
Schlafli Type : {2,9,4,6}
Number of vertices, edges, etc : 2, 18, 36, 24, 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 : {2,9,4,2}*576, {2,3,4,6}*576
4-fold quotients : {2,9,2,6}*432
6-fold quotients : {2,9,4,2}*288
8-fold quotients : {2,9,2,3}*216
9-fold quotients : {2,3,4,2}*192
12-fold quotients : {2,9,2,2}*144, {2,3,2,6}*144
18-fold quotients : {2,3,4,2}*96
24-fold quotients : {2,3,2,3}*72
36-fold quotients : {2,3,2,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 16, 17)( 19, 23)( 20, 25)
( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)( 39, 79)
( 40, 81)( 41, 80)( 42, 82)( 43, 75)( 44, 77)( 45, 76)( 46, 78)( 47, 83)
( 48, 85)( 49, 84)( 50, 86)( 51, 91)( 52, 93)( 53, 92)( 54, 94)( 55, 87)
( 56, 89)( 57, 88)( 58, 90)( 59, 95)( 60, 97)( 61, 96)( 62, 98)( 63,103)
( 64,105)( 65,104)( 66,106)( 67, 99)( 68,101)( 69,100)( 70,102)( 71,107)
( 72,109)( 73,108)( 74,110)(112,113)(115,119)(116,121)(117,120)(118,122)
(124,125)(127,131)(128,133)(129,132)(130,134)(136,137)(139,143)(140,145)
(141,144)(142,146)(147,187)(148,189)(149,188)(150,190)(151,183)(152,185)
(153,184)(154,186)(155,191)(156,193)(157,192)(158,194)(159,199)(160,201)
(161,200)(162,202)(163,195)(164,197)(165,196)(166,198)(167,203)(168,205)
(169,204)(170,206)(171,211)(172,213)(173,212)(174,214)(175,207)(176,209)
(177,208)(178,210)(179,215)(180,217)(181,216)(182,218);;
s2 := ( 3, 39)( 4, 40)( 5, 42)( 6, 41)( 7, 47)( 8, 48)( 9, 50)( 10, 49)
( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 51)( 16, 52)( 17, 54)( 18, 53)
( 19, 59)( 20, 60)( 21, 62)( 22, 61)( 23, 55)( 24, 56)( 25, 58)( 26, 57)
( 27, 63)( 28, 64)( 29, 66)( 30, 65)( 31, 71)( 32, 72)( 33, 74)( 34, 73)
( 35, 67)( 36, 68)( 37, 70)( 38, 69)( 75, 79)( 76, 80)( 77, 82)( 78, 81)
( 85, 86)( 87, 91)( 88, 92)( 89, 94)( 90, 93)( 97, 98)( 99,103)(100,104)
(101,106)(102,105)(109,110)(111,147)(112,148)(113,150)(114,149)(115,155)
(116,156)(117,158)(118,157)(119,151)(120,152)(121,154)(122,153)(123,159)
(124,160)(125,162)(126,161)(127,167)(128,168)(129,170)(130,169)(131,163)
(132,164)(133,166)(134,165)(135,171)(136,172)(137,174)(138,173)(139,179)
(140,180)(141,182)(142,181)(143,175)(144,176)(145,178)(146,177)(183,187)
(184,188)(185,190)(186,189)(193,194)(195,199)(196,200)(197,202)(198,201)
(205,206)(207,211)(208,212)(209,214)(210,213)(217,218);;
s3 := ( 3, 6)( 4, 5)( 7, 10)( 8, 9)( 11, 14)( 12, 13)( 15, 30)( 16, 29)
( 17, 28)( 18, 27)( 19, 34)( 20, 33)( 21, 32)( 22, 31)( 23, 38)( 24, 37)
( 25, 36)( 26, 35)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)( 48, 49)
( 51, 66)( 52, 65)( 53, 64)( 54, 63)( 55, 70)( 56, 69)( 57, 68)( 58, 67)
( 59, 74)( 60, 73)( 61, 72)( 62, 71)( 75, 78)( 76, 77)( 79, 82)( 80, 81)
( 83, 86)( 84, 85)( 87,102)( 88,101)( 89,100)( 90, 99)( 91,106)( 92,105)
( 93,104)( 94,103)( 95,110)( 96,109)( 97,108)( 98,107)(111,114)(112,113)
(115,118)(116,117)(119,122)(120,121)(123,138)(124,137)(125,136)(126,135)
(127,142)(128,141)(129,140)(130,139)(131,146)(132,145)(133,144)(134,143)
(147,150)(148,149)(151,154)(152,153)(155,158)(156,157)(159,174)(160,173)
(161,172)(162,171)(163,178)(164,177)(165,176)(166,175)(167,182)(168,181)
(169,180)(170,179)(183,186)(184,185)(187,190)(188,189)(191,194)(192,193)
(195,210)(196,209)(197,208)(198,207)(199,214)(200,213)(201,212)(202,211)
(203,218)(204,217)(205,216)(206,215);;
s4 := ( 3,123)( 4,124)( 5,125)( 6,126)( 7,127)( 8,128)( 9,129)( 10,130)
( 11,131)( 12,132)( 13,133)( 14,134)( 15,111)( 16,112)( 17,113)( 18,114)
( 19,115)( 20,116)( 21,117)( 22,118)( 23,119)( 24,120)( 25,121)( 26,122)
( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)( 33,141)( 34,142)
( 35,143)( 36,144)( 37,145)( 38,146)( 39,159)( 40,160)( 41,161)( 42,162)
( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)( 48,168)( 49,169)( 50,170)
( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)( 56,152)( 57,153)( 58,154)
( 59,155)( 60,156)( 61,157)( 62,158)( 63,171)( 64,172)( 65,173)( 66,174)
( 67,175)( 68,176)( 69,177)( 70,178)( 71,179)( 72,180)( 73,181)( 74,182)
( 75,195)( 76,196)( 77,197)( 78,198)( 79,199)( 80,200)( 81,201)( 82,202)
( 83,203)( 84,204)( 85,205)( 86,206)( 87,183)( 88,184)( 89,185)( 90,186)
( 91,187)( 92,188)( 93,189)( 94,190)( 95,191)( 96,192)( 97,193)( 98,194)
( 99,207)(100,208)(101,209)(102,210)(103,211)(104,212)(105,213)(106,214)
(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,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4,
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)!( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 16, 17)( 19, 23)
( 20, 25)( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)
( 39, 79)( 40, 81)( 41, 80)( 42, 82)( 43, 75)( 44, 77)( 45, 76)( 46, 78)
( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51, 91)( 52, 93)( 53, 92)( 54, 94)
( 55, 87)( 56, 89)( 57, 88)( 58, 90)( 59, 95)( 60, 97)( 61, 96)( 62, 98)
( 63,103)( 64,105)( 65,104)( 66,106)( 67, 99)( 68,101)( 69,100)( 70,102)
( 71,107)( 72,109)( 73,108)( 74,110)(112,113)(115,119)(116,121)(117,120)
(118,122)(124,125)(127,131)(128,133)(129,132)(130,134)(136,137)(139,143)
(140,145)(141,144)(142,146)(147,187)(148,189)(149,188)(150,190)(151,183)
(152,185)(153,184)(154,186)(155,191)(156,193)(157,192)(158,194)(159,199)
(160,201)(161,200)(162,202)(163,195)(164,197)(165,196)(166,198)(167,203)
(168,205)(169,204)(170,206)(171,211)(172,213)(173,212)(174,214)(175,207)
(176,209)(177,208)(178,210)(179,215)(180,217)(181,216)(182,218);
s2 := Sym(218)!( 3, 39)( 4, 40)( 5, 42)( 6, 41)( 7, 47)( 8, 48)( 9, 50)
( 10, 49)( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 51)( 16, 52)( 17, 54)
( 18, 53)( 19, 59)( 20, 60)( 21, 62)( 22, 61)( 23, 55)( 24, 56)( 25, 58)
( 26, 57)( 27, 63)( 28, 64)( 29, 66)( 30, 65)( 31, 71)( 32, 72)( 33, 74)
( 34, 73)( 35, 67)( 36, 68)( 37, 70)( 38, 69)( 75, 79)( 76, 80)( 77, 82)
( 78, 81)( 85, 86)( 87, 91)( 88, 92)( 89, 94)( 90, 93)( 97, 98)( 99,103)
(100,104)(101,106)(102,105)(109,110)(111,147)(112,148)(113,150)(114,149)
(115,155)(116,156)(117,158)(118,157)(119,151)(120,152)(121,154)(122,153)
(123,159)(124,160)(125,162)(126,161)(127,167)(128,168)(129,170)(130,169)
(131,163)(132,164)(133,166)(134,165)(135,171)(136,172)(137,174)(138,173)
(139,179)(140,180)(141,182)(142,181)(143,175)(144,176)(145,178)(146,177)
(183,187)(184,188)(185,190)(186,189)(193,194)(195,199)(196,200)(197,202)
(198,201)(205,206)(207,211)(208,212)(209,214)(210,213)(217,218);
s3 := Sym(218)!( 3, 6)( 4, 5)( 7, 10)( 8, 9)( 11, 14)( 12, 13)( 15, 30)
( 16, 29)( 17, 28)( 18, 27)( 19, 34)( 20, 33)( 21, 32)( 22, 31)( 23, 38)
( 24, 37)( 25, 36)( 26, 35)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)
( 48, 49)( 51, 66)( 52, 65)( 53, 64)( 54, 63)( 55, 70)( 56, 69)( 57, 68)
( 58, 67)( 59, 74)( 60, 73)( 61, 72)( 62, 71)( 75, 78)( 76, 77)( 79, 82)
( 80, 81)( 83, 86)( 84, 85)( 87,102)( 88,101)( 89,100)( 90, 99)( 91,106)
( 92,105)( 93,104)( 94,103)( 95,110)( 96,109)( 97,108)( 98,107)(111,114)
(112,113)(115,118)(116,117)(119,122)(120,121)(123,138)(124,137)(125,136)
(126,135)(127,142)(128,141)(129,140)(130,139)(131,146)(132,145)(133,144)
(134,143)(147,150)(148,149)(151,154)(152,153)(155,158)(156,157)(159,174)
(160,173)(161,172)(162,171)(163,178)(164,177)(165,176)(166,175)(167,182)
(168,181)(169,180)(170,179)(183,186)(184,185)(187,190)(188,189)(191,194)
(192,193)(195,210)(196,209)(197,208)(198,207)(199,214)(200,213)(201,212)
(202,211)(203,218)(204,217)(205,216)(206,215);
s4 := Sym(218)!( 3,123)( 4,124)( 5,125)( 6,126)( 7,127)( 8,128)( 9,129)
( 10,130)( 11,131)( 12,132)( 13,133)( 14,134)( 15,111)( 16,112)( 17,113)
( 18,114)( 19,115)( 20,116)( 21,117)( 22,118)( 23,119)( 24,120)( 25,121)
( 26,122)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)( 33,141)
( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,159)( 40,160)( 41,161)
( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)( 48,168)( 49,169)
( 50,170)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)( 56,152)( 57,153)
( 58,154)( 59,155)( 60,156)( 61,157)( 62,158)( 63,171)( 64,172)( 65,173)
( 66,174)( 67,175)( 68,176)( 69,177)( 70,178)( 71,179)( 72,180)( 73,181)
( 74,182)( 75,195)( 76,196)( 77,197)( 78,198)( 79,199)( 80,200)( 81,201)
( 82,202)( 83,203)( 84,204)( 85,205)( 86,206)( 87,183)( 88,184)( 89,185)
( 90,186)( 91,187)( 92,188)( 93,189)( 94,190)( 95,191)( 96,192)( 97,193)
( 98,194)( 99,207)(100,208)(101,209)(102,210)(103,211)(104,212)(105,213)
(106,214)(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, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s4*s3*s2*s3*s4*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope