Polytope of Type {2,6,4,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,4,9}*1728
if this polytope has a name.
Group : SmallGroup(1728,46115)
Rank : 5
Schlafli Type : {2,6,4,9}
Number of vertices, edges, etc : 2, 6, 24, 36, 18
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 : {2,2,4,9}*576, {2,6,4,3}*576
4-fold quotients : {2,6,2,9}*432
6-fold quotients : {2,2,4,9}*288
8-fold quotients : {2,3,2,9}*216
9-fold quotients : {2,2,4,3}*192
12-fold quotients : {2,2,2,9}*144, {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 := ( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 33)( 22, 34)
( 23, 35)( 24, 36)( 25, 37)( 26, 38)( 51, 63)( 52, 64)( 53, 65)( 54, 66)
( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 61, 73)( 62, 74)
( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)( 93,105)( 94,106)
( 95,107)( 96,108)( 97,109)( 98,110)(123,135)(124,136)(125,137)(126,138)
(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)(133,145)(134,146)
(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)(165,177)(166,178)
(167,179)(168,180)(169,181)(170,182)(195,207)(196,208)(197,209)(198,210)
(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)(205,217)(206,218);;
s2 := ( 3,125)( 4,126)( 5,123)( 6,124)( 7,129)( 8,130)( 9,127)( 10,128)
( 11,133)( 12,134)( 13,131)( 14,132)( 15,113)( 16,114)( 17,111)( 18,112)
( 19,117)( 20,118)( 21,115)( 22,116)( 23,121)( 24,122)( 25,119)( 26,120)
( 27,137)( 28,138)( 29,135)( 30,136)( 31,141)( 32,142)( 33,139)( 34,140)
( 35,145)( 36,146)( 37,143)( 38,144)( 39,161)( 40,162)( 41,159)( 42,160)
( 43,165)( 44,166)( 45,163)( 46,164)( 47,169)( 48,170)( 49,167)( 50,168)
( 51,149)( 52,150)( 53,147)( 54,148)( 55,153)( 56,154)( 57,151)( 58,152)
( 59,157)( 60,158)( 61,155)( 62,156)( 63,173)( 64,174)( 65,171)( 66,172)
( 67,177)( 68,178)( 69,175)( 70,176)( 71,181)( 72,182)( 73,179)( 74,180)
( 75,197)( 76,198)( 77,195)( 78,196)( 79,201)( 80,202)( 81,199)( 82,200)
( 83,205)( 84,206)( 85,203)( 86,204)( 87,185)( 88,186)( 89,183)( 90,184)
( 91,189)( 92,190)( 93,187)( 94,188)( 95,193)( 96,194)( 97,191)( 98,192)
( 99,209)(100,210)(101,207)(102,208)(103,213)(104,214)(105,211)(106,212)
(107,217)(108,218)(109,215)(110,216);;
s3 := ( 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);;
s4 := ( 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, 87)( 16, 90)( 17, 89)( 18, 88)
( 19, 95)( 20, 98)( 21, 97)( 22, 96)( 23, 91)( 24, 94)( 25, 93)( 26, 92)
( 27, 99)( 28,102)( 29,101)( 30,100)( 31,107)( 32,110)( 33,109)( 34,108)
( 35,103)( 36,106)( 37,105)( 38,104)( 40, 42)( 43, 47)( 44, 50)( 45, 49)
( 46, 48)( 52, 54)( 55, 59)( 56, 62)( 57, 61)( 58, 60)( 64, 66)( 67, 71)
( 68, 74)( 69, 73)( 70, 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)(123,195)
(124,198)(125,197)(126,196)(127,203)(128,206)(129,205)(130,204)(131,199)
(132,202)(133,201)(134,200)(135,207)(136,210)(137,209)(138,208)(139,215)
(140,218)(141,217)(142,216)(143,211)(144,214)(145,213)(146,212)(148,150)
(151,155)(152,158)(153,157)(154,156)(160,162)(163,167)(164,170)(165,169)
(166,168)(172,174)(175,179)(176,182)(177,181)(178,180);;
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*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,
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3,
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)!( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 33)
( 22, 34)( 23, 35)( 24, 36)( 25, 37)( 26, 38)( 51, 63)( 52, 64)( 53, 65)
( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 61, 73)
( 62, 74)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)( 93,105)
( 94,106)( 95,107)( 96,108)( 97,109)( 98,110)(123,135)(124,136)(125,137)
(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)(133,145)
(134,146)(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)(165,177)
(166,178)(167,179)(168,180)(169,181)(170,182)(195,207)(196,208)(197,209)
(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)(205,217)
(206,218);
s2 := Sym(218)!( 3,125)( 4,126)( 5,123)( 6,124)( 7,129)( 8,130)( 9,127)
( 10,128)( 11,133)( 12,134)( 13,131)( 14,132)( 15,113)( 16,114)( 17,111)
( 18,112)( 19,117)( 20,118)( 21,115)( 22,116)( 23,121)( 24,122)( 25,119)
( 26,120)( 27,137)( 28,138)( 29,135)( 30,136)( 31,141)( 32,142)( 33,139)
( 34,140)( 35,145)( 36,146)( 37,143)( 38,144)( 39,161)( 40,162)( 41,159)
( 42,160)( 43,165)( 44,166)( 45,163)( 46,164)( 47,169)( 48,170)( 49,167)
( 50,168)( 51,149)( 52,150)( 53,147)( 54,148)( 55,153)( 56,154)( 57,151)
( 58,152)( 59,157)( 60,158)( 61,155)( 62,156)( 63,173)( 64,174)( 65,171)
( 66,172)( 67,177)( 68,178)( 69,175)( 70,176)( 71,181)( 72,182)( 73,179)
( 74,180)( 75,197)( 76,198)( 77,195)( 78,196)( 79,201)( 80,202)( 81,199)
( 82,200)( 83,205)( 84,206)( 85,203)( 86,204)( 87,185)( 88,186)( 89,183)
( 90,184)( 91,189)( 92,190)( 93,187)( 94,188)( 95,193)( 96,194)( 97,191)
( 98,192)( 99,209)(100,210)(101,207)(102,208)(103,213)(104,214)(105,211)
(106,212)(107,217)(108,218)(109,215)(110,216);
s3 := 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);
s4 := 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, 87)( 16, 90)( 17, 89)
( 18, 88)( 19, 95)( 20, 98)( 21, 97)( 22, 96)( 23, 91)( 24, 94)( 25, 93)
( 26, 92)( 27, 99)( 28,102)( 29,101)( 30,100)( 31,107)( 32,110)( 33,109)
( 34,108)( 35,103)( 36,106)( 37,105)( 38,104)( 40, 42)( 43, 47)( 44, 50)
( 45, 49)( 46, 48)( 52, 54)( 55, 59)( 56, 62)( 57, 61)( 58, 60)( 64, 66)
( 67, 71)( 68, 74)( 69, 73)( 70, 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)
(123,195)(124,198)(125,197)(126,196)(127,203)(128,206)(129,205)(130,204)
(131,199)(132,202)(133,201)(134,200)(135,207)(136,210)(137,209)(138,208)
(139,215)(140,218)(141,217)(142,216)(143,211)(144,214)(145,213)(146,212)
(148,150)(151,155)(152,158)(153,157)(154,156)(160,162)(163,167)(164,170)
(165,169)(166,168)(172,174)(175,179)(176,182)(177,181)(178,180);
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*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,
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope