Polytope of Type {2,9,6,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,6,8}*1728
if this polytope has a name.
Group : SmallGroup(1728,17171)
Rank : 5
Schlafli Type : {2,9,6,8}
Number of vertices, edges, etc : 2, 9, 27, 24, 8
Order of s0s1s2s3s4 : 72
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) :
2-fold quotients : {2,9,6,4}*864
3-fold quotients : {2,9,2,8}*576, {2,3,6,8}*576
4-fold quotients : {2,9,6,2}*432
6-fold quotients : {2,9,2,4}*288, {2,3,6,4}*288
9-fold quotients : {2,3,2,8}*192
12-fold quotients : {2,9,2,2}*144, {2,3,6,2}*144
18-fold quotients : {2,3,2,4}*96
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)( 6, 9)( 7, 11)( 8, 10)( 12, 23)( 13, 22)( 14, 21)( 15, 29)
( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 39, 50)( 40, 49)( 41, 48)( 42, 56)( 43, 55)( 44, 54)( 45, 53)
( 46, 52)( 47, 51)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 66, 77)( 67, 76)
( 68, 75)( 69, 83)( 70, 82)( 71, 81)( 72, 80)( 73, 79)( 74, 78)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 93,104)( 94,103)( 95,102)( 96,110)( 97,109)
( 98,108)( 99,107)(100,106)(101,105)(112,113)(114,117)(115,119)(116,118)
(120,131)(121,130)(122,129)(123,137)(124,136)(125,135)(126,134)(127,133)
(128,132)(139,140)(141,144)(142,146)(143,145)(147,158)(148,157)(149,156)
(150,164)(151,163)(152,162)(153,161)(154,160)(155,159)(166,167)(168,171)
(169,173)(170,172)(174,185)(175,184)(176,183)(177,191)(178,190)(179,189)
(180,188)(181,187)(182,186)(193,194)(195,198)(196,200)(197,199)(201,212)
(202,211)(203,210)(204,218)(205,217)(206,216)(207,215)(208,214)(209,213);;
s2 := ( 3, 15)( 4, 17)( 5, 16)( 6, 12)( 7, 14)( 8, 13)( 9, 18)( 10, 20)
( 11, 19)( 21, 26)( 22, 25)( 23, 24)( 27, 29)( 30, 42)( 31, 44)( 32, 43)
( 33, 39)( 34, 41)( 35, 40)( 36, 45)( 37, 47)( 38, 46)( 48, 53)( 49, 52)
( 50, 51)( 54, 56)( 57, 69)( 58, 71)( 59, 70)( 60, 66)( 61, 68)( 62, 67)
( 63, 72)( 64, 74)( 65, 73)( 75, 80)( 76, 79)( 77, 78)( 81, 83)( 84, 96)
( 85, 98)( 86, 97)( 87, 93)( 88, 95)( 89, 94)( 90, 99)( 91,101)( 92,100)
(102,107)(103,106)(104,105)(108,110)(111,123)(112,125)(113,124)(114,120)
(115,122)(116,121)(117,126)(118,128)(119,127)(129,134)(130,133)(131,132)
(135,137)(138,150)(139,152)(140,151)(141,147)(142,149)(143,148)(144,153)
(145,155)(146,154)(156,161)(157,160)(158,159)(162,164)(165,177)(166,179)
(167,178)(168,174)(169,176)(170,175)(171,180)(172,182)(173,181)(183,188)
(184,187)(185,186)(189,191)(192,204)(193,206)(194,205)(195,201)(196,203)
(197,202)(198,207)(199,209)(200,208)(210,215)(211,214)(212,213)(216,218);;
s3 := ( 6, 9)( 7, 10)( 8, 11)( 15, 18)( 16, 19)( 17, 20)( 24, 27)( 25, 28)
( 26, 29)( 33, 36)( 34, 37)( 35, 38)( 42, 45)( 43, 46)( 44, 47)( 51, 54)
( 52, 55)( 53, 56)( 57, 84)( 58, 85)( 59, 86)( 60, 90)( 61, 91)( 62, 92)
( 63, 87)( 64, 88)( 65, 89)( 66, 93)( 67, 94)( 68, 95)( 69, 99)( 70,100)
( 71,101)( 72, 96)( 73, 97)( 74, 98)( 75,102)( 76,103)( 77,104)( 78,108)
( 79,109)( 80,110)( 81,105)( 82,106)( 83,107)(111,165)(112,166)(113,167)
(114,171)(115,172)(116,173)(117,168)(118,169)(119,170)(120,174)(121,175)
(122,176)(123,180)(124,181)(125,182)(126,177)(127,178)(128,179)(129,183)
(130,184)(131,185)(132,189)(133,190)(134,191)(135,186)(136,187)(137,188)
(138,192)(139,193)(140,194)(141,198)(142,199)(143,200)(144,195)(145,196)
(146,197)(147,201)(148,202)(149,203)(150,207)(151,208)(152,209)(153,204)
(154,205)(155,206)(156,210)(157,211)(158,212)(159,216)(160,217)(161,218)
(162,213)(163,214)(164,215);;
s4 := ( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)( 8,116)( 9,117)( 10,118)
( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)( 17,125)( 18,126)
( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)( 25,133)( 26,134)
( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)( 33,141)( 34,142)
( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)( 41,149)( 42,150)
( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)( 49,157)( 50,158)
( 51,159)( 52,160)( 53,161)( 54,162)( 55,163)( 56,164)( 57,192)( 58,193)
( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)( 65,200)( 66,201)
( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)( 73,208)( 74,209)
( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)( 81,216)( 82,217)
( 83,218)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)( 89,170)( 90,171)
( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)( 96,177)( 97,178)( 98,179)
( 99,180)(100,181)(101,182)(102,183)(103,184)(104,185)(105,186)(106,187)
(107,188)(108,189)(109,190)(110,191);;
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*s4*s3*s2*s3*s4*s3, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s4*s3*s4*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)( 6, 9)( 7, 11)( 8, 10)( 12, 23)( 13, 22)( 14, 21)
( 15, 29)( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 31, 32)( 33, 36)
( 34, 38)( 35, 37)( 39, 50)( 40, 49)( 41, 48)( 42, 56)( 43, 55)( 44, 54)
( 45, 53)( 46, 52)( 47, 51)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 66, 77)
( 67, 76)( 68, 75)( 69, 83)( 70, 82)( 71, 81)( 72, 80)( 73, 79)( 74, 78)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 93,104)( 94,103)( 95,102)( 96,110)
( 97,109)( 98,108)( 99,107)(100,106)(101,105)(112,113)(114,117)(115,119)
(116,118)(120,131)(121,130)(122,129)(123,137)(124,136)(125,135)(126,134)
(127,133)(128,132)(139,140)(141,144)(142,146)(143,145)(147,158)(148,157)
(149,156)(150,164)(151,163)(152,162)(153,161)(154,160)(155,159)(166,167)
(168,171)(169,173)(170,172)(174,185)(175,184)(176,183)(177,191)(178,190)
(179,189)(180,188)(181,187)(182,186)(193,194)(195,198)(196,200)(197,199)
(201,212)(202,211)(203,210)(204,218)(205,217)(206,216)(207,215)(208,214)
(209,213);
s2 := Sym(218)!( 3, 15)( 4, 17)( 5, 16)( 6, 12)( 7, 14)( 8, 13)( 9, 18)
( 10, 20)( 11, 19)( 21, 26)( 22, 25)( 23, 24)( 27, 29)( 30, 42)( 31, 44)
( 32, 43)( 33, 39)( 34, 41)( 35, 40)( 36, 45)( 37, 47)( 38, 46)( 48, 53)
( 49, 52)( 50, 51)( 54, 56)( 57, 69)( 58, 71)( 59, 70)( 60, 66)( 61, 68)
( 62, 67)( 63, 72)( 64, 74)( 65, 73)( 75, 80)( 76, 79)( 77, 78)( 81, 83)
( 84, 96)( 85, 98)( 86, 97)( 87, 93)( 88, 95)( 89, 94)( 90, 99)( 91,101)
( 92,100)(102,107)(103,106)(104,105)(108,110)(111,123)(112,125)(113,124)
(114,120)(115,122)(116,121)(117,126)(118,128)(119,127)(129,134)(130,133)
(131,132)(135,137)(138,150)(139,152)(140,151)(141,147)(142,149)(143,148)
(144,153)(145,155)(146,154)(156,161)(157,160)(158,159)(162,164)(165,177)
(166,179)(167,178)(168,174)(169,176)(170,175)(171,180)(172,182)(173,181)
(183,188)(184,187)(185,186)(189,191)(192,204)(193,206)(194,205)(195,201)
(196,203)(197,202)(198,207)(199,209)(200,208)(210,215)(211,214)(212,213)
(216,218);
s3 := Sym(218)!( 6, 9)( 7, 10)( 8, 11)( 15, 18)( 16, 19)( 17, 20)( 24, 27)
( 25, 28)( 26, 29)( 33, 36)( 34, 37)( 35, 38)( 42, 45)( 43, 46)( 44, 47)
( 51, 54)( 52, 55)( 53, 56)( 57, 84)( 58, 85)( 59, 86)( 60, 90)( 61, 91)
( 62, 92)( 63, 87)( 64, 88)( 65, 89)( 66, 93)( 67, 94)( 68, 95)( 69, 99)
( 70,100)( 71,101)( 72, 96)( 73, 97)( 74, 98)( 75,102)( 76,103)( 77,104)
( 78,108)( 79,109)( 80,110)( 81,105)( 82,106)( 83,107)(111,165)(112,166)
(113,167)(114,171)(115,172)(116,173)(117,168)(118,169)(119,170)(120,174)
(121,175)(122,176)(123,180)(124,181)(125,182)(126,177)(127,178)(128,179)
(129,183)(130,184)(131,185)(132,189)(133,190)(134,191)(135,186)(136,187)
(137,188)(138,192)(139,193)(140,194)(141,198)(142,199)(143,200)(144,195)
(145,196)(146,197)(147,201)(148,202)(149,203)(150,207)(151,208)(152,209)
(153,204)(154,205)(155,206)(156,210)(157,211)(158,212)(159,216)(160,217)
(161,218)(162,213)(163,214)(164,215);
s4 := Sym(218)!( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)( 8,116)( 9,117)
( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)( 17,125)
( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)( 25,133)
( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)( 33,141)
( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)( 41,149)
( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)( 49,157)
( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,163)( 56,164)( 57,192)
( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)( 65,200)
( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)( 73,208)
( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)( 81,216)
( 82,217)( 83,218)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)( 89,170)
( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)( 96,177)( 97,178)
( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)(104,185)(105,186)
(106,187)(107,188)(108,189)(109,190)(110,191);
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*s4*s3*s2*s3*s4*s3,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s4*s3*s4*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