Polytope of Type {2,8,6,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,8,6,9}*1728
if this polytope has a name.
Group : SmallGroup(1728,17171)
Rank : 5
Schlafli Type : {2,8,6,9}
Number of vertices, edges, etc : 2, 8, 24, 27, 9
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 : {2,4,6,9}*864
3-fold quotients : {2,8,2,9}*576, {2,8,6,3}*576
4-fold quotients : {2,2,6,9}*432
6-fold quotients : {2,4,2,9}*288, {2,4,6,3}*288
9-fold quotients : {2,8,2,3}*192
12-fold quotients : {2,2,2,9}*144, {2,2,6,3}*144
18-fold quotients : {2,4,2,3}*96
36-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 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);;
s2 := ( 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);;
s3 := ( 3, 6)( 4, 8)( 5, 7)( 10, 11)( 12, 26)( 13, 25)( 14, 24)( 15, 23)
( 16, 22)( 17, 21)( 18, 29)( 19, 28)( 20, 27)( 30, 33)( 31, 35)( 32, 34)
( 37, 38)( 39, 53)( 40, 52)( 41, 51)( 42, 50)( 43, 49)( 44, 48)( 45, 56)
( 46, 55)( 47, 54)( 57, 60)( 58, 62)( 59, 61)( 64, 65)( 66, 80)( 67, 79)
( 68, 78)( 69, 77)( 70, 76)( 71, 75)( 72, 83)( 73, 82)( 74, 81)( 84, 87)
( 85, 89)( 86, 88)( 91, 92)( 93,107)( 94,106)( 95,105)( 96,104)( 97,103)
( 98,102)( 99,110)(100,109)(101,108)(111,114)(112,116)(113,115)(118,119)
(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,137)(127,136)
(128,135)(138,141)(139,143)(140,142)(145,146)(147,161)(148,160)(149,159)
(150,158)(151,157)(152,156)(153,164)(154,163)(155,162)(165,168)(166,170)
(167,169)(172,173)(174,188)(175,187)(176,186)(177,185)(178,184)(179,183)
(180,191)(181,190)(182,189)(192,195)(193,197)(194,196)(199,200)(201,215)
(202,214)(203,213)(204,212)(205,211)(206,210)(207,218)(208,217)(209,216);;
s4 := ( 3, 12)( 4, 14)( 5, 13)( 6, 18)( 7, 20)( 8, 19)( 9, 15)( 10, 17)
( 11, 16)( 21, 23)( 24, 29)( 25, 28)( 26, 27)( 30, 39)( 31, 41)( 32, 40)
( 33, 45)( 34, 47)( 35, 46)( 36, 42)( 37, 44)( 38, 43)( 48, 50)( 51, 56)
( 52, 55)( 53, 54)( 57, 66)( 58, 68)( 59, 67)( 60, 72)( 61, 74)( 62, 73)
( 63, 69)( 64, 71)( 65, 70)( 75, 77)( 78, 83)( 79, 82)( 80, 81)( 84, 93)
( 85, 95)( 86, 94)( 87, 99)( 88,101)( 89,100)( 90, 96)( 91, 98)( 92, 97)
(102,104)(105,110)(106,109)(107,108)(111,120)(112,122)(113,121)(114,126)
(115,128)(116,127)(117,123)(118,125)(119,124)(129,131)(132,137)(133,136)
(134,135)(138,147)(139,149)(140,148)(141,153)(142,155)(143,154)(144,150)
(145,152)(146,151)(156,158)(159,164)(160,163)(161,162)(165,174)(166,176)
(167,175)(168,180)(169,182)(170,181)(171,177)(172,179)(173,178)(183,185)
(186,191)(187,190)(188,189)(192,201)(193,203)(194,202)(195,207)(196,209)
(197,208)(198,204)(199,206)(200,205)(210,212)(213,218)(214,217)(215,216);;
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, s4*s2*s3*s2*s3*s4*s2*s3*s2*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,
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)!( 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);
s2 := 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);
s3 := Sym(218)!( 3, 6)( 4, 8)( 5, 7)( 10, 11)( 12, 26)( 13, 25)( 14, 24)
( 15, 23)( 16, 22)( 17, 21)( 18, 29)( 19, 28)( 20, 27)( 30, 33)( 31, 35)
( 32, 34)( 37, 38)( 39, 53)( 40, 52)( 41, 51)( 42, 50)( 43, 49)( 44, 48)
( 45, 56)( 46, 55)( 47, 54)( 57, 60)( 58, 62)( 59, 61)( 64, 65)( 66, 80)
( 67, 79)( 68, 78)( 69, 77)( 70, 76)( 71, 75)( 72, 83)( 73, 82)( 74, 81)
( 84, 87)( 85, 89)( 86, 88)( 91, 92)( 93,107)( 94,106)( 95,105)( 96,104)
( 97,103)( 98,102)( 99,110)(100,109)(101,108)(111,114)(112,116)(113,115)
(118,119)(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,137)
(127,136)(128,135)(138,141)(139,143)(140,142)(145,146)(147,161)(148,160)
(149,159)(150,158)(151,157)(152,156)(153,164)(154,163)(155,162)(165,168)
(166,170)(167,169)(172,173)(174,188)(175,187)(176,186)(177,185)(178,184)
(179,183)(180,191)(181,190)(182,189)(192,195)(193,197)(194,196)(199,200)
(201,215)(202,214)(203,213)(204,212)(205,211)(206,210)(207,218)(208,217)
(209,216);
s4 := Sym(218)!( 3, 12)( 4, 14)( 5, 13)( 6, 18)( 7, 20)( 8, 19)( 9, 15)
( 10, 17)( 11, 16)( 21, 23)( 24, 29)( 25, 28)( 26, 27)( 30, 39)( 31, 41)
( 32, 40)( 33, 45)( 34, 47)( 35, 46)( 36, 42)( 37, 44)( 38, 43)( 48, 50)
( 51, 56)( 52, 55)( 53, 54)( 57, 66)( 58, 68)( 59, 67)( 60, 72)( 61, 74)
( 62, 73)( 63, 69)( 64, 71)( 65, 70)( 75, 77)( 78, 83)( 79, 82)( 80, 81)
( 84, 93)( 85, 95)( 86, 94)( 87, 99)( 88,101)( 89,100)( 90, 96)( 91, 98)
( 92, 97)(102,104)(105,110)(106,109)(107,108)(111,120)(112,122)(113,121)
(114,126)(115,128)(116,127)(117,123)(118,125)(119,124)(129,131)(132,137)
(133,136)(134,135)(138,147)(139,149)(140,148)(141,153)(142,155)(143,154)
(144,150)(145,152)(146,151)(156,158)(159,164)(160,163)(161,162)(165,174)
(166,176)(167,175)(168,180)(169,182)(170,181)(171,177)(172,179)(173,178)
(183,185)(186,191)(187,190)(188,189)(192,201)(193,203)(194,202)(195,207)
(196,209)(197,208)(198,204)(199,206)(200,205)(210,212)(213,218)(214,217)
(215,216);
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,
s4*s2*s3*s2*s3*s4*s2*s3*s2*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,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope