Polytope of Type {8,4,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,4,9}*1152
if this polytope has a name.
Group : SmallGroup(1152,154366)
Rank : 4
Schlafli Type : {8,4,9}
Number of vertices, edges, etc : 8, 32, 36, 18
Order of s0s1s2s3 : 72
Order of s0s1s2s3s2s1 : 2
Special Properties :
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 : {4,4,9}*576b
3-fold quotients : {8,4,3}*384
4-fold quotients : {8,2,9}*288, {2,4,9}*288
6-fold quotients : {4,4,3}*192b
8-fold quotients : {4,2,9}*144, {2,4,9}*144
12-fold quotients : {8,2,3}*96, {2,4,3}*96
16-fold quotients : {2,2,9}*72
24-fold quotients : {4,2,3}*48, {2,4,3}*48
48-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 73,109)( 74,110)( 75,111)( 76,112)( 77,113)( 78,114)( 79,115)( 80,116)
( 81,117)( 82,118)( 83,119)( 84,120)( 85,121)( 86,122)( 87,123)( 88,124)
( 89,125)( 90,126)( 91,127)( 92,128)( 93,129)( 94,130)( 95,131)( 96,132)
( 97,133)( 98,134)( 99,135)(100,136)(101,137)(102,138)(103,139)(104,140)
(105,141)(106,142)(107,143)(108,144)(145,217)(146,218)(147,219)(148,220)
(149,221)(150,222)(151,223)(152,224)(153,225)(154,226)(155,227)(156,228)
(157,229)(158,230)(159,231)(160,232)(161,233)(162,234)(163,235)(164,236)
(165,237)(166,238)(167,239)(168,240)(169,241)(170,242)(171,243)(172,244)
(173,245)(174,246)(175,247)(176,248)(177,249)(178,250)(179,251)(180,252)
(181,253)(182,254)(183,255)(184,256)(185,257)(186,258)(187,259)(188,260)
(189,261)(190,262)(191,263)(192,264)(193,265)(194,266)(195,267)(196,268)
(197,269)(198,270)(199,271)(200,272)(201,273)(202,274)(203,275)(204,276)
(205,277)(206,278)(207,279)(208,280)(209,281)(210,282)(211,283)(212,284)
(213,285)(214,286)(215,287)(216,288);;
s1 := ( 1,147)( 2,148)( 3,145)( 4,146)( 5,151)( 6,152)( 7,149)( 8,150)
( 9,155)( 10,156)( 11,153)( 12,154)( 13,159)( 14,160)( 15,157)( 16,158)
( 17,163)( 18,164)( 19,161)( 20,162)( 21,167)( 22,168)( 23,165)( 24,166)
( 25,171)( 26,172)( 27,169)( 28,170)( 29,175)( 30,176)( 31,173)( 32,174)
( 33,179)( 34,180)( 35,177)( 36,178)( 37,183)( 38,184)( 39,181)( 40,182)
( 41,187)( 42,188)( 43,185)( 44,186)( 45,191)( 46,192)( 47,189)( 48,190)
( 49,195)( 50,196)( 51,193)( 52,194)( 53,199)( 54,200)( 55,197)( 56,198)
( 57,203)( 58,204)( 59,201)( 60,202)( 61,207)( 62,208)( 63,205)( 64,206)
( 65,211)( 66,212)( 67,209)( 68,210)( 69,215)( 70,216)( 71,213)( 72,214)
( 73,255)( 74,256)( 75,253)( 76,254)( 77,259)( 78,260)( 79,257)( 80,258)
( 81,263)( 82,264)( 83,261)( 84,262)( 85,267)( 86,268)( 87,265)( 88,266)
( 89,271)( 90,272)( 91,269)( 92,270)( 93,275)( 94,276)( 95,273)( 96,274)
( 97,279)( 98,280)( 99,277)(100,278)(101,283)(102,284)(103,281)(104,282)
(105,287)(106,288)(107,285)(108,286)(109,219)(110,220)(111,217)(112,218)
(113,223)(114,224)(115,221)(116,222)(117,227)(118,228)(119,225)(120,226)
(121,231)(122,232)(123,229)(124,230)(125,235)(126,236)(127,233)(128,234)
(129,239)(130,240)(131,237)(132,238)(133,243)(134,244)(135,241)(136,242)
(137,247)(138,248)(139,245)(140,246)(141,251)(142,252)(143,249)(144,250);;
s2 := ( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 13, 33)( 14, 34)( 15, 36)
( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)( 23, 28)
( 24, 27)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 69)( 50, 70)
( 51, 72)( 52, 71)( 53, 65)( 54, 66)( 55, 68)( 56, 67)( 57, 61)( 58, 62)
( 59, 64)( 60, 63)( 75, 76)( 77, 81)( 78, 82)( 79, 84)( 80, 83)( 85,105)
( 86,106)( 87,108)( 88,107)( 89,101)( 90,102)( 91,104)( 92,103)( 93, 97)
( 94, 98)( 95,100)( 96, 99)(111,112)(113,117)(114,118)(115,120)(116,119)
(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)(127,140)(128,139)
(129,133)(130,134)(131,136)(132,135)(147,148)(149,153)(150,154)(151,156)
(152,155)(157,177)(158,178)(159,180)(160,179)(161,173)(162,174)(163,176)
(164,175)(165,169)(166,170)(167,172)(168,171)(183,184)(185,189)(186,190)
(187,192)(188,191)(193,213)(194,214)(195,216)(196,215)(197,209)(198,210)
(199,212)(200,211)(201,205)(202,206)(203,208)(204,207)(219,220)(221,225)
(222,226)(223,228)(224,227)(229,249)(230,250)(231,252)(232,251)(233,245)
(234,246)(235,248)(236,247)(237,241)(238,242)(239,244)(240,243)(255,256)
(257,261)(258,262)(259,264)(260,263)(265,285)(266,286)(267,288)(268,287)
(269,281)(270,282)(271,284)(272,283)(273,277)(274,278)(275,280)(276,279);;
s3 := ( 1, 13)( 2, 16)( 3, 15)( 4, 14)( 5, 21)( 6, 24)( 7, 23)( 8, 22)
( 9, 17)( 10, 20)( 11, 19)( 12, 18)( 25, 33)( 26, 36)( 27, 35)( 28, 34)
( 30, 32)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 57)( 42, 60)( 43, 59)
( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)( 61, 69)( 62, 72)( 63, 71)
( 64, 70)( 66, 68)( 73, 85)( 74, 88)( 75, 87)( 76, 86)( 77, 93)( 78, 96)
( 79, 95)( 80, 94)( 81, 89)( 82, 92)( 83, 91)( 84, 90)( 97,105)( 98,108)
( 99,107)(100,106)(102,104)(109,121)(110,124)(111,123)(112,122)(113,129)
(114,132)(115,131)(116,130)(117,125)(118,128)(119,127)(120,126)(133,141)
(134,144)(135,143)(136,142)(138,140)(145,157)(146,160)(147,159)(148,158)
(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)(155,163)(156,162)
(169,177)(170,180)(171,179)(172,178)(174,176)(181,193)(182,196)(183,195)
(184,194)(185,201)(186,204)(187,203)(188,202)(189,197)(190,200)(191,199)
(192,198)(205,213)(206,216)(207,215)(208,214)(210,212)(217,229)(218,232)
(219,231)(220,230)(221,237)(222,240)(223,239)(224,238)(225,233)(226,236)
(227,235)(228,234)(241,249)(242,252)(243,251)(244,250)(246,248)(253,265)
(254,268)(255,267)(256,266)(257,273)(258,276)(259,275)(260,274)(261,269)
(262,272)(263,271)(264,270)(277,285)(278,288)(279,287)(280,286)(282,284);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(288)!( 73,109)( 74,110)( 75,111)( 76,112)( 77,113)( 78,114)( 79,115)
( 80,116)( 81,117)( 82,118)( 83,119)( 84,120)( 85,121)( 86,122)( 87,123)
( 88,124)( 89,125)( 90,126)( 91,127)( 92,128)( 93,129)( 94,130)( 95,131)
( 96,132)( 97,133)( 98,134)( 99,135)(100,136)(101,137)(102,138)(103,139)
(104,140)(105,141)(106,142)(107,143)(108,144)(145,217)(146,218)(147,219)
(148,220)(149,221)(150,222)(151,223)(152,224)(153,225)(154,226)(155,227)
(156,228)(157,229)(158,230)(159,231)(160,232)(161,233)(162,234)(163,235)
(164,236)(165,237)(166,238)(167,239)(168,240)(169,241)(170,242)(171,243)
(172,244)(173,245)(174,246)(175,247)(176,248)(177,249)(178,250)(179,251)
(180,252)(181,253)(182,254)(183,255)(184,256)(185,257)(186,258)(187,259)
(188,260)(189,261)(190,262)(191,263)(192,264)(193,265)(194,266)(195,267)
(196,268)(197,269)(198,270)(199,271)(200,272)(201,273)(202,274)(203,275)
(204,276)(205,277)(206,278)(207,279)(208,280)(209,281)(210,282)(211,283)
(212,284)(213,285)(214,286)(215,287)(216,288);
s1 := Sym(288)!( 1,147)( 2,148)( 3,145)( 4,146)( 5,151)( 6,152)( 7,149)
( 8,150)( 9,155)( 10,156)( 11,153)( 12,154)( 13,159)( 14,160)( 15,157)
( 16,158)( 17,163)( 18,164)( 19,161)( 20,162)( 21,167)( 22,168)( 23,165)
( 24,166)( 25,171)( 26,172)( 27,169)( 28,170)( 29,175)( 30,176)( 31,173)
( 32,174)( 33,179)( 34,180)( 35,177)( 36,178)( 37,183)( 38,184)( 39,181)
( 40,182)( 41,187)( 42,188)( 43,185)( 44,186)( 45,191)( 46,192)( 47,189)
( 48,190)( 49,195)( 50,196)( 51,193)( 52,194)( 53,199)( 54,200)( 55,197)
( 56,198)( 57,203)( 58,204)( 59,201)( 60,202)( 61,207)( 62,208)( 63,205)
( 64,206)( 65,211)( 66,212)( 67,209)( 68,210)( 69,215)( 70,216)( 71,213)
( 72,214)( 73,255)( 74,256)( 75,253)( 76,254)( 77,259)( 78,260)( 79,257)
( 80,258)( 81,263)( 82,264)( 83,261)( 84,262)( 85,267)( 86,268)( 87,265)
( 88,266)( 89,271)( 90,272)( 91,269)( 92,270)( 93,275)( 94,276)( 95,273)
( 96,274)( 97,279)( 98,280)( 99,277)(100,278)(101,283)(102,284)(103,281)
(104,282)(105,287)(106,288)(107,285)(108,286)(109,219)(110,220)(111,217)
(112,218)(113,223)(114,224)(115,221)(116,222)(117,227)(118,228)(119,225)
(120,226)(121,231)(122,232)(123,229)(124,230)(125,235)(126,236)(127,233)
(128,234)(129,239)(130,240)(131,237)(132,238)(133,243)(134,244)(135,241)
(136,242)(137,247)(138,248)(139,245)(140,246)(141,251)(142,252)(143,249)
(144,250);
s2 := Sym(288)!( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 13, 33)( 14, 34)
( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)
( 23, 28)( 24, 27)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 69)
( 50, 70)( 51, 72)( 52, 71)( 53, 65)( 54, 66)( 55, 68)( 56, 67)( 57, 61)
( 58, 62)( 59, 64)( 60, 63)( 75, 76)( 77, 81)( 78, 82)( 79, 84)( 80, 83)
( 85,105)( 86,106)( 87,108)( 88,107)( 89,101)( 90,102)( 91,104)( 92,103)
( 93, 97)( 94, 98)( 95,100)( 96, 99)(111,112)(113,117)(114,118)(115,120)
(116,119)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)(127,140)
(128,139)(129,133)(130,134)(131,136)(132,135)(147,148)(149,153)(150,154)
(151,156)(152,155)(157,177)(158,178)(159,180)(160,179)(161,173)(162,174)
(163,176)(164,175)(165,169)(166,170)(167,172)(168,171)(183,184)(185,189)
(186,190)(187,192)(188,191)(193,213)(194,214)(195,216)(196,215)(197,209)
(198,210)(199,212)(200,211)(201,205)(202,206)(203,208)(204,207)(219,220)
(221,225)(222,226)(223,228)(224,227)(229,249)(230,250)(231,252)(232,251)
(233,245)(234,246)(235,248)(236,247)(237,241)(238,242)(239,244)(240,243)
(255,256)(257,261)(258,262)(259,264)(260,263)(265,285)(266,286)(267,288)
(268,287)(269,281)(270,282)(271,284)(272,283)(273,277)(274,278)(275,280)
(276,279);
s3 := Sym(288)!( 1, 13)( 2, 16)( 3, 15)( 4, 14)( 5, 21)( 6, 24)( 7, 23)
( 8, 22)( 9, 17)( 10, 20)( 11, 19)( 12, 18)( 25, 33)( 26, 36)( 27, 35)
( 28, 34)( 30, 32)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 57)( 42, 60)
( 43, 59)( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)( 61, 69)( 62, 72)
( 63, 71)( 64, 70)( 66, 68)( 73, 85)( 74, 88)( 75, 87)( 76, 86)( 77, 93)
( 78, 96)( 79, 95)( 80, 94)( 81, 89)( 82, 92)( 83, 91)( 84, 90)( 97,105)
( 98,108)( 99,107)(100,106)(102,104)(109,121)(110,124)(111,123)(112,122)
(113,129)(114,132)(115,131)(116,130)(117,125)(118,128)(119,127)(120,126)
(133,141)(134,144)(135,143)(136,142)(138,140)(145,157)(146,160)(147,159)
(148,158)(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)(155,163)
(156,162)(169,177)(170,180)(171,179)(172,178)(174,176)(181,193)(182,196)
(183,195)(184,194)(185,201)(186,204)(187,203)(188,202)(189,197)(190,200)
(191,199)(192,198)(205,213)(206,216)(207,215)(208,214)(210,212)(217,229)
(218,232)(219,231)(220,230)(221,237)(222,240)(223,239)(224,238)(225,233)
(226,236)(227,235)(228,234)(241,249)(242,252)(243,251)(244,250)(246,248)
(253,265)(254,268)(255,267)(256,266)(257,273)(258,276)(259,275)(260,274)
(261,269)(262,272)(263,271)(264,270)(277,285)(278,288)(279,287)(280,286)
(282,284);
poly := sub<Sym(288)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope