Polytope of Type {4,8,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,8,9}*1152
if this polytope has a name.
Group : SmallGroup(1152,154295)
Rank : 4
Schlafli Type : {4,8,9}
Number of vertices, edges, etc : 4, 32, 72, 18
Order of s0s1s2s3 : 36
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, {2,8,9}*576
3-fold quotients : {4,8,3}*384
4-fold quotients : {2,4,9}*288
6-fold quotients : {4,4,3}*192b, {2,8,3}*192
8-fold quotients : {4,2,9}*144, {2,4,9}*144
12-fold quotients : {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 := (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 := ( 1,149)( 2,150)( 3,152)( 4,151)( 5,146)( 6,145)( 7,147)( 8,148)
( 9,157)( 10,158)( 11,160)( 12,159)( 13,154)( 14,153)( 15,155)( 16,156)
( 17,165)( 18,166)( 19,168)( 20,167)( 21,162)( 22,161)( 23,163)( 24,164)
( 25,173)( 26,174)( 27,176)( 28,175)( 29,170)( 30,169)( 31,171)( 32,172)
( 33,181)( 34,182)( 35,184)( 36,183)( 37,178)( 38,177)( 39,179)( 40,180)
( 41,189)( 42,190)( 43,192)( 44,191)( 45,186)( 46,185)( 47,187)( 48,188)
( 49,197)( 50,198)( 51,200)( 52,199)( 53,194)( 54,193)( 55,195)( 56,196)
( 57,205)( 58,206)( 59,208)( 60,207)( 61,202)( 62,201)( 63,203)( 64,204)
( 65,213)( 66,214)( 67,216)( 68,215)( 69,210)( 70,209)( 71,211)( 72,212)
( 73,221)( 74,222)( 75,224)( 76,223)( 77,218)( 78,217)( 79,219)( 80,220)
( 81,229)( 82,230)( 83,232)( 84,231)( 85,226)( 86,225)( 87,227)( 88,228)
( 89,237)( 90,238)( 91,240)( 92,239)( 93,234)( 94,233)( 95,235)( 96,236)
( 97,245)( 98,246)( 99,248)(100,247)(101,242)(102,241)(103,243)(104,244)
(105,253)(106,254)(107,256)(108,255)(109,250)(110,249)(111,251)(112,252)
(113,261)(114,262)(115,264)(116,263)(117,258)(118,257)(119,259)(120,260)
(121,269)(122,270)(123,272)(124,271)(125,266)(126,265)(127,267)(128,268)
(129,277)(130,278)(131,280)(132,279)(133,274)(134,273)(135,275)(136,276)
(137,285)(138,286)(139,288)(140,287)(141,282)(142,281)(143,283)(144,284);;
s2 := ( 3, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)( 13, 23)
( 14, 24)( 15, 21)( 16, 22)( 25, 65)( 26, 66)( 27, 68)( 28, 67)( 29, 71)
( 30, 72)( 31, 69)( 32, 70)( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 37, 63)
( 38, 64)( 39, 61)( 40, 62)( 41, 49)( 42, 50)( 43, 52)( 44, 51)( 45, 55)
( 46, 56)( 47, 53)( 48, 54)( 75, 76)( 77, 79)( 78, 80)( 81, 89)( 82, 90)
( 83, 92)( 84, 91)( 85, 95)( 86, 96)( 87, 93)( 88, 94)( 97,137)( 98,138)
( 99,140)(100,139)(101,143)(102,144)(103,141)(104,142)(105,129)(106,130)
(107,132)(108,131)(109,135)(110,136)(111,133)(112,134)(113,121)(114,122)
(115,124)(116,123)(117,127)(118,128)(119,125)(120,126)(145,146)(149,152)
(150,151)(153,162)(154,161)(155,163)(156,164)(157,168)(158,167)(159,166)
(160,165)(169,210)(170,209)(171,211)(172,212)(173,216)(174,215)(175,214)
(176,213)(177,202)(178,201)(179,203)(180,204)(181,208)(182,207)(183,206)
(184,205)(185,194)(186,193)(187,195)(188,196)(189,200)(190,199)(191,198)
(192,197)(217,218)(221,224)(222,223)(225,234)(226,233)(227,235)(228,236)
(229,240)(230,239)(231,238)(232,237)(241,282)(242,281)(243,283)(244,284)
(245,288)(246,287)(247,286)(248,285)(249,274)(250,273)(251,275)(252,276)
(253,280)(254,279)(255,278)(256,277)(257,266)(258,265)(259,267)(260,268)
(261,272)(262,271)(263,270)(264,269);;
s3 := ( 1, 25)( 2, 26)( 3, 31)( 4, 32)( 5, 30)( 6, 29)( 7, 27)( 8, 28)
( 9, 41)( 10, 42)( 11, 47)( 12, 48)( 13, 46)( 14, 45)( 15, 43)( 16, 44)
( 17, 33)( 18, 34)( 19, 39)( 20, 40)( 21, 38)( 22, 37)( 23, 35)( 24, 36)
( 49, 65)( 50, 66)( 51, 71)( 52, 72)( 53, 70)( 54, 69)( 55, 67)( 56, 68)
( 59, 63)( 60, 64)( 61, 62)( 73, 97)( 74, 98)( 75,103)( 76,104)( 77,102)
( 78,101)( 79, 99)( 80,100)( 81,113)( 82,114)( 83,119)( 84,120)( 85,118)
( 86,117)( 87,115)( 88,116)( 89,105)( 90,106)( 91,111)( 92,112)( 93,110)
( 94,109)( 95,107)( 96,108)(121,137)(122,138)(123,143)(124,144)(125,142)
(126,141)(127,139)(128,140)(131,135)(132,136)(133,134)(145,170)(146,169)
(147,176)(148,175)(149,173)(150,174)(151,172)(152,171)(153,186)(154,185)
(155,192)(156,191)(157,189)(158,190)(159,188)(160,187)(161,178)(162,177)
(163,184)(164,183)(165,181)(166,182)(167,180)(168,179)(193,210)(194,209)
(195,216)(196,215)(197,213)(198,214)(199,212)(200,211)(201,202)(203,208)
(204,207)(217,242)(218,241)(219,248)(220,247)(221,245)(222,246)(223,244)
(224,243)(225,258)(226,257)(227,264)(228,263)(229,261)(230,262)(231,260)
(232,259)(233,250)(234,249)(235,256)(236,255)(237,253)(238,254)(239,252)
(240,251)(265,282)(266,281)(267,288)(268,287)(269,285)(270,286)(271,284)
(272,283)(273,274)(275,280)(276,279);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2,
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)!(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)!( 1,149)( 2,150)( 3,152)( 4,151)( 5,146)( 6,145)( 7,147)
( 8,148)( 9,157)( 10,158)( 11,160)( 12,159)( 13,154)( 14,153)( 15,155)
( 16,156)( 17,165)( 18,166)( 19,168)( 20,167)( 21,162)( 22,161)( 23,163)
( 24,164)( 25,173)( 26,174)( 27,176)( 28,175)( 29,170)( 30,169)( 31,171)
( 32,172)( 33,181)( 34,182)( 35,184)( 36,183)( 37,178)( 38,177)( 39,179)
( 40,180)( 41,189)( 42,190)( 43,192)( 44,191)( 45,186)( 46,185)( 47,187)
( 48,188)( 49,197)( 50,198)( 51,200)( 52,199)( 53,194)( 54,193)( 55,195)
( 56,196)( 57,205)( 58,206)( 59,208)( 60,207)( 61,202)( 62,201)( 63,203)
( 64,204)( 65,213)( 66,214)( 67,216)( 68,215)( 69,210)( 70,209)( 71,211)
( 72,212)( 73,221)( 74,222)( 75,224)( 76,223)( 77,218)( 78,217)( 79,219)
( 80,220)( 81,229)( 82,230)( 83,232)( 84,231)( 85,226)( 86,225)( 87,227)
( 88,228)( 89,237)( 90,238)( 91,240)( 92,239)( 93,234)( 94,233)( 95,235)
( 96,236)( 97,245)( 98,246)( 99,248)(100,247)(101,242)(102,241)(103,243)
(104,244)(105,253)(106,254)(107,256)(108,255)(109,250)(110,249)(111,251)
(112,252)(113,261)(114,262)(115,264)(116,263)(117,258)(118,257)(119,259)
(120,260)(121,269)(122,270)(123,272)(124,271)(125,266)(126,265)(127,267)
(128,268)(129,277)(130,278)(131,280)(132,279)(133,274)(134,273)(135,275)
(136,276)(137,285)(138,286)(139,288)(140,287)(141,282)(142,281)(143,283)
(144,284);
s2 := Sym(288)!( 3, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)
( 13, 23)( 14, 24)( 15, 21)( 16, 22)( 25, 65)( 26, 66)( 27, 68)( 28, 67)
( 29, 71)( 30, 72)( 31, 69)( 32, 70)( 33, 57)( 34, 58)( 35, 60)( 36, 59)
( 37, 63)( 38, 64)( 39, 61)( 40, 62)( 41, 49)( 42, 50)( 43, 52)( 44, 51)
( 45, 55)( 46, 56)( 47, 53)( 48, 54)( 75, 76)( 77, 79)( 78, 80)( 81, 89)
( 82, 90)( 83, 92)( 84, 91)( 85, 95)( 86, 96)( 87, 93)( 88, 94)( 97,137)
( 98,138)( 99,140)(100,139)(101,143)(102,144)(103,141)(104,142)(105,129)
(106,130)(107,132)(108,131)(109,135)(110,136)(111,133)(112,134)(113,121)
(114,122)(115,124)(116,123)(117,127)(118,128)(119,125)(120,126)(145,146)
(149,152)(150,151)(153,162)(154,161)(155,163)(156,164)(157,168)(158,167)
(159,166)(160,165)(169,210)(170,209)(171,211)(172,212)(173,216)(174,215)
(175,214)(176,213)(177,202)(178,201)(179,203)(180,204)(181,208)(182,207)
(183,206)(184,205)(185,194)(186,193)(187,195)(188,196)(189,200)(190,199)
(191,198)(192,197)(217,218)(221,224)(222,223)(225,234)(226,233)(227,235)
(228,236)(229,240)(230,239)(231,238)(232,237)(241,282)(242,281)(243,283)
(244,284)(245,288)(246,287)(247,286)(248,285)(249,274)(250,273)(251,275)
(252,276)(253,280)(254,279)(255,278)(256,277)(257,266)(258,265)(259,267)
(260,268)(261,272)(262,271)(263,270)(264,269);
s3 := Sym(288)!( 1, 25)( 2, 26)( 3, 31)( 4, 32)( 5, 30)( 6, 29)( 7, 27)
( 8, 28)( 9, 41)( 10, 42)( 11, 47)( 12, 48)( 13, 46)( 14, 45)( 15, 43)
( 16, 44)( 17, 33)( 18, 34)( 19, 39)( 20, 40)( 21, 38)( 22, 37)( 23, 35)
( 24, 36)( 49, 65)( 50, 66)( 51, 71)( 52, 72)( 53, 70)( 54, 69)( 55, 67)
( 56, 68)( 59, 63)( 60, 64)( 61, 62)( 73, 97)( 74, 98)( 75,103)( 76,104)
( 77,102)( 78,101)( 79, 99)( 80,100)( 81,113)( 82,114)( 83,119)( 84,120)
( 85,118)( 86,117)( 87,115)( 88,116)( 89,105)( 90,106)( 91,111)( 92,112)
( 93,110)( 94,109)( 95,107)( 96,108)(121,137)(122,138)(123,143)(124,144)
(125,142)(126,141)(127,139)(128,140)(131,135)(132,136)(133,134)(145,170)
(146,169)(147,176)(148,175)(149,173)(150,174)(151,172)(152,171)(153,186)
(154,185)(155,192)(156,191)(157,189)(158,190)(159,188)(160,187)(161,178)
(162,177)(163,184)(164,183)(165,181)(166,182)(167,180)(168,179)(193,210)
(194,209)(195,216)(196,215)(197,213)(198,214)(199,212)(200,211)(201,202)
(203,208)(204,207)(217,242)(218,241)(219,248)(220,247)(221,245)(222,246)
(223,244)(224,243)(225,258)(226,257)(227,264)(228,263)(229,261)(230,262)
(231,260)(232,259)(233,250)(234,249)(235,256)(236,255)(237,253)(238,254)
(239,252)(240,251)(265,282)(266,281)(267,288)(268,287)(269,285)(270,286)
(271,284)(272,283)(273,274)(275,280)(276,279);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope