Polytope of Type {4,9,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,9,4}*576b
if this polytope has a name.
Group : SmallGroup(576,8263)
Rank : 4
Schlafli Type : {4,9,4}
Number of vertices, edges, etc : 8, 36, 36, 4
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,9,4,2} of size 1152
Vertex Figure Of :
{2,4,9,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,9,4}*288
3-fold quotients : {4,3,4}*192b
4-fold quotients : {2,9,4}*144
6-fold quotients : {4,3,4}*96
12-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,9,4}*1152, {4,9,4}*1152, {4,18,4}*1152d, {4,18,4}*1152e
3-fold covers : {4,27,4}*1728b, {12,9,4}*1728
Permutation Representation (GAP) :
s0 := ( 1,153)( 2,154)( 3,155)( 4,156)( 5,157)( 6,158)( 7,159)( 8,160)
( 9,145)( 10,146)( 11,147)( 12,148)( 13,149)( 14,150)( 15,151)( 16,152)
( 17,169)( 18,170)( 19,171)( 20,172)( 21,173)( 22,174)( 23,175)( 24,176)
( 25,161)( 26,162)( 27,163)( 28,164)( 29,165)( 30,166)( 31,167)( 32,168)
( 33,185)( 34,186)( 35,187)( 36,188)( 37,189)( 38,190)( 39,191)( 40,192)
( 41,177)( 42,178)( 43,179)( 44,180)( 45,181)( 46,182)( 47,183)( 48,184)
( 49,201)( 50,202)( 51,203)( 52,204)( 53,205)( 54,206)( 55,207)( 56,208)
( 57,193)( 58,194)( 59,195)( 60,196)( 61,197)( 62,198)( 63,199)( 64,200)
( 65,217)( 66,218)( 67,219)( 68,220)( 69,221)( 70,222)( 71,223)( 72,224)
( 73,209)( 74,210)( 75,211)( 76,212)( 77,213)( 78,214)( 79,215)( 80,216)
( 81,233)( 82,234)( 83,235)( 84,236)( 85,237)( 86,238)( 87,239)( 88,240)
( 89,225)( 90,226)( 91,227)( 92,228)( 93,229)( 94,230)( 95,231)( 96,232)
( 97,249)( 98,250)( 99,251)(100,252)(101,253)(102,254)(103,255)(104,256)
(105,241)(106,242)(107,243)(108,244)(109,245)(110,246)(111,247)(112,248)
(113,265)(114,266)(115,267)(116,268)(117,269)(118,270)(119,271)(120,272)
(121,257)(122,258)(123,259)(124,260)(125,261)(126,262)(127,263)(128,264)
(129,281)(130,282)(131,283)(132,284)(133,285)(134,286)(135,287)(136,288)
(137,273)(138,274)(139,275)(140,276)(141,277)(142,278)(143,279)(144,280);;
s1 := ( 1, 49)( 2, 52)( 3, 51)( 4, 50)( 5, 57)( 6, 60)( 7, 59)( 8, 58)
( 9, 53)( 10, 56)( 11, 55)( 12, 54)( 13, 61)( 14, 64)( 15, 63)( 16, 62)
( 17, 81)( 18, 84)( 19, 83)( 20, 82)( 21, 89)( 22, 92)( 23, 91)( 24, 90)
( 25, 85)( 26, 88)( 27, 87)( 28, 86)( 29, 93)( 30, 96)( 31, 95)( 32, 94)
( 33, 65)( 34, 68)( 35, 67)( 36, 66)( 37, 73)( 38, 76)( 39, 75)( 40, 74)
( 41, 69)( 42, 72)( 43, 71)( 44, 70)( 45, 77)( 46, 80)( 47, 79)( 48, 78)
( 97,113)( 98,116)( 99,115)(100,114)(101,121)(102,124)(103,123)(104,122)
(105,117)(106,120)(107,119)(108,118)(109,125)(110,128)(111,127)(112,126)
(130,132)(133,137)(134,140)(135,139)(136,138)(142,144)(145,193)(146,196)
(147,195)(148,194)(149,201)(150,204)(151,203)(152,202)(153,197)(154,200)
(155,199)(156,198)(157,205)(158,208)(159,207)(160,206)(161,225)(162,228)
(163,227)(164,226)(165,233)(166,236)(167,235)(168,234)(169,229)(170,232)
(171,231)(172,230)(173,237)(174,240)(175,239)(176,238)(177,209)(178,212)
(179,211)(180,210)(181,217)(182,220)(183,219)(184,218)(185,213)(186,216)
(187,215)(188,214)(189,221)(190,224)(191,223)(192,222)(241,257)(242,260)
(243,259)(244,258)(245,265)(246,268)(247,267)(248,266)(249,261)(250,264)
(251,263)(252,262)(253,269)(254,272)(255,271)(256,270)(274,276)(277,281)
(278,284)(279,283)(280,282)(286,288);;
s2 := ( 3, 4)( 5, 13)( 6, 14)( 7, 16)( 8, 15)( 11, 12)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 45)( 22, 46)( 23, 48)( 24, 47)( 25, 41)( 26, 42)
( 27, 44)( 28, 43)( 29, 37)( 30, 38)( 31, 40)( 32, 39)( 49,113)( 50,114)
( 51,116)( 52,115)( 53,125)( 54,126)( 55,128)( 56,127)( 57,121)( 58,122)
( 59,124)( 60,123)( 61,117)( 62,118)( 63,120)( 64,119)( 65, 97)( 66, 98)
( 67,100)( 68, 99)( 69,109)( 70,110)( 71,112)( 72,111)( 73,105)( 74,106)
( 75,108)( 76,107)( 77,101)( 78,102)( 79,104)( 80,103)( 81,129)( 82,130)
( 83,132)( 84,131)( 85,141)( 86,142)( 87,144)( 88,143)( 89,137)( 90,138)
( 91,140)( 92,139)( 93,133)( 94,134)( 95,136)( 96,135)(147,148)(149,157)
(150,158)(151,160)(152,159)(155,156)(161,177)(162,178)(163,180)(164,179)
(165,189)(166,190)(167,192)(168,191)(169,185)(170,186)(171,188)(172,187)
(173,181)(174,182)(175,184)(176,183)(193,257)(194,258)(195,260)(196,259)
(197,269)(198,270)(199,272)(200,271)(201,265)(202,266)(203,268)(204,267)
(205,261)(206,262)(207,264)(208,263)(209,241)(210,242)(211,244)(212,243)
(213,253)(214,254)(215,256)(216,255)(217,249)(218,250)(219,252)(220,251)
(221,245)(222,246)(223,248)(224,247)(225,273)(226,274)(227,276)(228,275)
(229,285)(230,286)(231,288)(232,287)(233,281)(234,282)(235,284)(236,283)
(237,277)(238,278)(239,280)(240,279);;
s3 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288);;
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,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*s2*s1*s2,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
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(288)!( 1,153)( 2,154)( 3,155)( 4,156)( 5,157)( 6,158)( 7,159)
( 8,160)( 9,145)( 10,146)( 11,147)( 12,148)( 13,149)( 14,150)( 15,151)
( 16,152)( 17,169)( 18,170)( 19,171)( 20,172)( 21,173)( 22,174)( 23,175)
( 24,176)( 25,161)( 26,162)( 27,163)( 28,164)( 29,165)( 30,166)( 31,167)
( 32,168)( 33,185)( 34,186)( 35,187)( 36,188)( 37,189)( 38,190)( 39,191)
( 40,192)( 41,177)( 42,178)( 43,179)( 44,180)( 45,181)( 46,182)( 47,183)
( 48,184)( 49,201)( 50,202)( 51,203)( 52,204)( 53,205)( 54,206)( 55,207)
( 56,208)( 57,193)( 58,194)( 59,195)( 60,196)( 61,197)( 62,198)( 63,199)
( 64,200)( 65,217)( 66,218)( 67,219)( 68,220)( 69,221)( 70,222)( 71,223)
( 72,224)( 73,209)( 74,210)( 75,211)( 76,212)( 77,213)( 78,214)( 79,215)
( 80,216)( 81,233)( 82,234)( 83,235)( 84,236)( 85,237)( 86,238)( 87,239)
( 88,240)( 89,225)( 90,226)( 91,227)( 92,228)( 93,229)( 94,230)( 95,231)
( 96,232)( 97,249)( 98,250)( 99,251)(100,252)(101,253)(102,254)(103,255)
(104,256)(105,241)(106,242)(107,243)(108,244)(109,245)(110,246)(111,247)
(112,248)(113,265)(114,266)(115,267)(116,268)(117,269)(118,270)(119,271)
(120,272)(121,257)(122,258)(123,259)(124,260)(125,261)(126,262)(127,263)
(128,264)(129,281)(130,282)(131,283)(132,284)(133,285)(134,286)(135,287)
(136,288)(137,273)(138,274)(139,275)(140,276)(141,277)(142,278)(143,279)
(144,280);
s1 := Sym(288)!( 1, 49)( 2, 52)( 3, 51)( 4, 50)( 5, 57)( 6, 60)( 7, 59)
( 8, 58)( 9, 53)( 10, 56)( 11, 55)( 12, 54)( 13, 61)( 14, 64)( 15, 63)
( 16, 62)( 17, 81)( 18, 84)( 19, 83)( 20, 82)( 21, 89)( 22, 92)( 23, 91)
( 24, 90)( 25, 85)( 26, 88)( 27, 87)( 28, 86)( 29, 93)( 30, 96)( 31, 95)
( 32, 94)( 33, 65)( 34, 68)( 35, 67)( 36, 66)( 37, 73)( 38, 76)( 39, 75)
( 40, 74)( 41, 69)( 42, 72)( 43, 71)( 44, 70)( 45, 77)( 46, 80)( 47, 79)
( 48, 78)( 97,113)( 98,116)( 99,115)(100,114)(101,121)(102,124)(103,123)
(104,122)(105,117)(106,120)(107,119)(108,118)(109,125)(110,128)(111,127)
(112,126)(130,132)(133,137)(134,140)(135,139)(136,138)(142,144)(145,193)
(146,196)(147,195)(148,194)(149,201)(150,204)(151,203)(152,202)(153,197)
(154,200)(155,199)(156,198)(157,205)(158,208)(159,207)(160,206)(161,225)
(162,228)(163,227)(164,226)(165,233)(166,236)(167,235)(168,234)(169,229)
(170,232)(171,231)(172,230)(173,237)(174,240)(175,239)(176,238)(177,209)
(178,212)(179,211)(180,210)(181,217)(182,220)(183,219)(184,218)(185,213)
(186,216)(187,215)(188,214)(189,221)(190,224)(191,223)(192,222)(241,257)
(242,260)(243,259)(244,258)(245,265)(246,268)(247,267)(248,266)(249,261)
(250,264)(251,263)(252,262)(253,269)(254,272)(255,271)(256,270)(274,276)
(277,281)(278,284)(279,283)(280,282)(286,288);
s2 := Sym(288)!( 3, 4)( 5, 13)( 6, 14)( 7, 16)( 8, 15)( 11, 12)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 45)( 22, 46)( 23, 48)( 24, 47)( 25, 41)
( 26, 42)( 27, 44)( 28, 43)( 29, 37)( 30, 38)( 31, 40)( 32, 39)( 49,113)
( 50,114)( 51,116)( 52,115)( 53,125)( 54,126)( 55,128)( 56,127)( 57,121)
( 58,122)( 59,124)( 60,123)( 61,117)( 62,118)( 63,120)( 64,119)( 65, 97)
( 66, 98)( 67,100)( 68, 99)( 69,109)( 70,110)( 71,112)( 72,111)( 73,105)
( 74,106)( 75,108)( 76,107)( 77,101)( 78,102)( 79,104)( 80,103)( 81,129)
( 82,130)( 83,132)( 84,131)( 85,141)( 86,142)( 87,144)( 88,143)( 89,137)
( 90,138)( 91,140)( 92,139)( 93,133)( 94,134)( 95,136)( 96,135)(147,148)
(149,157)(150,158)(151,160)(152,159)(155,156)(161,177)(162,178)(163,180)
(164,179)(165,189)(166,190)(167,192)(168,191)(169,185)(170,186)(171,188)
(172,187)(173,181)(174,182)(175,184)(176,183)(193,257)(194,258)(195,260)
(196,259)(197,269)(198,270)(199,272)(200,271)(201,265)(202,266)(203,268)
(204,267)(205,261)(206,262)(207,264)(208,263)(209,241)(210,242)(211,244)
(212,243)(213,253)(214,254)(215,256)(216,255)(217,249)(218,250)(219,252)
(220,251)(221,245)(222,246)(223,248)(224,247)(225,273)(226,274)(227,276)
(228,275)(229,285)(230,286)(231,288)(232,287)(233,281)(234,282)(235,284)
(236,283)(237,277)(238,278)(239,280)(240,279);
s3 := Sym(288)!( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288);
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, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s2*s1*s3*s2*s3*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope