Polytope of Type {8,6,3,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6,3,4}*1152
if this polytope has a name.
Group : SmallGroup(1152,156074)
Rank : 5
Schlafli Type : {8,6,3,4}
Number of vertices, edges, etc : 8, 24, 9, 6, 4
Order of s0s1s2s3s4 : 24
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Universal
Non-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 : {4,6,3,4}*576
3-fold quotients : {8,2,3,4}*384
4-fold quotients : {2,6,3,4}*288
6-fold quotients : {4,2,3,4}*192
12-fold quotients : {2,2,3,4}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,145)( 2,146)( 3,147)( 4,148)( 5,149)( 6,150)( 7,151)( 8,152)
( 9,153)( 10,154)( 11,155)( 12,156)( 13,157)( 14,158)( 15,159)( 16,160)
( 17,161)( 18,162)( 19,163)( 20,164)( 21,165)( 22,166)( 23,167)( 24,168)
( 25,169)( 26,170)( 27,171)( 28,172)( 29,173)( 30,174)( 31,175)( 32,176)
( 33,177)( 34,178)( 35,179)( 36,180)( 37,181)( 38,182)( 39,183)( 40,184)
( 41,185)( 42,186)( 43,187)( 44,188)( 45,189)( 46,190)( 47,191)( 48,192)
( 49,193)( 50,194)( 51,195)( 52,196)( 53,197)( 54,198)( 55,199)( 56,200)
( 57,201)( 58,202)( 59,203)( 60,204)( 61,205)( 62,206)( 63,207)( 64,208)
( 65,209)( 66,210)( 67,211)( 68,212)( 69,213)( 70,214)( 71,215)( 72,216)
( 73,253)( 74,254)( 75,255)( 76,256)( 77,257)( 78,258)( 79,259)( 80,260)
( 81,261)( 82,262)( 83,263)( 84,264)( 85,265)( 86,266)( 87,267)( 88,268)
( 89,269)( 90,270)( 91,271)( 92,272)( 93,273)( 94,274)( 95,275)( 96,276)
( 97,277)( 98,278)( 99,279)(100,280)(101,281)(102,282)(103,283)(104,284)
(105,285)(106,286)(107,287)(108,288)(109,217)(110,218)(111,219)(112,220)
(113,221)(114,222)(115,223)(116,224)(117,225)(118,226)(119,227)(120,228)
(121,229)(122,230)(123,231)(124,232)(125,233)(126,234)(127,235)(128,236)
(129,237)(130,238)(131,239)(132,240)(133,241)(134,242)(135,243)(136,244)
(137,245)(138,246)(139,247)(140,248)(141,249)(142,250)(143,251)(144,252);;
s1 := ( 5, 9)( 6, 10)( 7, 11)( 8, 12)( 17, 21)( 18, 22)( 19, 23)( 20, 24)
( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 69)( 66, 70)( 67, 71)( 68, 72)
( 73,109)( 74,110)( 75,111)( 76,112)( 77,117)( 78,118)( 79,119)( 80,120)
( 81,113)( 82,114)( 83,115)( 84,116)( 85,121)( 86,122)( 87,123)( 88,124)
( 89,129)( 90,130)( 91,131)( 92,132)( 93,125)( 94,126)( 95,127)( 96,128)
( 97,133)( 98,134)( 99,135)(100,136)(101,141)(102,142)(103,143)(104,144)
(105,137)(106,138)(107,139)(108,140)(145,217)(146,218)(147,219)(148,220)
(149,225)(150,226)(151,227)(152,228)(153,221)(154,222)(155,223)(156,224)
(157,229)(158,230)(159,231)(160,232)(161,237)(162,238)(163,239)(164,240)
(165,233)(166,234)(167,235)(168,236)(169,241)(170,242)(171,243)(172,244)
(173,249)(174,250)(175,251)(176,252)(177,245)(178,246)(179,247)(180,248)
(181,253)(182,254)(183,255)(184,256)(185,261)(186,262)(187,263)(188,264)
(189,257)(190,258)(191,259)(192,260)(193,265)(194,266)(195,267)(196,268)
(197,273)(198,274)(199,275)(200,276)(201,269)(202,270)(203,271)(204,272)
(205,277)(206,278)(207,279)(208,280)(209,285)(210,286)(211,287)(212,288)
(213,281)(214,282)(215,283)(216,284);;
s2 := ( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 11, 12)( 13, 29)( 14, 30)( 15, 32)
( 16, 31)( 17, 25)( 18, 26)( 19, 28)( 20, 27)( 21, 33)( 22, 34)( 23, 36)
( 24, 35)( 37, 41)( 38, 42)( 39, 44)( 40, 43)( 47, 48)( 49, 65)( 50, 66)
( 51, 68)( 52, 67)( 53, 61)( 54, 62)( 55, 64)( 56, 63)( 57, 69)( 58, 70)
( 59, 72)( 60, 71)( 73, 77)( 74, 78)( 75, 80)( 76, 79)( 83, 84)( 85,101)
( 86,102)( 87,104)( 88,103)( 89, 97)( 90, 98)( 91,100)( 92, 99)( 93,105)
( 94,106)( 95,108)( 96,107)(109,113)(110,114)(111,116)(112,115)(119,120)
(121,137)(122,138)(123,140)(124,139)(125,133)(126,134)(127,136)(128,135)
(129,141)(130,142)(131,144)(132,143)(145,149)(146,150)(147,152)(148,151)
(155,156)(157,173)(158,174)(159,176)(160,175)(161,169)(162,170)(163,172)
(164,171)(165,177)(166,178)(167,180)(168,179)(181,185)(182,186)(183,188)
(184,187)(191,192)(193,209)(194,210)(195,212)(196,211)(197,205)(198,206)
(199,208)(200,207)(201,213)(202,214)(203,216)(204,215)(217,221)(218,222)
(219,224)(220,223)(227,228)(229,245)(230,246)(231,248)(232,247)(233,241)
(234,242)(235,244)(236,243)(237,249)(238,250)(239,252)(240,251)(253,257)
(254,258)(255,260)(256,259)(263,264)(265,281)(266,282)(267,284)(268,283)
(269,277)(270,278)(271,280)(272,279)(273,285)(274,286)(275,288)(276,287);;
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)( 26, 28)( 29, 33)( 30, 36)( 31, 35)
( 32, 34)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 57)( 42, 60)( 43, 59)
( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)( 62, 64)( 65, 69)( 66, 72)
( 67, 71)( 68, 70)( 73, 85)( 74, 88)( 75, 87)( 76, 86)( 77, 93)( 78, 96)
( 79, 95)( 80, 94)( 81, 89)( 82, 92)( 83, 91)( 84, 90)( 98,100)(101,105)
(102,108)(103,107)(104,106)(109,121)(110,124)(111,123)(112,122)(113,129)
(114,132)(115,131)(116,130)(117,125)(118,128)(119,127)(120,126)(134,136)
(137,141)(138,144)(139,143)(140,142)(145,157)(146,160)(147,159)(148,158)
(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)(155,163)(156,162)
(170,172)(173,177)(174,180)(175,179)(176,178)(181,193)(182,196)(183,195)
(184,194)(185,201)(186,204)(187,203)(188,202)(189,197)(190,200)(191,199)
(192,198)(206,208)(209,213)(210,216)(211,215)(212,214)(217,229)(218,232)
(219,231)(220,230)(221,237)(222,240)(223,239)(224,238)(225,233)(226,236)
(227,235)(228,234)(242,244)(245,249)(246,252)(247,251)(248,250)(253,265)
(254,268)(255,267)(256,266)(257,273)(258,276)(259,275)(260,274)(261,269)
(262,272)(263,271)(264,270)(278,280)(281,285)(282,288)(283,287)(284,286);;
s4 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)
(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)
(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)
(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)
(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)
(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)
(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)
(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)
(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288);;
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*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*s2*s3*s2*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s3*s4*s3*s4*s3*s4*s3*s4,
s2*s4*s3*s2*s4*s3*s2*s4*s3, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(288)!( 1,145)( 2,146)( 3,147)( 4,148)( 5,149)( 6,150)( 7,151)
( 8,152)( 9,153)( 10,154)( 11,155)( 12,156)( 13,157)( 14,158)( 15,159)
( 16,160)( 17,161)( 18,162)( 19,163)( 20,164)( 21,165)( 22,166)( 23,167)
( 24,168)( 25,169)( 26,170)( 27,171)( 28,172)( 29,173)( 30,174)( 31,175)
( 32,176)( 33,177)( 34,178)( 35,179)( 36,180)( 37,181)( 38,182)( 39,183)
( 40,184)( 41,185)( 42,186)( 43,187)( 44,188)( 45,189)( 46,190)( 47,191)
( 48,192)( 49,193)( 50,194)( 51,195)( 52,196)( 53,197)( 54,198)( 55,199)
( 56,200)( 57,201)( 58,202)( 59,203)( 60,204)( 61,205)( 62,206)( 63,207)
( 64,208)( 65,209)( 66,210)( 67,211)( 68,212)( 69,213)( 70,214)( 71,215)
( 72,216)( 73,253)( 74,254)( 75,255)( 76,256)( 77,257)( 78,258)( 79,259)
( 80,260)( 81,261)( 82,262)( 83,263)( 84,264)( 85,265)( 86,266)( 87,267)
( 88,268)( 89,269)( 90,270)( 91,271)( 92,272)( 93,273)( 94,274)( 95,275)
( 96,276)( 97,277)( 98,278)( 99,279)(100,280)(101,281)(102,282)(103,283)
(104,284)(105,285)(106,286)(107,287)(108,288)(109,217)(110,218)(111,219)
(112,220)(113,221)(114,222)(115,223)(116,224)(117,225)(118,226)(119,227)
(120,228)(121,229)(122,230)(123,231)(124,232)(125,233)(126,234)(127,235)
(128,236)(129,237)(130,238)(131,239)(132,240)(133,241)(134,242)(135,243)
(136,244)(137,245)(138,246)(139,247)(140,248)(141,249)(142,250)(143,251)
(144,252);
s1 := Sym(288)!( 5, 9)( 6, 10)( 7, 11)( 8, 12)( 17, 21)( 18, 22)( 19, 23)
( 20, 24)( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 41, 45)( 42, 46)( 43, 47)
( 44, 48)( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 69)( 66, 70)( 67, 71)
( 68, 72)( 73,109)( 74,110)( 75,111)( 76,112)( 77,117)( 78,118)( 79,119)
( 80,120)( 81,113)( 82,114)( 83,115)( 84,116)( 85,121)( 86,122)( 87,123)
( 88,124)( 89,129)( 90,130)( 91,131)( 92,132)( 93,125)( 94,126)( 95,127)
( 96,128)( 97,133)( 98,134)( 99,135)(100,136)(101,141)(102,142)(103,143)
(104,144)(105,137)(106,138)(107,139)(108,140)(145,217)(146,218)(147,219)
(148,220)(149,225)(150,226)(151,227)(152,228)(153,221)(154,222)(155,223)
(156,224)(157,229)(158,230)(159,231)(160,232)(161,237)(162,238)(163,239)
(164,240)(165,233)(166,234)(167,235)(168,236)(169,241)(170,242)(171,243)
(172,244)(173,249)(174,250)(175,251)(176,252)(177,245)(178,246)(179,247)
(180,248)(181,253)(182,254)(183,255)(184,256)(185,261)(186,262)(187,263)
(188,264)(189,257)(190,258)(191,259)(192,260)(193,265)(194,266)(195,267)
(196,268)(197,273)(198,274)(199,275)(200,276)(201,269)(202,270)(203,271)
(204,272)(205,277)(206,278)(207,279)(208,280)(209,285)(210,286)(211,287)
(212,288)(213,281)(214,282)(215,283)(216,284);
s2 := Sym(288)!( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 11, 12)( 13, 29)( 14, 30)
( 15, 32)( 16, 31)( 17, 25)( 18, 26)( 19, 28)( 20, 27)( 21, 33)( 22, 34)
( 23, 36)( 24, 35)( 37, 41)( 38, 42)( 39, 44)( 40, 43)( 47, 48)( 49, 65)
( 50, 66)( 51, 68)( 52, 67)( 53, 61)( 54, 62)( 55, 64)( 56, 63)( 57, 69)
( 58, 70)( 59, 72)( 60, 71)( 73, 77)( 74, 78)( 75, 80)( 76, 79)( 83, 84)
( 85,101)( 86,102)( 87,104)( 88,103)( 89, 97)( 90, 98)( 91,100)( 92, 99)
( 93,105)( 94,106)( 95,108)( 96,107)(109,113)(110,114)(111,116)(112,115)
(119,120)(121,137)(122,138)(123,140)(124,139)(125,133)(126,134)(127,136)
(128,135)(129,141)(130,142)(131,144)(132,143)(145,149)(146,150)(147,152)
(148,151)(155,156)(157,173)(158,174)(159,176)(160,175)(161,169)(162,170)
(163,172)(164,171)(165,177)(166,178)(167,180)(168,179)(181,185)(182,186)
(183,188)(184,187)(191,192)(193,209)(194,210)(195,212)(196,211)(197,205)
(198,206)(199,208)(200,207)(201,213)(202,214)(203,216)(204,215)(217,221)
(218,222)(219,224)(220,223)(227,228)(229,245)(230,246)(231,248)(232,247)
(233,241)(234,242)(235,244)(236,243)(237,249)(238,250)(239,252)(240,251)
(253,257)(254,258)(255,260)(256,259)(263,264)(265,281)(266,282)(267,284)
(268,283)(269,277)(270,278)(271,280)(272,279)(273,285)(274,286)(275,288)
(276,287);
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)( 26, 28)( 29, 33)( 30, 36)
( 31, 35)( 32, 34)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 57)( 42, 60)
( 43, 59)( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)( 62, 64)( 65, 69)
( 66, 72)( 67, 71)( 68, 70)( 73, 85)( 74, 88)( 75, 87)( 76, 86)( 77, 93)
( 78, 96)( 79, 95)( 80, 94)( 81, 89)( 82, 92)( 83, 91)( 84, 90)( 98,100)
(101,105)(102,108)(103,107)(104,106)(109,121)(110,124)(111,123)(112,122)
(113,129)(114,132)(115,131)(116,130)(117,125)(118,128)(119,127)(120,126)
(134,136)(137,141)(138,144)(139,143)(140,142)(145,157)(146,160)(147,159)
(148,158)(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)(155,163)
(156,162)(170,172)(173,177)(174,180)(175,179)(176,178)(181,193)(182,196)
(183,195)(184,194)(185,201)(186,204)(187,203)(188,202)(189,197)(190,200)
(191,199)(192,198)(206,208)(209,213)(210,216)(211,215)(212,214)(217,229)
(218,232)(219,231)(220,230)(221,237)(222,240)(223,239)(224,238)(225,233)
(226,236)(227,235)(228,234)(242,244)(245,249)(246,252)(247,251)(248,250)
(253,265)(254,268)(255,267)(256,266)(257,273)(258,276)(259,275)(260,274)
(261,269)(262,272)(263,271)(264,270)(278,280)(281,285)(282,288)(283,287)
(284,286);
s4 := Sym(288)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)
(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)
(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)
(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)
(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)
(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)
(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)
(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)
(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)
(287,288);
poly := sub<Sym(288)|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*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*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s3*s4*s3*s4*s3*s4*s3*s4, s2*s4*s3*s2*s4*s3*s2*s4*s3,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope