Polytope of Type {8,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6,4}*768b
if this polytope has a name.
Group : SmallGroup(768,1089292)
Rank : 4
Schlafli Type : {8,6,4}
Number of vertices, edges, etc : 16, 48, 24, 4
Order of s0s1s2s3 : 12
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 : {4,6,4}*384b, {8,6,2}*384c
4-fold quotients : {4,6,4}*192c, {4,6,2}*192
8-fold quotients : {2,6,4}*96a, {4,3,2}*96, {4,6,2}*96b, {4,6,2}*96c
16-fold quotients : {4,3,2}*48, {2,6,2}*48
24-fold quotients : {2,2,4}*32
32-fold quotients : {2,3,2}*24
48-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,199)( 2,200)( 3,197)( 4,198)( 5,196)( 6,195)( 7,194)( 8,193)
( 9,207)( 10,208)( 11,205)( 12,206)( 13,204)( 14,203)( 15,202)( 16,201)
( 17,215)( 18,216)( 19,213)( 20,214)( 21,212)( 22,211)( 23,210)( 24,209)
( 25,223)( 26,224)( 27,221)( 28,222)( 29,220)( 30,219)( 31,218)( 32,217)
( 33,231)( 34,232)( 35,229)( 36,230)( 37,228)( 38,227)( 39,226)( 40,225)
( 41,239)( 42,240)( 43,237)( 44,238)( 45,236)( 46,235)( 47,234)( 48,233)
( 49,248)( 50,247)( 51,246)( 52,245)( 53,243)( 54,244)( 55,241)( 56,242)
( 57,256)( 58,255)( 59,254)( 60,253)( 61,251)( 62,252)( 63,249)( 64,250)
( 65,264)( 66,263)( 67,262)( 68,261)( 69,259)( 70,260)( 71,257)( 72,258)
( 73,272)( 74,271)( 75,270)( 76,269)( 77,267)( 78,268)( 79,265)( 80,266)
( 81,280)( 82,279)( 83,278)( 84,277)( 85,275)( 86,276)( 87,273)( 88,274)
( 89,288)( 90,287)( 91,286)( 92,285)( 93,283)( 94,284)( 95,281)( 96,282)
( 97,295)( 98,296)( 99,293)(100,294)(101,292)(102,291)(103,290)(104,289)
(105,303)(106,304)(107,301)(108,302)(109,300)(110,299)(111,298)(112,297)
(113,311)(114,312)(115,309)(116,310)(117,308)(118,307)(119,306)(120,305)
(121,319)(122,320)(123,317)(124,318)(125,316)(126,315)(127,314)(128,313)
(129,327)(130,328)(131,325)(132,326)(133,324)(134,323)(135,322)(136,321)
(137,335)(138,336)(139,333)(140,334)(141,332)(142,331)(143,330)(144,329)
(145,344)(146,343)(147,342)(148,341)(149,339)(150,340)(151,337)(152,338)
(153,352)(154,351)(155,350)(156,349)(157,347)(158,348)(159,345)(160,346)
(161,360)(162,359)(163,358)(164,357)(165,355)(166,356)(167,353)(168,354)
(169,368)(170,367)(171,366)(172,365)(173,363)(174,364)(175,361)(176,362)
(177,376)(178,375)(179,374)(180,373)(181,371)(182,372)(183,369)(184,370)
(185,384)(186,383)(187,382)(188,381)(189,379)(190,380)(191,377)(192,378);;
s1 := ( 3, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)( 13, 23)
( 14, 24)( 15, 21)( 16, 22)( 27, 28)( 29, 31)( 30, 32)( 33, 41)( 34, 42)
( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 51, 52)( 53, 55)
( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)( 62, 72)( 63, 69)
( 64, 70)( 75, 76)( 77, 79)( 78, 80)( 81, 89)( 82, 90)( 83, 92)( 84, 91)
( 85, 95)( 86, 96)( 87, 93)( 88, 94)( 99,100)(101,103)(102,104)(105,113)
(106,114)(107,116)(108,115)(109,119)(110,120)(111,117)(112,118)(123,124)
(125,127)(126,128)(129,137)(130,138)(131,140)(132,139)(133,143)(134,144)
(135,141)(136,142)(147,148)(149,151)(150,152)(153,161)(154,162)(155,164)
(156,163)(157,167)(158,168)(159,165)(160,166)(171,172)(173,175)(174,176)
(177,185)(178,186)(179,188)(180,187)(181,191)(182,192)(183,189)(184,190)
(195,196)(197,199)(198,200)(201,209)(202,210)(203,212)(204,211)(205,215)
(206,216)(207,213)(208,214)(219,220)(221,223)(222,224)(225,233)(226,234)
(227,236)(228,235)(229,239)(230,240)(231,237)(232,238)(243,244)(245,247)
(246,248)(249,257)(250,258)(251,260)(252,259)(253,263)(254,264)(255,261)
(256,262)(267,268)(269,271)(270,272)(273,281)(274,282)(275,284)(276,283)
(277,287)(278,288)(279,285)(280,286)(291,292)(293,295)(294,296)(297,305)
(298,306)(299,308)(300,307)(301,311)(302,312)(303,309)(304,310)(315,316)
(317,319)(318,320)(321,329)(322,330)(323,332)(324,331)(325,335)(326,336)
(327,333)(328,334)(339,340)(341,343)(342,344)(345,353)(346,354)(347,356)
(348,355)(349,359)(350,360)(351,357)(352,358)(363,364)(365,367)(366,368)
(369,377)(370,378)(371,380)(372,379)(373,383)(374,384)(375,381)(376,382);;
s2 := ( 1, 65)( 2, 66)( 3, 70)( 4, 69)( 5, 68)( 6, 67)( 7, 72)( 8, 71)
( 9, 57)( 10, 58)( 11, 62)( 12, 61)( 13, 60)( 14, 59)( 15, 64)( 16, 63)
( 17, 49)( 18, 50)( 19, 54)( 20, 53)( 21, 52)( 22, 51)( 23, 56)( 24, 55)
( 25, 89)( 26, 90)( 27, 94)( 28, 93)( 29, 92)( 30, 91)( 31, 96)( 32, 95)
( 33, 81)( 34, 82)( 35, 86)( 36, 85)( 37, 84)( 38, 83)( 39, 88)( 40, 87)
( 41, 73)( 42, 74)( 43, 78)( 44, 77)( 45, 76)( 46, 75)( 47, 80)( 48, 79)
( 97,161)( 98,162)( 99,166)(100,165)(101,164)(102,163)(103,168)(104,167)
(105,153)(106,154)(107,158)(108,157)(109,156)(110,155)(111,160)(112,159)
(113,145)(114,146)(115,150)(116,149)(117,148)(118,147)(119,152)(120,151)
(121,185)(122,186)(123,190)(124,189)(125,188)(126,187)(127,192)(128,191)
(129,177)(130,178)(131,182)(132,181)(133,180)(134,179)(135,184)(136,183)
(137,169)(138,170)(139,174)(140,173)(141,172)(142,171)(143,176)(144,175)
(193,257)(194,258)(195,262)(196,261)(197,260)(198,259)(199,264)(200,263)
(201,249)(202,250)(203,254)(204,253)(205,252)(206,251)(207,256)(208,255)
(209,241)(210,242)(211,246)(212,245)(213,244)(214,243)(215,248)(216,247)
(217,281)(218,282)(219,286)(220,285)(221,284)(222,283)(223,288)(224,287)
(225,273)(226,274)(227,278)(228,277)(229,276)(230,275)(231,280)(232,279)
(233,265)(234,266)(235,270)(236,269)(237,268)(238,267)(239,272)(240,271)
(289,353)(290,354)(291,358)(292,357)(293,356)(294,355)(295,360)(296,359)
(297,345)(298,346)(299,350)(300,349)(301,348)(302,347)(303,352)(304,351)
(305,337)(306,338)(307,342)(308,341)(309,340)(310,339)(311,344)(312,343)
(313,377)(314,378)(315,382)(316,381)(317,380)(318,379)(319,384)(320,383)
(321,369)(322,370)(323,374)(324,373)(325,372)(326,371)(327,376)(328,375)
(329,361)(330,362)(331,366)(332,365)(333,364)(334,363)(335,368)(336,367);;
s3 := ( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,101)( 6,102)( 7,103)( 8,104)
( 9,105)( 10,106)( 11,107)( 12,108)( 13,109)( 14,110)( 15,111)( 16,112)
( 17,113)( 18,114)( 19,115)( 20,116)( 21,117)( 22,118)( 23,119)( 24,120)
( 25,121)( 26,122)( 27,123)( 28,124)( 29,125)( 30,126)( 31,127)( 32,128)
( 33,129)( 34,130)( 35,131)( 36,132)( 37,133)( 38,134)( 39,135)( 40,136)
( 41,137)( 42,138)( 43,139)( 44,140)( 45,141)( 46,142)( 47,143)( 48,144)
( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)( 56,176)
( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)( 64,184)
( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)( 72,192)
( 73,145)( 74,146)( 75,147)( 76,148)( 77,149)( 78,150)( 79,151)( 80,152)
( 81,153)( 82,154)( 83,155)( 84,156)( 85,157)( 86,158)( 87,159)( 88,160)
( 89,161)( 90,162)( 91,163)( 92,164)( 93,165)( 94,166)( 95,167)( 96,168)
(193,289)(194,290)(195,291)(196,292)(197,293)(198,294)(199,295)(200,296)
(201,297)(202,298)(203,299)(204,300)(205,301)(206,302)(207,303)(208,304)
(209,305)(210,306)(211,307)(212,308)(213,309)(214,310)(215,311)(216,312)
(217,313)(218,314)(219,315)(220,316)(221,317)(222,318)(223,319)(224,320)
(225,321)(226,322)(227,323)(228,324)(229,325)(230,326)(231,327)(232,328)
(233,329)(234,330)(235,331)(236,332)(237,333)(238,334)(239,335)(240,336)
(241,361)(242,362)(243,363)(244,364)(245,365)(246,366)(247,367)(248,368)
(249,369)(250,370)(251,371)(252,372)(253,373)(254,374)(255,375)(256,376)
(257,377)(258,378)(259,379)(260,380)(261,381)(262,382)(263,383)(264,384)
(265,337)(266,338)(267,339)(268,340)(269,341)(270,342)(271,343)(272,344)
(273,345)(274,346)(275,347)(276,348)(277,349)(278,350)(279,351)(280,352)
(281,353)(282,354)(283,355)(284,356)(285,357)(286,358)(287,359)(288,360);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(384)!( 1,199)( 2,200)( 3,197)( 4,198)( 5,196)( 6,195)( 7,194)
( 8,193)( 9,207)( 10,208)( 11,205)( 12,206)( 13,204)( 14,203)( 15,202)
( 16,201)( 17,215)( 18,216)( 19,213)( 20,214)( 21,212)( 22,211)( 23,210)
( 24,209)( 25,223)( 26,224)( 27,221)( 28,222)( 29,220)( 30,219)( 31,218)
( 32,217)( 33,231)( 34,232)( 35,229)( 36,230)( 37,228)( 38,227)( 39,226)
( 40,225)( 41,239)( 42,240)( 43,237)( 44,238)( 45,236)( 46,235)( 47,234)
( 48,233)( 49,248)( 50,247)( 51,246)( 52,245)( 53,243)( 54,244)( 55,241)
( 56,242)( 57,256)( 58,255)( 59,254)( 60,253)( 61,251)( 62,252)( 63,249)
( 64,250)( 65,264)( 66,263)( 67,262)( 68,261)( 69,259)( 70,260)( 71,257)
( 72,258)( 73,272)( 74,271)( 75,270)( 76,269)( 77,267)( 78,268)( 79,265)
( 80,266)( 81,280)( 82,279)( 83,278)( 84,277)( 85,275)( 86,276)( 87,273)
( 88,274)( 89,288)( 90,287)( 91,286)( 92,285)( 93,283)( 94,284)( 95,281)
( 96,282)( 97,295)( 98,296)( 99,293)(100,294)(101,292)(102,291)(103,290)
(104,289)(105,303)(106,304)(107,301)(108,302)(109,300)(110,299)(111,298)
(112,297)(113,311)(114,312)(115,309)(116,310)(117,308)(118,307)(119,306)
(120,305)(121,319)(122,320)(123,317)(124,318)(125,316)(126,315)(127,314)
(128,313)(129,327)(130,328)(131,325)(132,326)(133,324)(134,323)(135,322)
(136,321)(137,335)(138,336)(139,333)(140,334)(141,332)(142,331)(143,330)
(144,329)(145,344)(146,343)(147,342)(148,341)(149,339)(150,340)(151,337)
(152,338)(153,352)(154,351)(155,350)(156,349)(157,347)(158,348)(159,345)
(160,346)(161,360)(162,359)(163,358)(164,357)(165,355)(166,356)(167,353)
(168,354)(169,368)(170,367)(171,366)(172,365)(173,363)(174,364)(175,361)
(176,362)(177,376)(178,375)(179,374)(180,373)(181,371)(182,372)(183,369)
(184,370)(185,384)(186,383)(187,382)(188,381)(189,379)(190,380)(191,377)
(192,378);
s1 := Sym(384)!( 3, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)
( 13, 23)( 14, 24)( 15, 21)( 16, 22)( 27, 28)( 29, 31)( 30, 32)( 33, 41)
( 34, 42)( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 51, 52)
( 53, 55)( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)( 62, 72)
( 63, 69)( 64, 70)( 75, 76)( 77, 79)( 78, 80)( 81, 89)( 82, 90)( 83, 92)
( 84, 91)( 85, 95)( 86, 96)( 87, 93)( 88, 94)( 99,100)(101,103)(102,104)
(105,113)(106,114)(107,116)(108,115)(109,119)(110,120)(111,117)(112,118)
(123,124)(125,127)(126,128)(129,137)(130,138)(131,140)(132,139)(133,143)
(134,144)(135,141)(136,142)(147,148)(149,151)(150,152)(153,161)(154,162)
(155,164)(156,163)(157,167)(158,168)(159,165)(160,166)(171,172)(173,175)
(174,176)(177,185)(178,186)(179,188)(180,187)(181,191)(182,192)(183,189)
(184,190)(195,196)(197,199)(198,200)(201,209)(202,210)(203,212)(204,211)
(205,215)(206,216)(207,213)(208,214)(219,220)(221,223)(222,224)(225,233)
(226,234)(227,236)(228,235)(229,239)(230,240)(231,237)(232,238)(243,244)
(245,247)(246,248)(249,257)(250,258)(251,260)(252,259)(253,263)(254,264)
(255,261)(256,262)(267,268)(269,271)(270,272)(273,281)(274,282)(275,284)
(276,283)(277,287)(278,288)(279,285)(280,286)(291,292)(293,295)(294,296)
(297,305)(298,306)(299,308)(300,307)(301,311)(302,312)(303,309)(304,310)
(315,316)(317,319)(318,320)(321,329)(322,330)(323,332)(324,331)(325,335)
(326,336)(327,333)(328,334)(339,340)(341,343)(342,344)(345,353)(346,354)
(347,356)(348,355)(349,359)(350,360)(351,357)(352,358)(363,364)(365,367)
(366,368)(369,377)(370,378)(371,380)(372,379)(373,383)(374,384)(375,381)
(376,382);
s2 := Sym(384)!( 1, 65)( 2, 66)( 3, 70)( 4, 69)( 5, 68)( 6, 67)( 7, 72)
( 8, 71)( 9, 57)( 10, 58)( 11, 62)( 12, 61)( 13, 60)( 14, 59)( 15, 64)
( 16, 63)( 17, 49)( 18, 50)( 19, 54)( 20, 53)( 21, 52)( 22, 51)( 23, 56)
( 24, 55)( 25, 89)( 26, 90)( 27, 94)( 28, 93)( 29, 92)( 30, 91)( 31, 96)
( 32, 95)( 33, 81)( 34, 82)( 35, 86)( 36, 85)( 37, 84)( 38, 83)( 39, 88)
( 40, 87)( 41, 73)( 42, 74)( 43, 78)( 44, 77)( 45, 76)( 46, 75)( 47, 80)
( 48, 79)( 97,161)( 98,162)( 99,166)(100,165)(101,164)(102,163)(103,168)
(104,167)(105,153)(106,154)(107,158)(108,157)(109,156)(110,155)(111,160)
(112,159)(113,145)(114,146)(115,150)(116,149)(117,148)(118,147)(119,152)
(120,151)(121,185)(122,186)(123,190)(124,189)(125,188)(126,187)(127,192)
(128,191)(129,177)(130,178)(131,182)(132,181)(133,180)(134,179)(135,184)
(136,183)(137,169)(138,170)(139,174)(140,173)(141,172)(142,171)(143,176)
(144,175)(193,257)(194,258)(195,262)(196,261)(197,260)(198,259)(199,264)
(200,263)(201,249)(202,250)(203,254)(204,253)(205,252)(206,251)(207,256)
(208,255)(209,241)(210,242)(211,246)(212,245)(213,244)(214,243)(215,248)
(216,247)(217,281)(218,282)(219,286)(220,285)(221,284)(222,283)(223,288)
(224,287)(225,273)(226,274)(227,278)(228,277)(229,276)(230,275)(231,280)
(232,279)(233,265)(234,266)(235,270)(236,269)(237,268)(238,267)(239,272)
(240,271)(289,353)(290,354)(291,358)(292,357)(293,356)(294,355)(295,360)
(296,359)(297,345)(298,346)(299,350)(300,349)(301,348)(302,347)(303,352)
(304,351)(305,337)(306,338)(307,342)(308,341)(309,340)(310,339)(311,344)
(312,343)(313,377)(314,378)(315,382)(316,381)(317,380)(318,379)(319,384)
(320,383)(321,369)(322,370)(323,374)(324,373)(325,372)(326,371)(327,376)
(328,375)(329,361)(330,362)(331,366)(332,365)(333,364)(334,363)(335,368)
(336,367);
s3 := Sym(384)!( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,101)( 6,102)( 7,103)
( 8,104)( 9,105)( 10,106)( 11,107)( 12,108)( 13,109)( 14,110)( 15,111)
( 16,112)( 17,113)( 18,114)( 19,115)( 20,116)( 21,117)( 22,118)( 23,119)
( 24,120)( 25,121)( 26,122)( 27,123)( 28,124)( 29,125)( 30,126)( 31,127)
( 32,128)( 33,129)( 34,130)( 35,131)( 36,132)( 37,133)( 38,134)( 39,135)
( 40,136)( 41,137)( 42,138)( 43,139)( 44,140)( 45,141)( 46,142)( 47,143)
( 48,144)( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)
( 56,176)( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)
( 64,184)( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)
( 72,192)( 73,145)( 74,146)( 75,147)( 76,148)( 77,149)( 78,150)( 79,151)
( 80,152)( 81,153)( 82,154)( 83,155)( 84,156)( 85,157)( 86,158)( 87,159)
( 88,160)( 89,161)( 90,162)( 91,163)( 92,164)( 93,165)( 94,166)( 95,167)
( 96,168)(193,289)(194,290)(195,291)(196,292)(197,293)(198,294)(199,295)
(200,296)(201,297)(202,298)(203,299)(204,300)(205,301)(206,302)(207,303)
(208,304)(209,305)(210,306)(211,307)(212,308)(213,309)(214,310)(215,311)
(216,312)(217,313)(218,314)(219,315)(220,316)(221,317)(222,318)(223,319)
(224,320)(225,321)(226,322)(227,323)(228,324)(229,325)(230,326)(231,327)
(232,328)(233,329)(234,330)(235,331)(236,332)(237,333)(238,334)(239,335)
(240,336)(241,361)(242,362)(243,363)(244,364)(245,365)(246,366)(247,367)
(248,368)(249,369)(250,370)(251,371)(252,372)(253,373)(254,374)(255,375)
(256,376)(257,377)(258,378)(259,379)(260,380)(261,381)(262,382)(263,383)
(264,384)(265,337)(266,338)(267,339)(268,340)(269,341)(270,342)(271,343)
(272,344)(273,345)(274,346)(275,347)(276,348)(277,349)(278,350)(279,351)
(280,352)(281,353)(282,354)(283,355)(284,356)(285,357)(286,358)(287,359)
(288,360);
poly := sub<Sym(384)|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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1 >;
References : None.
to this polytope