Polytope of Type {8,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6,4}*768a
if this polytope has a name.
Group : SmallGroup(768,1089270)
Rank : 4
Schlafli Type : {8,6,4}
Number of vertices, edges, etc : 8, 48, 24, 8
Order of s0s1s2s3 : 24
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 : {8,6,4}*384b, {4,6,4}*384a
4-fold quotients : {8,6,2}*192, {4,6,4}*192b, {2,6,4}*192
8-fold quotients : {4,6,2}*96a, {2,3,4}*96, {2,6,4}*96b, {2,6,4}*96c
12-fold quotients : {8,2,2}*64
16-fold quotients : {2,3,4}*48, {2,6,2}*48
24-fold quotients : {4,2,2}*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, 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)
( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,121)( 38,122)( 39,123)( 40,124)
( 41,125)( 42,126)( 43,127)( 44,128)( 45,129)( 46,130)( 47,131)( 48,132)
( 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)
(217,325)(218,326)(219,327)(220,328)(221,329)(222,330)(223,331)(224,332)
(225,333)(226,334)(227,335)(228,336)(229,313)(230,314)(231,315)(232,316)
(233,317)(234,318)(235,319)(236,320)(237,321)(238,322)(239,323)(240,324)
(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);;
s1 := ( 1, 49)( 2, 50)( 3, 52)( 4, 51)( 5, 57)( 6, 58)( 7, 60)( 8, 59)
( 9, 53)( 10, 54)( 11, 56)( 12, 55)( 13, 61)( 14, 62)( 15, 64)( 16, 63)
( 17, 69)( 18, 70)( 19, 72)( 20, 71)( 21, 65)( 22, 66)( 23, 68)( 24, 67)
( 25, 85)( 26, 86)( 27, 88)( 28, 87)( 29, 93)( 30, 94)( 31, 96)( 32, 95)
( 33, 89)( 34, 90)( 35, 92)( 36, 91)( 37, 73)( 38, 74)( 39, 76)( 40, 75)
( 41, 81)( 42, 82)( 43, 84)( 44, 83)( 45, 77)( 46, 78)( 47, 80)( 48, 79)
( 97,145)( 98,146)( 99,148)(100,147)(101,153)(102,154)(103,156)(104,155)
(105,149)(106,150)(107,152)(108,151)(109,157)(110,158)(111,160)(112,159)
(113,165)(114,166)(115,168)(116,167)(117,161)(118,162)(119,164)(120,163)
(121,181)(122,182)(123,184)(124,183)(125,189)(126,190)(127,192)(128,191)
(129,185)(130,186)(131,188)(132,187)(133,169)(134,170)(135,172)(136,171)
(137,177)(138,178)(139,180)(140,179)(141,173)(142,174)(143,176)(144,175)
(193,241)(194,242)(195,244)(196,243)(197,249)(198,250)(199,252)(200,251)
(201,245)(202,246)(203,248)(204,247)(205,253)(206,254)(207,256)(208,255)
(209,261)(210,262)(211,264)(212,263)(213,257)(214,258)(215,260)(216,259)
(217,277)(218,278)(219,280)(220,279)(221,285)(222,286)(223,288)(224,287)
(225,281)(226,282)(227,284)(228,283)(229,265)(230,266)(231,268)(232,267)
(233,273)(234,274)(235,276)(236,275)(237,269)(238,270)(239,272)(240,271)
(289,337)(290,338)(291,340)(292,339)(293,345)(294,346)(295,348)(296,347)
(297,341)(298,342)(299,344)(300,343)(301,349)(302,350)(303,352)(304,351)
(305,357)(306,358)(307,360)(308,359)(309,353)(310,354)(311,356)(312,355)
(313,373)(314,374)(315,376)(316,375)(317,381)(318,382)(319,384)(320,383)
(321,377)(322,378)(323,380)(324,379)(325,361)(326,362)(327,364)(328,363)
(329,369)(330,370)(331,372)(332,371)(333,365)(334,366)(335,368)(336,367);;
s2 := ( 1, 9)( 2, 11)( 3, 10)( 4, 12)( 6, 7)( 13, 21)( 14, 23)( 15, 22)
( 16, 24)( 18, 19)( 25, 33)( 26, 35)( 27, 34)( 28, 36)( 30, 31)( 37, 45)
( 38, 47)( 39, 46)( 40, 48)( 42, 43)( 49, 57)( 50, 59)( 51, 58)( 52, 60)
( 54, 55)( 61, 69)( 62, 71)( 63, 70)( 64, 72)( 66, 67)( 73, 81)( 74, 83)
( 75, 82)( 76, 84)( 78, 79)( 85, 93)( 86, 95)( 87, 94)( 88, 96)( 90, 91)
( 97,105)( 98,107)( 99,106)(100,108)(102,103)(109,117)(110,119)(111,118)
(112,120)(114,115)(121,129)(122,131)(123,130)(124,132)(126,127)(133,141)
(134,143)(135,142)(136,144)(138,139)(145,153)(146,155)(147,154)(148,156)
(150,151)(157,165)(158,167)(159,166)(160,168)(162,163)(169,177)(170,179)
(171,178)(172,180)(174,175)(181,189)(182,191)(183,190)(184,192)(186,187)
(193,201)(194,203)(195,202)(196,204)(198,199)(205,213)(206,215)(207,214)
(208,216)(210,211)(217,225)(218,227)(219,226)(220,228)(222,223)(229,237)
(230,239)(231,238)(232,240)(234,235)(241,249)(242,251)(243,250)(244,252)
(246,247)(253,261)(254,263)(255,262)(256,264)(258,259)(265,273)(266,275)
(267,274)(268,276)(270,271)(277,285)(278,287)(279,286)(280,288)(282,283)
(289,297)(290,299)(291,298)(292,300)(294,295)(301,309)(302,311)(303,310)
(304,312)(306,307)(313,321)(314,323)(315,322)(316,324)(318,319)(325,333)
(326,335)(327,334)(328,336)(330,331)(337,345)(338,347)(339,346)(340,348)
(342,343)(349,357)(350,359)(351,358)(352,360)(354,355)(361,369)(362,371)
(363,370)(364,372)(366,367)(373,381)(374,383)(375,382)(376,384)(378,379);;
s3 := ( 1,194)( 2,193)( 3,196)( 4,195)( 5,198)( 6,197)( 7,200)( 8,199)
( 9,202)( 10,201)( 11,204)( 12,203)( 13,206)( 14,205)( 15,208)( 16,207)
( 17,210)( 18,209)( 19,212)( 20,211)( 21,214)( 22,213)( 23,216)( 24,215)
( 25,218)( 26,217)( 27,220)( 28,219)( 29,222)( 30,221)( 31,224)( 32,223)
( 33,226)( 34,225)( 35,228)( 36,227)( 37,230)( 38,229)( 39,232)( 40,231)
( 41,234)( 42,233)( 43,236)( 44,235)( 45,238)( 46,237)( 47,240)( 48,239)
( 49,242)( 50,241)( 51,244)( 52,243)( 53,246)( 54,245)( 55,248)( 56,247)
( 57,250)( 58,249)( 59,252)( 60,251)( 61,254)( 62,253)( 63,256)( 64,255)
( 65,258)( 66,257)( 67,260)( 68,259)( 69,262)( 70,261)( 71,264)( 72,263)
( 73,266)( 74,265)( 75,268)( 76,267)( 77,270)( 78,269)( 79,272)( 80,271)
( 81,274)( 82,273)( 83,276)( 84,275)( 85,278)( 86,277)( 87,280)( 88,279)
( 89,282)( 90,281)( 91,284)( 92,283)( 93,286)( 94,285)( 95,288)( 96,287)
( 97,290)( 98,289)( 99,292)(100,291)(101,294)(102,293)(103,296)(104,295)
(105,298)(106,297)(107,300)(108,299)(109,302)(110,301)(111,304)(112,303)
(113,306)(114,305)(115,308)(116,307)(117,310)(118,309)(119,312)(120,311)
(121,314)(122,313)(123,316)(124,315)(125,318)(126,317)(127,320)(128,319)
(129,322)(130,321)(131,324)(132,323)(133,326)(134,325)(135,328)(136,327)
(137,330)(138,329)(139,332)(140,331)(141,334)(142,333)(143,336)(144,335)
(145,338)(146,337)(147,340)(148,339)(149,342)(150,341)(151,344)(152,343)
(153,346)(154,345)(155,348)(156,347)(157,350)(158,349)(159,352)(160,351)
(161,354)(162,353)(163,356)(164,355)(165,358)(166,357)(167,360)(168,359)
(169,362)(170,361)(171,364)(172,363)(173,366)(174,365)(175,368)(176,367)
(177,370)(178,369)(179,372)(180,371)(181,374)(182,373)(183,376)(184,375)
(185,378)(186,377)(187,380)(188,379)(189,382)(190,381)(191,384)(192,383);;
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,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*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(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)( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)
( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,121)( 38,122)( 39,123)
( 40,124)( 41,125)( 42,126)( 43,127)( 44,128)( 45,129)( 46,130)( 47,131)
( 48,132)( 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)(217,325)(218,326)(219,327)(220,328)(221,329)(222,330)(223,331)
(224,332)(225,333)(226,334)(227,335)(228,336)(229,313)(230,314)(231,315)
(232,316)(233,317)(234,318)(235,319)(236,320)(237,321)(238,322)(239,323)
(240,324)(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);
s1 := Sym(384)!( 1, 49)( 2, 50)( 3, 52)( 4, 51)( 5, 57)( 6, 58)( 7, 60)
( 8, 59)( 9, 53)( 10, 54)( 11, 56)( 12, 55)( 13, 61)( 14, 62)( 15, 64)
( 16, 63)( 17, 69)( 18, 70)( 19, 72)( 20, 71)( 21, 65)( 22, 66)( 23, 68)
( 24, 67)( 25, 85)( 26, 86)( 27, 88)( 28, 87)( 29, 93)( 30, 94)( 31, 96)
( 32, 95)( 33, 89)( 34, 90)( 35, 92)( 36, 91)( 37, 73)( 38, 74)( 39, 76)
( 40, 75)( 41, 81)( 42, 82)( 43, 84)( 44, 83)( 45, 77)( 46, 78)( 47, 80)
( 48, 79)( 97,145)( 98,146)( 99,148)(100,147)(101,153)(102,154)(103,156)
(104,155)(105,149)(106,150)(107,152)(108,151)(109,157)(110,158)(111,160)
(112,159)(113,165)(114,166)(115,168)(116,167)(117,161)(118,162)(119,164)
(120,163)(121,181)(122,182)(123,184)(124,183)(125,189)(126,190)(127,192)
(128,191)(129,185)(130,186)(131,188)(132,187)(133,169)(134,170)(135,172)
(136,171)(137,177)(138,178)(139,180)(140,179)(141,173)(142,174)(143,176)
(144,175)(193,241)(194,242)(195,244)(196,243)(197,249)(198,250)(199,252)
(200,251)(201,245)(202,246)(203,248)(204,247)(205,253)(206,254)(207,256)
(208,255)(209,261)(210,262)(211,264)(212,263)(213,257)(214,258)(215,260)
(216,259)(217,277)(218,278)(219,280)(220,279)(221,285)(222,286)(223,288)
(224,287)(225,281)(226,282)(227,284)(228,283)(229,265)(230,266)(231,268)
(232,267)(233,273)(234,274)(235,276)(236,275)(237,269)(238,270)(239,272)
(240,271)(289,337)(290,338)(291,340)(292,339)(293,345)(294,346)(295,348)
(296,347)(297,341)(298,342)(299,344)(300,343)(301,349)(302,350)(303,352)
(304,351)(305,357)(306,358)(307,360)(308,359)(309,353)(310,354)(311,356)
(312,355)(313,373)(314,374)(315,376)(316,375)(317,381)(318,382)(319,384)
(320,383)(321,377)(322,378)(323,380)(324,379)(325,361)(326,362)(327,364)
(328,363)(329,369)(330,370)(331,372)(332,371)(333,365)(334,366)(335,368)
(336,367);
s2 := Sym(384)!( 1, 9)( 2, 11)( 3, 10)( 4, 12)( 6, 7)( 13, 21)( 14, 23)
( 15, 22)( 16, 24)( 18, 19)( 25, 33)( 26, 35)( 27, 34)( 28, 36)( 30, 31)
( 37, 45)( 38, 47)( 39, 46)( 40, 48)( 42, 43)( 49, 57)( 50, 59)( 51, 58)
( 52, 60)( 54, 55)( 61, 69)( 62, 71)( 63, 70)( 64, 72)( 66, 67)( 73, 81)
( 74, 83)( 75, 82)( 76, 84)( 78, 79)( 85, 93)( 86, 95)( 87, 94)( 88, 96)
( 90, 91)( 97,105)( 98,107)( 99,106)(100,108)(102,103)(109,117)(110,119)
(111,118)(112,120)(114,115)(121,129)(122,131)(123,130)(124,132)(126,127)
(133,141)(134,143)(135,142)(136,144)(138,139)(145,153)(146,155)(147,154)
(148,156)(150,151)(157,165)(158,167)(159,166)(160,168)(162,163)(169,177)
(170,179)(171,178)(172,180)(174,175)(181,189)(182,191)(183,190)(184,192)
(186,187)(193,201)(194,203)(195,202)(196,204)(198,199)(205,213)(206,215)
(207,214)(208,216)(210,211)(217,225)(218,227)(219,226)(220,228)(222,223)
(229,237)(230,239)(231,238)(232,240)(234,235)(241,249)(242,251)(243,250)
(244,252)(246,247)(253,261)(254,263)(255,262)(256,264)(258,259)(265,273)
(266,275)(267,274)(268,276)(270,271)(277,285)(278,287)(279,286)(280,288)
(282,283)(289,297)(290,299)(291,298)(292,300)(294,295)(301,309)(302,311)
(303,310)(304,312)(306,307)(313,321)(314,323)(315,322)(316,324)(318,319)
(325,333)(326,335)(327,334)(328,336)(330,331)(337,345)(338,347)(339,346)
(340,348)(342,343)(349,357)(350,359)(351,358)(352,360)(354,355)(361,369)
(362,371)(363,370)(364,372)(366,367)(373,381)(374,383)(375,382)(376,384)
(378,379);
s3 := Sym(384)!( 1,194)( 2,193)( 3,196)( 4,195)( 5,198)( 6,197)( 7,200)
( 8,199)( 9,202)( 10,201)( 11,204)( 12,203)( 13,206)( 14,205)( 15,208)
( 16,207)( 17,210)( 18,209)( 19,212)( 20,211)( 21,214)( 22,213)( 23,216)
( 24,215)( 25,218)( 26,217)( 27,220)( 28,219)( 29,222)( 30,221)( 31,224)
( 32,223)( 33,226)( 34,225)( 35,228)( 36,227)( 37,230)( 38,229)( 39,232)
( 40,231)( 41,234)( 42,233)( 43,236)( 44,235)( 45,238)( 46,237)( 47,240)
( 48,239)( 49,242)( 50,241)( 51,244)( 52,243)( 53,246)( 54,245)( 55,248)
( 56,247)( 57,250)( 58,249)( 59,252)( 60,251)( 61,254)( 62,253)( 63,256)
( 64,255)( 65,258)( 66,257)( 67,260)( 68,259)( 69,262)( 70,261)( 71,264)
( 72,263)( 73,266)( 74,265)( 75,268)( 76,267)( 77,270)( 78,269)( 79,272)
( 80,271)( 81,274)( 82,273)( 83,276)( 84,275)( 85,278)( 86,277)( 87,280)
( 88,279)( 89,282)( 90,281)( 91,284)( 92,283)( 93,286)( 94,285)( 95,288)
( 96,287)( 97,290)( 98,289)( 99,292)(100,291)(101,294)(102,293)(103,296)
(104,295)(105,298)(106,297)(107,300)(108,299)(109,302)(110,301)(111,304)
(112,303)(113,306)(114,305)(115,308)(116,307)(117,310)(118,309)(119,312)
(120,311)(121,314)(122,313)(123,316)(124,315)(125,318)(126,317)(127,320)
(128,319)(129,322)(130,321)(131,324)(132,323)(133,326)(134,325)(135,328)
(136,327)(137,330)(138,329)(139,332)(140,331)(141,334)(142,333)(143,336)
(144,335)(145,338)(146,337)(147,340)(148,339)(149,342)(150,341)(151,344)
(152,343)(153,346)(154,345)(155,348)(156,347)(157,350)(158,349)(159,352)
(160,351)(161,354)(162,353)(163,356)(164,355)(165,358)(166,357)(167,360)
(168,359)(169,362)(170,361)(171,364)(172,363)(173,366)(174,365)(175,368)
(176,367)(177,370)(178,369)(179,372)(180,371)(181,374)(182,373)(183,376)
(184,375)(185,378)(186,377)(187,380)(188,379)(189,382)(190,381)(191,384)
(192,383);
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,
s0*s1*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope