Polytope of Type {6,8,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,8,4}*768c
if this polytope has a name.
Group : SmallGroup(768,1089251)
Rank : 4
Schlafli Type : {6,8,4}
Number of vertices, edges, etc : 12, 48, 32, 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 : {3,8,4}*384, {6,4,4}*384d, {6,8,2}*384b
4-fold quotients : {3,4,4}*192b, {3,8,2}*192, {6,4,2}*192
8-fold quotients : {6,2,4}*96, {3,4,2}*96, {6,4,2}*96b, {6,4,2}*96c
16-fold quotients : {3,2,4}*48, {3,4,2}*48, {6,2,2}*48
24-fold quotients : {2,2,4}*32
32-fold quotients : {3,2,2}*24
48-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 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)( 49, 50)( 53, 56)
( 54, 55)( 57, 66)( 58, 65)( 59, 67)( 60, 68)( 61, 72)( 62, 71)( 63, 70)
( 64, 69)( 73, 74)( 77, 80)( 78, 79)( 81, 90)( 82, 89)( 83, 91)( 84, 92)
( 85, 96)( 86, 95)( 87, 94)( 88, 93)( 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)(145,146)(149,152)(150,151)(153,162)(154,161)(155,163)
(156,164)(157,168)(158,167)(159,166)(160,165)(169,170)(173,176)(174,175)
(177,186)(178,185)(179,187)(180,188)(181,192)(182,191)(183,190)(184,189)
(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)(241,242)(245,248)
(246,247)(249,258)(250,257)(251,259)(252,260)(253,264)(254,263)(255,262)
(256,261)(265,266)(269,272)(270,271)(273,282)(274,281)(275,283)(276,284)
(277,288)(278,287)(279,286)(280,285)(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)(337,338)(341,344)(342,343)(345,354)(346,353)(347,355)
(348,356)(349,360)(350,359)(351,358)(352,357)(361,362)(365,368)(366,367)
(369,378)(370,377)(371,379)(372,380)(373,384)(374,383)(375,382)(376,381);;
s1 := ( 1,209)( 2,210)( 3,214)( 4,213)( 5,212)( 6,211)( 7,216)( 8,215)
( 9,201)( 10,202)( 11,206)( 12,205)( 13,204)( 14,203)( 15,208)( 16,207)
( 17,193)( 18,194)( 19,198)( 20,197)( 21,196)( 22,195)( 23,200)( 24,199)
( 25,233)( 26,234)( 27,238)( 28,237)( 29,236)( 30,235)( 31,240)( 32,239)
( 33,225)( 34,226)( 35,230)( 36,229)( 37,228)( 38,227)( 39,232)( 40,231)
( 41,217)( 42,218)( 43,222)( 44,221)( 45,220)( 46,219)( 47,224)( 48,223)
( 49,258)( 50,257)( 51,261)( 52,262)( 53,259)( 54,260)( 55,263)( 56,264)
( 57,250)( 58,249)( 59,253)( 60,254)( 61,251)( 62,252)( 63,255)( 64,256)
( 65,242)( 66,241)( 67,245)( 68,246)( 69,243)( 70,244)( 71,247)( 72,248)
( 73,282)( 74,281)( 75,285)( 76,286)( 77,283)( 78,284)( 79,287)( 80,288)
( 81,274)( 82,273)( 83,277)( 84,278)( 85,275)( 86,276)( 87,279)( 88,280)
( 89,266)( 90,265)( 91,269)( 92,270)( 93,267)( 94,268)( 95,271)( 96,272)
( 97,305)( 98,306)( 99,310)(100,309)(101,308)(102,307)(103,312)(104,311)
(105,297)(106,298)(107,302)(108,301)(109,300)(110,299)(111,304)(112,303)
(113,289)(114,290)(115,294)(116,293)(117,292)(118,291)(119,296)(120,295)
(121,329)(122,330)(123,334)(124,333)(125,332)(126,331)(127,336)(128,335)
(129,321)(130,322)(131,326)(132,325)(133,324)(134,323)(135,328)(136,327)
(137,313)(138,314)(139,318)(140,317)(141,316)(142,315)(143,320)(144,319)
(145,354)(146,353)(147,357)(148,358)(149,355)(150,356)(151,359)(152,360)
(153,346)(154,345)(155,349)(156,350)(157,347)(158,348)(159,351)(160,352)
(161,338)(162,337)(163,341)(164,342)(165,339)(166,340)(167,343)(168,344)
(169,378)(170,377)(171,381)(172,382)(173,379)(174,380)(175,383)(176,384)
(177,370)(178,369)(179,373)(180,374)(181,371)(182,372)(183,375)(184,376)
(185,362)(186,361)(187,365)(188,366)(189,363)(190,364)(191,367)(192,368);;
s2 := ( 1, 52)( 2, 51)( 3, 49)( 4, 50)( 5, 56)( 6, 55)( 7, 53)( 8, 54)
( 9, 60)( 10, 59)( 11, 57)( 12, 58)( 13, 64)( 14, 63)( 15, 61)( 16, 62)
( 17, 68)( 18, 67)( 19, 65)( 20, 66)( 21, 72)( 22, 71)( 23, 69)( 24, 70)
( 25, 76)( 26, 75)( 27, 73)( 28, 74)( 29, 80)( 30, 79)( 31, 77)( 32, 78)
( 33, 84)( 34, 83)( 35, 81)( 36, 82)( 37, 88)( 38, 87)( 39, 85)( 40, 86)
( 41, 92)( 42, 91)( 43, 89)( 44, 90)( 45, 96)( 46, 95)( 47, 93)( 48, 94)
( 97,148)( 98,147)( 99,145)(100,146)(101,152)(102,151)(103,149)(104,150)
(105,156)(106,155)(107,153)(108,154)(109,160)(110,159)(111,157)(112,158)
(113,164)(114,163)(115,161)(116,162)(117,168)(118,167)(119,165)(120,166)
(121,172)(122,171)(123,169)(124,170)(125,176)(126,175)(127,173)(128,174)
(129,180)(130,179)(131,177)(132,178)(133,184)(134,183)(135,181)(136,182)
(137,188)(138,187)(139,185)(140,186)(141,192)(142,191)(143,189)(144,190)
(193,244)(194,243)(195,241)(196,242)(197,248)(198,247)(199,245)(200,246)
(201,252)(202,251)(203,249)(204,250)(205,256)(206,255)(207,253)(208,254)
(209,260)(210,259)(211,257)(212,258)(213,264)(214,263)(215,261)(216,262)
(217,268)(218,267)(219,265)(220,266)(221,272)(222,271)(223,269)(224,270)
(225,276)(226,275)(227,273)(228,274)(229,280)(230,279)(231,277)(232,278)
(233,284)(234,283)(235,281)(236,282)(237,288)(238,287)(239,285)(240,286)
(289,340)(290,339)(291,337)(292,338)(293,344)(294,343)(295,341)(296,342)
(297,348)(298,347)(299,345)(300,346)(301,352)(302,351)(303,349)(304,350)
(305,356)(306,355)(307,353)(308,354)(309,360)(310,359)(311,357)(312,358)
(313,364)(314,363)(315,361)(316,362)(317,368)(318,367)(319,365)(320,366)
(321,372)(322,371)(323,369)(324,370)(325,376)(326,375)(327,373)(328,374)
(329,380)(330,379)(331,377)(332,378)(333,384)(334,383)(335,381)(336,382);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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)( 49, 50)
( 53, 56)( 54, 55)( 57, 66)( 58, 65)( 59, 67)( 60, 68)( 61, 72)( 62, 71)
( 63, 70)( 64, 69)( 73, 74)( 77, 80)( 78, 79)( 81, 90)( 82, 89)( 83, 91)
( 84, 92)( 85, 96)( 86, 95)( 87, 94)( 88, 93)( 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)(145,146)(149,152)(150,151)(153,162)(154,161)
(155,163)(156,164)(157,168)(158,167)(159,166)(160,165)(169,170)(173,176)
(174,175)(177,186)(178,185)(179,187)(180,188)(181,192)(182,191)(183,190)
(184,189)(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)(241,242)
(245,248)(246,247)(249,258)(250,257)(251,259)(252,260)(253,264)(254,263)
(255,262)(256,261)(265,266)(269,272)(270,271)(273,282)(274,281)(275,283)
(276,284)(277,288)(278,287)(279,286)(280,285)(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)(337,338)(341,344)(342,343)(345,354)(346,353)
(347,355)(348,356)(349,360)(350,359)(351,358)(352,357)(361,362)(365,368)
(366,367)(369,378)(370,377)(371,379)(372,380)(373,384)(374,383)(375,382)
(376,381);
s1 := Sym(384)!( 1,209)( 2,210)( 3,214)( 4,213)( 5,212)( 6,211)( 7,216)
( 8,215)( 9,201)( 10,202)( 11,206)( 12,205)( 13,204)( 14,203)( 15,208)
( 16,207)( 17,193)( 18,194)( 19,198)( 20,197)( 21,196)( 22,195)( 23,200)
( 24,199)( 25,233)( 26,234)( 27,238)( 28,237)( 29,236)( 30,235)( 31,240)
( 32,239)( 33,225)( 34,226)( 35,230)( 36,229)( 37,228)( 38,227)( 39,232)
( 40,231)( 41,217)( 42,218)( 43,222)( 44,221)( 45,220)( 46,219)( 47,224)
( 48,223)( 49,258)( 50,257)( 51,261)( 52,262)( 53,259)( 54,260)( 55,263)
( 56,264)( 57,250)( 58,249)( 59,253)( 60,254)( 61,251)( 62,252)( 63,255)
( 64,256)( 65,242)( 66,241)( 67,245)( 68,246)( 69,243)( 70,244)( 71,247)
( 72,248)( 73,282)( 74,281)( 75,285)( 76,286)( 77,283)( 78,284)( 79,287)
( 80,288)( 81,274)( 82,273)( 83,277)( 84,278)( 85,275)( 86,276)( 87,279)
( 88,280)( 89,266)( 90,265)( 91,269)( 92,270)( 93,267)( 94,268)( 95,271)
( 96,272)( 97,305)( 98,306)( 99,310)(100,309)(101,308)(102,307)(103,312)
(104,311)(105,297)(106,298)(107,302)(108,301)(109,300)(110,299)(111,304)
(112,303)(113,289)(114,290)(115,294)(116,293)(117,292)(118,291)(119,296)
(120,295)(121,329)(122,330)(123,334)(124,333)(125,332)(126,331)(127,336)
(128,335)(129,321)(130,322)(131,326)(132,325)(133,324)(134,323)(135,328)
(136,327)(137,313)(138,314)(139,318)(140,317)(141,316)(142,315)(143,320)
(144,319)(145,354)(146,353)(147,357)(148,358)(149,355)(150,356)(151,359)
(152,360)(153,346)(154,345)(155,349)(156,350)(157,347)(158,348)(159,351)
(160,352)(161,338)(162,337)(163,341)(164,342)(165,339)(166,340)(167,343)
(168,344)(169,378)(170,377)(171,381)(172,382)(173,379)(174,380)(175,383)
(176,384)(177,370)(178,369)(179,373)(180,374)(181,371)(182,372)(183,375)
(184,376)(185,362)(186,361)(187,365)(188,366)(189,363)(190,364)(191,367)
(192,368);
s2 := Sym(384)!( 1, 52)( 2, 51)( 3, 49)( 4, 50)( 5, 56)( 6, 55)( 7, 53)
( 8, 54)( 9, 60)( 10, 59)( 11, 57)( 12, 58)( 13, 64)( 14, 63)( 15, 61)
( 16, 62)( 17, 68)( 18, 67)( 19, 65)( 20, 66)( 21, 72)( 22, 71)( 23, 69)
( 24, 70)( 25, 76)( 26, 75)( 27, 73)( 28, 74)( 29, 80)( 30, 79)( 31, 77)
( 32, 78)( 33, 84)( 34, 83)( 35, 81)( 36, 82)( 37, 88)( 38, 87)( 39, 85)
( 40, 86)( 41, 92)( 42, 91)( 43, 89)( 44, 90)( 45, 96)( 46, 95)( 47, 93)
( 48, 94)( 97,148)( 98,147)( 99,145)(100,146)(101,152)(102,151)(103,149)
(104,150)(105,156)(106,155)(107,153)(108,154)(109,160)(110,159)(111,157)
(112,158)(113,164)(114,163)(115,161)(116,162)(117,168)(118,167)(119,165)
(120,166)(121,172)(122,171)(123,169)(124,170)(125,176)(126,175)(127,173)
(128,174)(129,180)(130,179)(131,177)(132,178)(133,184)(134,183)(135,181)
(136,182)(137,188)(138,187)(139,185)(140,186)(141,192)(142,191)(143,189)
(144,190)(193,244)(194,243)(195,241)(196,242)(197,248)(198,247)(199,245)
(200,246)(201,252)(202,251)(203,249)(204,250)(205,256)(206,255)(207,253)
(208,254)(209,260)(210,259)(211,257)(212,258)(213,264)(214,263)(215,261)
(216,262)(217,268)(218,267)(219,265)(220,266)(221,272)(222,271)(223,269)
(224,270)(225,276)(226,275)(227,273)(228,274)(229,280)(230,279)(231,277)
(232,278)(233,284)(234,283)(235,281)(236,282)(237,288)(238,287)(239,285)
(240,286)(289,340)(290,339)(291,337)(292,338)(293,344)(294,343)(295,341)
(296,342)(297,348)(298,347)(299,345)(300,346)(301,352)(302,351)(303,349)
(304,350)(305,356)(306,355)(307,353)(308,354)(309,360)(310,359)(311,357)
(312,358)(313,364)(314,363)(315,361)(316,362)(317,368)(318,367)(319,365)
(320,366)(321,372)(322,371)(323,369)(324,370)(325,376)(326,375)(327,373)
(328,374)(329,380)(330,379)(331,377)(332,378)(333,384)(334,383)(335,381)
(336,382);
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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 >;
References : None.
to this polytope