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Polytope of Type {8,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6,4}*768c
if this polytope has a name.
Group : SmallGroup(768,1089307)
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}*384b
   4-fold quotients : {4,6,4}*192c, {8,3,2}*192, {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)
( 49,247)( 50,248)( 51,245)( 52,246)( 53,244)( 54,243)( 55,242)( 56,241)
( 57,255)( 58,256)( 59,253)( 60,254)( 61,252)( 62,251)( 63,250)( 64,249)
( 65,263)( 66,264)( 67,261)( 68,262)( 69,260)( 70,259)( 71,258)( 72,257)
( 73,271)( 74,272)( 75,269)( 76,270)( 77,268)( 78,267)( 79,266)( 80,265)
( 81,279)( 82,280)( 83,277)( 84,278)( 85,276)( 86,275)( 87,274)( 88,273)
( 89,287)( 90,288)( 91,285)( 92,286)( 93,284)( 94,283)( 95,282)( 96,281)
( 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)
(145,343)(146,344)(147,341)(148,342)(149,340)(150,339)(151,338)(152,337)
(153,351)(154,352)(155,349)(156,350)(157,348)(158,347)(159,346)(160,345)
(161,359)(162,360)(163,357)(164,358)(165,356)(166,355)(167,354)(168,353)
(169,367)(170,368)(171,365)(172,366)(173,364)(174,363)(175,362)(176,361)
(177,375)(178,376)(179,373)(180,374)(181,372)(182,371)(183,370)(184,369)
(185,383)(186,384)(187,381)(188,382)(189,380)(190,379)(191,378)(192,377);;
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)
(193,194)(197,200)(198,199)(201,210)(202,209)(203,211)(204,212)(205,216)
(206,215)(207,214)(208,213)(217,218)(221,224)(222,223)(225,234)(226,233)
(227,235)(228,236)(229,240)(230,239)(231,238)(232,237)(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)(289,290)(293,296)(294,295)(297,306)
(298,305)(299,307)(300,308)(301,312)(302,311)(303,310)(304,309)(313,314)
(317,320)(318,319)(321,330)(322,329)(323,331)(324,332)(325,336)(326,335)
(327,334)(328,333)(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);;
s2 := (  1,113)(  2,114)(  3,118)(  4,117)(  5,116)(  6,115)(  7,120)(  8,119)
(  9,105)( 10,106)( 11,110)( 12,109)( 13,108)( 14,107)( 15,112)( 16,111)
( 17, 97)( 18, 98)( 19,102)( 20,101)( 21,100)( 22, 99)( 23,104)( 24,103)
( 25,137)( 26,138)( 27,142)( 28,141)( 29,140)( 30,139)( 31,144)( 32,143)
( 33,129)( 34,130)( 35,134)( 36,133)( 37,132)( 38,131)( 39,136)( 40,135)
( 41,121)( 42,122)( 43,126)( 44,125)( 45,124)( 46,123)( 47,128)( 48,127)
( 49,185)( 50,186)( 51,190)( 52,189)( 53,188)( 54,187)( 55,192)( 56,191)
( 57,177)( 58,178)( 59,182)( 60,181)( 61,180)( 62,179)( 63,184)( 64,183)
( 65,169)( 66,170)( 67,174)( 68,173)( 69,172)( 70,171)( 71,176)( 72,175)
( 73,161)( 74,162)( 75,166)( 76,165)( 77,164)( 78,163)( 79,168)( 80,167)
( 81,153)( 82,154)( 83,158)( 84,157)( 85,156)( 86,155)( 87,160)( 88,159)
( 89,145)( 90,146)( 91,150)( 92,149)( 93,148)( 94,147)( 95,152)( 96,151)
(193,306)(194,305)(195,309)(196,310)(197,307)(198,308)(199,311)(200,312)
(201,298)(202,297)(203,301)(204,302)(205,299)(206,300)(207,303)(208,304)
(209,290)(210,289)(211,293)(212,294)(213,291)(214,292)(215,295)(216,296)
(217,330)(218,329)(219,333)(220,334)(221,331)(222,332)(223,335)(224,336)
(225,322)(226,321)(227,325)(228,326)(229,323)(230,324)(231,327)(232,328)
(233,314)(234,313)(235,317)(236,318)(237,315)(238,316)(239,319)(240,320)
(241,378)(242,377)(243,381)(244,382)(245,379)(246,380)(247,383)(248,384)
(249,370)(250,369)(251,373)(252,374)(253,371)(254,372)(255,375)(256,376)
(257,362)(258,361)(259,365)(260,366)(261,363)(262,364)(263,367)(264,368)
(265,354)(266,353)(267,357)(268,358)(269,355)(270,356)(271,359)(272,360)
(273,346)(274,345)(275,349)(276,350)(277,347)(278,348)(279,351)(280,352)
(281,338)(282,337)(283,341)(284,342)(285,339)(286,340)(287,343)(288,344);;
s3 := (  1, 49)(  2, 50)(  3, 51)(  4, 52)(  5, 53)(  6, 54)(  7, 55)(  8, 56)
(  9, 57)( 10, 58)( 11, 59)( 12, 60)( 13, 61)( 14, 62)( 15, 63)( 16, 64)
( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)( 24, 72)
( 25, 73)( 26, 74)( 27, 75)( 28, 76)( 29, 77)( 30, 78)( 31, 79)( 32, 80)
( 33, 81)( 34, 82)( 35, 83)( 36, 84)( 37, 85)( 38, 86)( 39, 87)( 40, 88)
( 41, 89)( 42, 90)( 43, 91)( 44, 92)( 45, 93)( 46, 94)( 47, 95)( 48, 96)
( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)
(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)
(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)
(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)
(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)(136,184)
(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)(144,192)
(193,241)(194,242)(195,243)(196,244)(197,245)(198,246)(199,247)(200,248)
(201,249)(202,250)(203,251)(204,252)(205,253)(206,254)(207,255)(208,256)
(209,257)(210,258)(211,259)(212,260)(213,261)(214,262)(215,263)(216,264)
(217,265)(218,266)(219,267)(220,268)(221,269)(222,270)(223,271)(224,272)
(225,273)(226,274)(227,275)(228,276)(229,277)(230,278)(231,279)(232,280)
(233,281)(234,282)(235,283)(236,284)(237,285)(238,286)(239,287)(240,288)
(289,337)(290,338)(291,339)(292,340)(293,341)(294,342)(295,343)(296,344)
(297,345)(298,346)(299,347)(300,348)(301,349)(302,350)(303,351)(304,352)
(305,353)(306,354)(307,355)(308,356)(309,357)(310,358)(311,359)(312,360)
(313,361)(314,362)(315,363)(316,364)(317,365)(318,366)(319,367)(320,368)
(321,369)(322,370)(323,371)(324,372)(325,373)(326,374)(327,375)(328,376)
(329,377)(330,378)(331,379)(332,380)(333,381)(334,382)(335,383)(336,384);;
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*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)( 49,247)( 50,248)( 51,245)( 52,246)( 53,244)( 54,243)( 55,242)
( 56,241)( 57,255)( 58,256)( 59,253)( 60,254)( 61,252)( 62,251)( 63,250)
( 64,249)( 65,263)( 66,264)( 67,261)( 68,262)( 69,260)( 70,259)( 71,258)
( 72,257)( 73,271)( 74,272)( 75,269)( 76,270)( 77,268)( 78,267)( 79,266)
( 80,265)( 81,279)( 82,280)( 83,277)( 84,278)( 85,276)( 86,275)( 87,274)
( 88,273)( 89,287)( 90,288)( 91,285)( 92,286)( 93,284)( 94,283)( 95,282)
( 96,281)( 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)(145,343)(146,344)(147,341)(148,342)(149,340)(150,339)(151,338)
(152,337)(153,351)(154,352)(155,349)(156,350)(157,348)(158,347)(159,346)
(160,345)(161,359)(162,360)(163,357)(164,358)(165,356)(166,355)(167,354)
(168,353)(169,367)(170,368)(171,365)(172,366)(173,364)(174,363)(175,362)
(176,361)(177,375)(178,376)(179,373)(180,374)(181,372)(182,371)(183,370)
(184,369)(185,383)(186,384)(187,381)(188,382)(189,380)(190,379)(191,378)
(192,377);
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)(193,194)(197,200)(198,199)(201,210)(202,209)(203,211)(204,212)
(205,216)(206,215)(207,214)(208,213)(217,218)(221,224)(222,223)(225,234)
(226,233)(227,235)(228,236)(229,240)(230,239)(231,238)(232,237)(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)(289,290)(293,296)(294,295)
(297,306)(298,305)(299,307)(300,308)(301,312)(302,311)(303,310)(304,309)
(313,314)(317,320)(318,319)(321,330)(322,329)(323,331)(324,332)(325,336)
(326,335)(327,334)(328,333)(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);
s2 := Sym(384)!(  1,113)(  2,114)(  3,118)(  4,117)(  5,116)(  6,115)(  7,120)
(  8,119)(  9,105)( 10,106)( 11,110)( 12,109)( 13,108)( 14,107)( 15,112)
( 16,111)( 17, 97)( 18, 98)( 19,102)( 20,101)( 21,100)( 22, 99)( 23,104)
( 24,103)( 25,137)( 26,138)( 27,142)( 28,141)( 29,140)( 30,139)( 31,144)
( 32,143)( 33,129)( 34,130)( 35,134)( 36,133)( 37,132)( 38,131)( 39,136)
( 40,135)( 41,121)( 42,122)( 43,126)( 44,125)( 45,124)( 46,123)( 47,128)
( 48,127)( 49,185)( 50,186)( 51,190)( 52,189)( 53,188)( 54,187)( 55,192)
( 56,191)( 57,177)( 58,178)( 59,182)( 60,181)( 61,180)( 62,179)( 63,184)
( 64,183)( 65,169)( 66,170)( 67,174)( 68,173)( 69,172)( 70,171)( 71,176)
( 72,175)( 73,161)( 74,162)( 75,166)( 76,165)( 77,164)( 78,163)( 79,168)
( 80,167)( 81,153)( 82,154)( 83,158)( 84,157)( 85,156)( 86,155)( 87,160)
( 88,159)( 89,145)( 90,146)( 91,150)( 92,149)( 93,148)( 94,147)( 95,152)
( 96,151)(193,306)(194,305)(195,309)(196,310)(197,307)(198,308)(199,311)
(200,312)(201,298)(202,297)(203,301)(204,302)(205,299)(206,300)(207,303)
(208,304)(209,290)(210,289)(211,293)(212,294)(213,291)(214,292)(215,295)
(216,296)(217,330)(218,329)(219,333)(220,334)(221,331)(222,332)(223,335)
(224,336)(225,322)(226,321)(227,325)(228,326)(229,323)(230,324)(231,327)
(232,328)(233,314)(234,313)(235,317)(236,318)(237,315)(238,316)(239,319)
(240,320)(241,378)(242,377)(243,381)(244,382)(245,379)(246,380)(247,383)
(248,384)(249,370)(250,369)(251,373)(252,374)(253,371)(254,372)(255,375)
(256,376)(257,362)(258,361)(259,365)(260,366)(261,363)(262,364)(263,367)
(264,368)(265,354)(266,353)(267,357)(268,358)(269,355)(270,356)(271,359)
(272,360)(273,346)(274,345)(275,349)(276,350)(277,347)(278,348)(279,351)
(280,352)(281,338)(282,337)(283,341)(284,342)(285,339)(286,340)(287,343)
(288,344);
s3 := Sym(384)!(  1, 49)(  2, 50)(  3, 51)(  4, 52)(  5, 53)(  6, 54)(  7, 55)
(  8, 56)(  9, 57)( 10, 58)( 11, 59)( 12, 60)( 13, 61)( 14, 62)( 15, 63)
( 16, 64)( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)
( 24, 72)( 25, 73)( 26, 74)( 27, 75)( 28, 76)( 29, 77)( 30, 78)( 31, 79)
( 32, 80)( 33, 81)( 34, 82)( 35, 83)( 36, 84)( 37, 85)( 38, 86)( 39, 87)
( 40, 88)( 41, 89)( 42, 90)( 43, 91)( 44, 92)( 45, 93)( 46, 94)( 47, 95)
( 48, 96)( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)
(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)
(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)
(120,168)(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)
(128,176)(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)
(136,184)(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)
(144,192)(193,241)(194,242)(195,243)(196,244)(197,245)(198,246)(199,247)
(200,248)(201,249)(202,250)(203,251)(204,252)(205,253)(206,254)(207,255)
(208,256)(209,257)(210,258)(211,259)(212,260)(213,261)(214,262)(215,263)
(216,264)(217,265)(218,266)(219,267)(220,268)(221,269)(222,270)(223,271)
(224,272)(225,273)(226,274)(227,275)(228,276)(229,277)(230,278)(231,279)
(232,280)(233,281)(234,282)(235,283)(236,284)(237,285)(238,286)(239,287)
(240,288)(289,337)(290,338)(291,339)(292,340)(293,341)(294,342)(295,343)
(296,344)(297,345)(298,346)(299,347)(300,348)(301,349)(302,350)(303,351)
(304,352)(305,353)(306,354)(307,355)(308,356)(309,357)(310,358)(311,359)
(312,360)(313,361)(314,362)(315,363)(316,364)(317,365)(318,366)(319,367)
(320,368)(321,369)(322,370)(323,371)(324,372)(325,373)(326,374)(327,375)
(328,376)(329,377)(330,378)(331,379)(332,380)(333,381)(334,382)(335,383)
(336,384);
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope