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Polytope of Type {4,8,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,8,6}*768c
if this polytope has a name.
Group : SmallGroup(768,1089251)
Rank : 4
Schlafli Type : {4,8,6}
Number of vertices, edges, etc : 4, 32, 48, 12
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 : {4,8,3}*384, {4,4,6}*384d, {2,8,6}*384b
   4-fold quotients : {4,4,3}*192b, {2,8,3}*192, {2,4,6}*192
   8-fold quotients : {4,2,6}*96, {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
   16-fold quotients : {4,2,3}*48, {2,4,3}*48, {2,2,6}*48
   24-fold quotients : {4,2,2}*32
   32-fold quotients : {2,2,3}*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)
( 25,121)( 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);;
s1 := (  1, 55)(  2, 56)(  3, 53)(  4, 54)(  5, 52)(  6, 51)(  7, 50)(  8, 49)
(  9, 63)( 10, 64)( 11, 61)( 12, 62)( 13, 60)( 14, 59)( 15, 58)( 16, 57)
( 17, 71)( 18, 72)( 19, 69)( 20, 70)( 21, 68)( 22, 67)( 23, 66)( 24, 65)
( 25, 79)( 26, 80)( 27, 77)( 28, 78)( 29, 76)( 30, 75)( 31, 74)( 32, 73)
( 33, 87)( 34, 88)( 35, 85)( 36, 86)( 37, 84)( 38, 83)( 39, 82)( 40, 81)
( 41, 95)( 42, 96)( 43, 93)( 44, 94)( 45, 92)( 46, 91)( 47, 90)( 48, 89)
( 97,151)( 98,152)( 99,149)(100,150)(101,148)(102,147)(103,146)(104,145)
(105,159)(106,160)(107,157)(108,158)(109,156)(110,155)(111,154)(112,153)
(113,167)(114,168)(115,165)(116,166)(117,164)(118,163)(119,162)(120,161)
(121,175)(122,176)(123,173)(124,174)(125,172)(126,171)(127,170)(128,169)
(129,183)(130,184)(131,181)(132,182)(133,180)(134,179)(135,178)(136,177)
(137,191)(138,192)(139,189)(140,190)(141,188)(142,187)(143,186)(144,185)
(193,247)(194,248)(195,245)(196,246)(197,244)(198,243)(199,242)(200,241)
(201,255)(202,256)(203,253)(204,254)(205,252)(206,251)(207,250)(208,249)
(209,263)(210,264)(211,261)(212,262)(213,260)(214,259)(215,258)(216,257)
(217,271)(218,272)(219,269)(220,270)(221,268)(222,267)(223,266)(224,265)
(225,279)(226,280)(227,277)(228,278)(229,276)(230,275)(231,274)(232,273)
(233,287)(234,288)(235,285)(236,286)(237,284)(238,283)(239,282)(240,281)
(289,343)(290,344)(291,341)(292,342)(293,340)(294,339)(295,338)(296,337)
(297,351)(298,352)(299,349)(300,350)(301,348)(302,347)(303,346)(304,345)
(305,359)(306,360)(307,357)(308,358)(309,356)(310,355)(311,354)(312,353)
(313,367)(314,368)(315,365)(316,366)(317,364)(318,363)(319,362)(320,361)
(321,375)(322,376)(323,373)(324,374)(325,372)(326,371)(327,370)(328,369)
(329,383)(330,384)(331,381)(332,382)(333,380)(334,379)(335,378)(336,377);;
s2 := (  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);;
s3 := (  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);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2 ];;
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)( 25,121)( 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);
s1 := Sym(384)!(  1, 55)(  2, 56)(  3, 53)(  4, 54)(  5, 52)(  6, 51)(  7, 50)
(  8, 49)(  9, 63)( 10, 64)( 11, 61)( 12, 62)( 13, 60)( 14, 59)( 15, 58)
( 16, 57)( 17, 71)( 18, 72)( 19, 69)( 20, 70)( 21, 68)( 22, 67)( 23, 66)
( 24, 65)( 25, 79)( 26, 80)( 27, 77)( 28, 78)( 29, 76)( 30, 75)( 31, 74)
( 32, 73)( 33, 87)( 34, 88)( 35, 85)( 36, 86)( 37, 84)( 38, 83)( 39, 82)
( 40, 81)( 41, 95)( 42, 96)( 43, 93)( 44, 94)( 45, 92)( 46, 91)( 47, 90)
( 48, 89)( 97,151)( 98,152)( 99,149)(100,150)(101,148)(102,147)(103,146)
(104,145)(105,159)(106,160)(107,157)(108,158)(109,156)(110,155)(111,154)
(112,153)(113,167)(114,168)(115,165)(116,166)(117,164)(118,163)(119,162)
(120,161)(121,175)(122,176)(123,173)(124,174)(125,172)(126,171)(127,170)
(128,169)(129,183)(130,184)(131,181)(132,182)(133,180)(134,179)(135,178)
(136,177)(137,191)(138,192)(139,189)(140,190)(141,188)(142,187)(143,186)
(144,185)(193,247)(194,248)(195,245)(196,246)(197,244)(198,243)(199,242)
(200,241)(201,255)(202,256)(203,253)(204,254)(205,252)(206,251)(207,250)
(208,249)(209,263)(210,264)(211,261)(212,262)(213,260)(214,259)(215,258)
(216,257)(217,271)(218,272)(219,269)(220,270)(221,268)(222,267)(223,266)
(224,265)(225,279)(226,280)(227,277)(228,278)(229,276)(230,275)(231,274)
(232,273)(233,287)(234,288)(235,285)(236,286)(237,284)(238,283)(239,282)
(240,281)(289,343)(290,344)(291,341)(292,342)(293,340)(294,339)(295,338)
(296,337)(297,351)(298,352)(299,349)(300,350)(301,348)(302,347)(303,346)
(304,345)(305,359)(306,360)(307,357)(308,358)(309,356)(310,355)(311,354)
(312,353)(313,367)(314,368)(315,365)(316,366)(317,364)(318,363)(319,362)
(320,361)(321,375)(322,376)(323,373)(324,374)(325,372)(326,371)(327,370)
(328,369)(329,383)(330,384)(331,381)(332,382)(333,380)(334,379)(335,378)
(336,377);
s2 := 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);
s3 := 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);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2 >; 
 
References : None.
to this polytope