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