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