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