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Polytope of Type {4,6,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,8}*768b
if this polytope has a name.
Group : SmallGroup(768,1089292)
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}*384c
   4-fold quotients : {4,6,4}*192b, {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)
(193,241)(194,242)(195,244)(196,243)(197,247)(198,248)(199,245)(200,246)
(201,257)(202,258)(203,260)(204,259)(205,263)(206,264)(207,261)(208,262)
(209,249)(210,250)(211,252)(212,251)(213,255)(214,256)(215,253)(216,254)
(217,265)(218,266)(219,268)(220,267)(221,271)(222,272)(223,269)(224,270)
(225,281)(226,282)(227,284)(228,283)(229,287)(230,288)(231,285)(232,286)
(233,273)(234,274)(235,276)(236,275)(237,279)(238,280)(239,277)(240,278)
(289,337)(290,338)(291,340)(292,339)(293,343)(294,344)(295,341)(296,342)
(297,353)(298,354)(299,356)(300,355)(301,359)(302,360)(303,357)(304,358)
(305,345)(306,346)(307,348)(308,347)(309,351)(310,352)(311,349)(312,350)
(313,361)(314,362)(315,364)(316,363)(317,367)(318,368)(319,365)(320,366)
(321,377)(322,378)(323,380)(324,379)(325,383)(326,384)(327,381)(328,382)
(329,369)(330,370)(331,372)(332,371)(333,375)(334,376)(335,373)(336,374);;
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)
(193,209)(194,210)(195,214)(196,213)(197,212)(198,211)(199,216)(200,215)
(203,206)(204,205)(207,208)(217,233)(218,234)(219,238)(220,237)(221,236)
(222,235)(223,240)(224,239)(227,230)(228,229)(231,232)(241,257)(242,258)
(243,262)(244,261)(245,260)(246,259)(247,264)(248,263)(251,254)(252,253)
(255,256)(265,281)(266,282)(267,286)(268,285)(269,284)(270,283)(271,288)
(272,287)(275,278)(276,277)(279,280)(289,305)(290,306)(291,310)(292,309)
(293,308)(294,307)(295,312)(296,311)(299,302)(300,301)(303,304)(313,329)
(314,330)(315,334)(316,333)(317,332)(318,331)(319,336)(320,335)(323,326)
(324,325)(327,328)(337,353)(338,354)(339,358)(340,357)(341,356)(342,355)
(343,360)(344,359)(347,350)(348,349)(351,352)(361,377)(362,378)(363,382)
(364,381)(365,380)(366,379)(367,384)(368,383)(371,374)(372,373)(375,376);;
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)
( 49,243)( 50,244)( 51,242)( 52,241)( 53,247)( 54,248)( 55,246)( 56,245)
( 57,251)( 58,252)( 59,250)( 60,249)( 61,255)( 62,256)( 63,254)( 64,253)
( 65,259)( 66,260)( 67,258)( 68,257)( 69,263)( 70,264)( 71,262)( 72,261)
( 73,267)( 74,268)( 75,266)( 76,265)( 77,271)( 78,272)( 79,270)( 80,269)
( 81,275)( 82,276)( 83,274)( 84,273)( 85,279)( 86,280)( 87,278)( 88,277)
( 89,283)( 90,284)( 91,282)( 92,281)( 93,287)( 94,288)( 95,286)( 96,285)
( 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)
(145,339)(146,340)(147,338)(148,337)(149,343)(150,344)(151,342)(152,341)
(153,347)(154,348)(155,346)(156,345)(157,351)(158,352)(159,350)(160,349)
(161,355)(162,356)(163,354)(164,353)(165,359)(166,360)(167,358)(168,357)
(169,363)(170,364)(171,362)(172,361)(173,367)(174,368)(175,366)(176,365)
(177,371)(178,372)(179,370)(180,369)(181,375)(182,376)(183,374)(184,373)
(185,379)(186,380)(187,378)(188,377)(189,383)(190,384)(191,382)(192,381);;
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, 
s3*s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*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, 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)(193,241)(194,242)(195,244)(196,243)(197,247)(198,248)(199,245)
(200,246)(201,257)(202,258)(203,260)(204,259)(205,263)(206,264)(207,261)
(208,262)(209,249)(210,250)(211,252)(212,251)(213,255)(214,256)(215,253)
(216,254)(217,265)(218,266)(219,268)(220,267)(221,271)(222,272)(223,269)
(224,270)(225,281)(226,282)(227,284)(228,283)(229,287)(230,288)(231,285)
(232,286)(233,273)(234,274)(235,276)(236,275)(237,279)(238,280)(239,277)
(240,278)(289,337)(290,338)(291,340)(292,339)(293,343)(294,344)(295,341)
(296,342)(297,353)(298,354)(299,356)(300,355)(301,359)(302,360)(303,357)
(304,358)(305,345)(306,346)(307,348)(308,347)(309,351)(310,352)(311,349)
(312,350)(313,361)(314,362)(315,364)(316,363)(317,367)(318,368)(319,365)
(320,366)(321,377)(322,378)(323,380)(324,379)(325,383)(326,384)(327,381)
(328,382)(329,369)(330,370)(331,372)(332,371)(333,375)(334,376)(335,373)
(336,374);
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)(193,209)(194,210)(195,214)(196,213)(197,212)(198,211)(199,216)
(200,215)(203,206)(204,205)(207,208)(217,233)(218,234)(219,238)(220,237)
(221,236)(222,235)(223,240)(224,239)(227,230)(228,229)(231,232)(241,257)
(242,258)(243,262)(244,261)(245,260)(246,259)(247,264)(248,263)(251,254)
(252,253)(255,256)(265,281)(266,282)(267,286)(268,285)(269,284)(270,283)
(271,288)(272,287)(275,278)(276,277)(279,280)(289,305)(290,306)(291,310)
(292,309)(293,308)(294,307)(295,312)(296,311)(299,302)(300,301)(303,304)
(313,329)(314,330)(315,334)(316,333)(317,332)(318,331)(319,336)(320,335)
(323,326)(324,325)(327,328)(337,353)(338,354)(339,358)(340,357)(341,356)
(342,355)(343,360)(344,359)(347,350)(348,349)(351,352)(361,377)(362,378)
(363,382)(364,381)(365,380)(366,379)(367,384)(368,383)(371,374)(372,373)
(375,376);
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)( 49,243)( 50,244)( 51,242)( 52,241)( 53,247)( 54,248)( 55,246)
( 56,245)( 57,251)( 58,252)( 59,250)( 60,249)( 61,255)( 62,256)( 63,254)
( 64,253)( 65,259)( 66,260)( 67,258)( 68,257)( 69,263)( 70,264)( 71,262)
( 72,261)( 73,267)( 74,268)( 75,266)( 76,265)( 77,271)( 78,272)( 79,270)
( 80,269)( 81,275)( 82,276)( 83,274)( 84,273)( 85,279)( 86,280)( 87,278)
( 88,277)( 89,283)( 90,284)( 91,282)( 92,281)( 93,287)( 94,288)( 95,286)
( 96,285)( 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)(145,339)(146,340)(147,338)(148,337)(149,343)(150,344)(151,342)
(152,341)(153,347)(154,348)(155,346)(156,345)(157,351)(158,352)(159,350)
(160,349)(161,355)(162,356)(163,354)(164,353)(165,359)(166,360)(167,358)
(168,357)(169,363)(170,364)(171,362)(172,361)(173,367)(174,368)(175,366)
(176,365)(177,371)(178,372)(179,370)(180,369)(181,375)(182,376)(183,374)
(184,373)(185,379)(186,380)(187,378)(188,377)(189,383)(190,384)(191,382)
(192,381);
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, 
s3*s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2 >; 
 
References : None.
to this polytope