Polytope of Type {4,6,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,8}*768e
if this polytope has a name.
Group : SmallGroup(768,1090195)
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
Non-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 : {4,3,8}*384, {4,6,4}*384c
4-fold quotients : {4,3,4}*192a, {4,6,4}*192d, {4,6,4}*192e
8-fold quotients : {4,6,2}*96c, {4,3,4}*96
16-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 25)( 2, 26)( 3, 27)( 4, 28)( 5, 29)( 6, 30)( 7, 31)( 8, 32)
( 9, 17)( 10, 18)( 11, 19)( 12, 20)( 13, 21)( 14, 22)( 15, 23)( 16, 24)
( 33, 57)( 34, 58)( 35, 59)( 36, 60)( 37, 61)( 38, 62)( 39, 63)( 40, 64)
( 41, 49)( 42, 50)( 43, 51)( 44, 52)( 45, 53)( 46, 54)( 47, 55)( 48, 56)
( 65, 89)( 66, 90)( 67, 91)( 68, 92)( 69, 93)( 70, 94)( 71, 95)( 72, 96)
( 73, 81)( 74, 82)( 75, 83)( 76, 84)( 77, 85)( 78, 86)( 79, 87)( 80, 88)
( 97,121)( 98,122)( 99,123)(100,124)(101,125)(102,126)(103,127)(104,128)
(105,113)(106,114)(107,115)(108,116)(109,117)(110,118)(111,119)(112,120)
(129,153)(130,154)(131,155)(132,156)(133,157)(134,158)(135,159)(136,160)
(137,145)(138,146)(139,147)(140,148)(141,149)(142,150)(143,151)(144,152)
(161,185)(162,186)(163,187)(164,188)(165,189)(166,190)(167,191)(168,192)
(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,183)(176,184)
(193,217)(194,218)(195,219)(196,220)(197,221)(198,222)(199,223)(200,224)
(201,209)(202,210)(203,211)(204,212)(205,213)(206,214)(207,215)(208,216)
(225,249)(226,250)(227,251)(228,252)(229,253)(230,254)(231,255)(232,256)
(233,241)(234,242)(235,243)(236,244)(237,245)(238,246)(239,247)(240,248)
(257,281)(258,282)(259,283)(260,284)(261,285)(262,286)(263,287)(264,288)
(265,273)(266,274)(267,275)(268,276)(269,277)(270,278)(271,279)(272,280)
(289,313)(290,314)(291,315)(292,316)(293,317)(294,318)(295,319)(296,320)
(297,305)(298,306)(299,307)(300,308)(301,309)(302,310)(303,311)(304,312)
(321,345)(322,346)(323,347)(324,348)(325,349)(326,350)(327,351)(328,352)
(329,337)(330,338)(331,339)(332,340)(333,341)(334,342)(335,343)(336,344)
(353,377)(354,378)(355,379)(356,380)(357,381)(358,382)(359,383)(360,384)
(361,369)(362,370)(363,371)(364,372)(365,373)(366,374)(367,375)(368,376);;
s1 := ( 3, 4)( 5, 7)( 6, 8)( 11, 12)( 13, 15)( 14, 16)( 17, 25)( 18, 26)
( 19, 28)( 20, 27)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)( 34, 66)
( 35, 68)( 36, 67)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)( 42, 74)
( 43, 76)( 44, 75)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)( 50, 90)
( 51, 92)( 52, 91)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)( 58, 82)
( 59, 84)( 60, 83)( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 97, 98)(101,104)
(102,103)(105,106)(109,112)(110,111)(113,122)(114,121)(115,123)(116,124)
(117,128)(118,127)(119,126)(120,125)(129,162)(130,161)(131,163)(132,164)
(133,168)(134,167)(135,166)(136,165)(137,170)(138,169)(139,171)(140,172)
(141,176)(142,175)(143,174)(144,173)(145,186)(146,185)(147,187)(148,188)
(149,192)(150,191)(151,190)(152,189)(153,178)(154,177)(155,179)(156,180)
(157,184)(158,183)(159,182)(160,181)(195,196)(197,199)(198,200)(203,204)
(205,207)(206,208)(209,217)(210,218)(211,220)(212,219)(213,223)(214,224)
(215,221)(216,222)(225,257)(226,258)(227,260)(228,259)(229,263)(230,264)
(231,261)(232,262)(233,265)(234,266)(235,268)(236,267)(237,271)(238,272)
(239,269)(240,270)(241,281)(242,282)(243,284)(244,283)(245,287)(246,288)
(247,285)(248,286)(249,273)(250,274)(251,276)(252,275)(253,279)(254,280)
(255,277)(256,278)(289,290)(293,296)(294,295)(297,298)(301,304)(302,303)
(305,314)(306,313)(307,315)(308,316)(309,320)(310,319)(311,318)(312,317)
(321,354)(322,353)(323,355)(324,356)(325,360)(326,359)(327,358)(328,357)
(329,362)(330,361)(331,363)(332,364)(333,368)(334,367)(335,366)(336,365)
(337,378)(338,377)(339,379)(340,380)(341,384)(342,383)(343,382)(344,381)
(345,370)(346,369)(347,371)(348,372)(349,376)(350,375)(351,374)(352,373);;
s2 := ( 1,257)( 2,258)( 3,262)( 4,261)( 5,260)( 6,259)( 7,264)( 8,263)
( 9,273)( 10,274)( 11,278)( 12,277)( 13,276)( 14,275)( 15,280)( 16,279)
( 17,265)( 18,266)( 19,270)( 20,269)( 21,268)( 22,267)( 23,272)( 24,271)
( 25,281)( 26,282)( 27,286)( 28,285)( 29,284)( 30,283)( 31,288)( 32,287)
( 33,225)( 34,226)( 35,230)( 36,229)( 37,228)( 38,227)( 39,232)( 40,231)
( 41,241)( 42,242)( 43,246)( 44,245)( 45,244)( 46,243)( 47,248)( 48,247)
( 49,233)( 50,234)( 51,238)( 52,237)( 53,236)( 54,235)( 55,240)( 56,239)
( 57,249)( 58,250)( 59,254)( 60,253)( 61,252)( 62,251)( 63,256)( 64,255)
( 65,193)( 66,194)( 67,198)( 68,197)( 69,196)( 70,195)( 71,200)( 72,199)
( 73,209)( 74,210)( 75,214)( 76,213)( 77,212)( 78,211)( 79,216)( 80,215)
( 81,201)( 82,202)( 83,206)( 84,205)( 85,204)( 86,203)( 87,208)( 88,207)
( 89,217)( 90,218)( 91,222)( 92,221)( 93,220)( 94,219)( 95,224)( 96,223)
( 97,354)( 98,353)( 99,357)(100,358)(101,355)(102,356)(103,359)(104,360)
(105,370)(106,369)(107,373)(108,374)(109,371)(110,372)(111,375)(112,376)
(113,362)(114,361)(115,365)(116,366)(117,363)(118,364)(119,367)(120,368)
(121,378)(122,377)(123,381)(124,382)(125,379)(126,380)(127,383)(128,384)
(129,322)(130,321)(131,325)(132,326)(133,323)(134,324)(135,327)(136,328)
(137,338)(138,337)(139,341)(140,342)(141,339)(142,340)(143,343)(144,344)
(145,330)(146,329)(147,333)(148,334)(149,331)(150,332)(151,335)(152,336)
(153,346)(154,345)(155,349)(156,350)(157,347)(158,348)(159,351)(160,352)
(161,290)(162,289)(163,293)(164,294)(165,291)(166,292)(167,295)(168,296)
(169,306)(170,305)(171,309)(172,310)(173,307)(174,308)(175,311)(176,312)
(177,298)(178,297)(179,301)(180,302)(181,299)(182,300)(183,303)(184,304)
(185,314)(186,313)(187,317)(188,318)(189,315)(190,316)(191,319)(192,320);;
s3 := ( 1,100)( 2, 99)( 3, 97)( 4, 98)( 5,104)( 6,103)( 7,101)( 8,102)
( 9,108)( 10,107)( 11,105)( 12,106)( 13,112)( 14,111)( 15,109)( 16,110)
( 17,116)( 18,115)( 19,113)( 20,114)( 21,120)( 22,119)( 23,117)( 24,118)
( 25,124)( 26,123)( 27,121)( 28,122)( 29,128)( 30,127)( 31,125)( 32,126)
( 33,132)( 34,131)( 35,129)( 36,130)( 37,136)( 38,135)( 39,133)( 40,134)
( 41,140)( 42,139)( 43,137)( 44,138)( 45,144)( 46,143)( 47,141)( 48,142)
( 49,148)( 50,147)( 51,145)( 52,146)( 53,152)( 54,151)( 55,149)( 56,150)
( 57,156)( 58,155)( 59,153)( 60,154)( 61,160)( 62,159)( 63,157)( 64,158)
( 65,164)( 66,163)( 67,161)( 68,162)( 69,168)( 70,167)( 71,165)( 72,166)
( 73,172)( 74,171)( 75,169)( 76,170)( 77,176)( 78,175)( 79,173)( 80,174)
( 81,180)( 82,179)( 83,177)( 84,178)( 85,184)( 86,183)( 87,181)( 88,182)
( 89,188)( 90,187)( 91,185)( 92,186)( 93,192)( 94,191)( 95,189)( 96,190)
(193,292)(194,291)(195,289)(196,290)(197,296)(198,295)(199,293)(200,294)
(201,300)(202,299)(203,297)(204,298)(205,304)(206,303)(207,301)(208,302)
(209,308)(210,307)(211,305)(212,306)(213,312)(214,311)(215,309)(216,310)
(217,316)(218,315)(219,313)(220,314)(221,320)(222,319)(223,317)(224,318)
(225,324)(226,323)(227,321)(228,322)(229,328)(230,327)(231,325)(232,326)
(233,332)(234,331)(235,329)(236,330)(237,336)(238,335)(239,333)(240,334)
(241,340)(242,339)(243,337)(244,338)(245,344)(246,343)(247,341)(248,342)
(249,348)(250,347)(251,345)(252,346)(253,352)(254,351)(255,349)(256,350)
(257,356)(258,355)(259,353)(260,354)(261,360)(262,359)(263,357)(264,358)
(265,364)(266,363)(267,361)(268,362)(269,368)(270,367)(271,365)(272,366)
(273,372)(274,371)(275,369)(276,370)(277,376)(278,375)(279,373)(280,374)
(281,380)(282,379)(283,377)(284,378)(285,384)(286,383)(287,381)(288,382);;
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*s0*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, 25)( 2, 26)( 3, 27)( 4, 28)( 5, 29)( 6, 30)( 7, 31)
( 8, 32)( 9, 17)( 10, 18)( 11, 19)( 12, 20)( 13, 21)( 14, 22)( 15, 23)
( 16, 24)( 33, 57)( 34, 58)( 35, 59)( 36, 60)( 37, 61)( 38, 62)( 39, 63)
( 40, 64)( 41, 49)( 42, 50)( 43, 51)( 44, 52)( 45, 53)( 46, 54)( 47, 55)
( 48, 56)( 65, 89)( 66, 90)( 67, 91)( 68, 92)( 69, 93)( 70, 94)( 71, 95)
( 72, 96)( 73, 81)( 74, 82)( 75, 83)( 76, 84)( 77, 85)( 78, 86)( 79, 87)
( 80, 88)( 97,121)( 98,122)( 99,123)(100,124)(101,125)(102,126)(103,127)
(104,128)(105,113)(106,114)(107,115)(108,116)(109,117)(110,118)(111,119)
(112,120)(129,153)(130,154)(131,155)(132,156)(133,157)(134,158)(135,159)
(136,160)(137,145)(138,146)(139,147)(140,148)(141,149)(142,150)(143,151)
(144,152)(161,185)(162,186)(163,187)(164,188)(165,189)(166,190)(167,191)
(168,192)(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,183)
(176,184)(193,217)(194,218)(195,219)(196,220)(197,221)(198,222)(199,223)
(200,224)(201,209)(202,210)(203,211)(204,212)(205,213)(206,214)(207,215)
(208,216)(225,249)(226,250)(227,251)(228,252)(229,253)(230,254)(231,255)
(232,256)(233,241)(234,242)(235,243)(236,244)(237,245)(238,246)(239,247)
(240,248)(257,281)(258,282)(259,283)(260,284)(261,285)(262,286)(263,287)
(264,288)(265,273)(266,274)(267,275)(268,276)(269,277)(270,278)(271,279)
(272,280)(289,313)(290,314)(291,315)(292,316)(293,317)(294,318)(295,319)
(296,320)(297,305)(298,306)(299,307)(300,308)(301,309)(302,310)(303,311)
(304,312)(321,345)(322,346)(323,347)(324,348)(325,349)(326,350)(327,351)
(328,352)(329,337)(330,338)(331,339)(332,340)(333,341)(334,342)(335,343)
(336,344)(353,377)(354,378)(355,379)(356,380)(357,381)(358,382)(359,383)
(360,384)(361,369)(362,370)(363,371)(364,372)(365,373)(366,374)(367,375)
(368,376);
s1 := Sym(384)!( 3, 4)( 5, 7)( 6, 8)( 11, 12)( 13, 15)( 14, 16)( 17, 25)
( 18, 26)( 19, 28)( 20, 27)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)
( 34, 66)( 35, 68)( 36, 67)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)
( 42, 74)( 43, 76)( 44, 75)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)
( 50, 90)( 51, 92)( 52, 91)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)
( 58, 82)( 59, 84)( 60, 83)( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 97, 98)
(101,104)(102,103)(105,106)(109,112)(110,111)(113,122)(114,121)(115,123)
(116,124)(117,128)(118,127)(119,126)(120,125)(129,162)(130,161)(131,163)
(132,164)(133,168)(134,167)(135,166)(136,165)(137,170)(138,169)(139,171)
(140,172)(141,176)(142,175)(143,174)(144,173)(145,186)(146,185)(147,187)
(148,188)(149,192)(150,191)(151,190)(152,189)(153,178)(154,177)(155,179)
(156,180)(157,184)(158,183)(159,182)(160,181)(195,196)(197,199)(198,200)
(203,204)(205,207)(206,208)(209,217)(210,218)(211,220)(212,219)(213,223)
(214,224)(215,221)(216,222)(225,257)(226,258)(227,260)(228,259)(229,263)
(230,264)(231,261)(232,262)(233,265)(234,266)(235,268)(236,267)(237,271)
(238,272)(239,269)(240,270)(241,281)(242,282)(243,284)(244,283)(245,287)
(246,288)(247,285)(248,286)(249,273)(250,274)(251,276)(252,275)(253,279)
(254,280)(255,277)(256,278)(289,290)(293,296)(294,295)(297,298)(301,304)
(302,303)(305,314)(306,313)(307,315)(308,316)(309,320)(310,319)(311,318)
(312,317)(321,354)(322,353)(323,355)(324,356)(325,360)(326,359)(327,358)
(328,357)(329,362)(330,361)(331,363)(332,364)(333,368)(334,367)(335,366)
(336,365)(337,378)(338,377)(339,379)(340,380)(341,384)(342,383)(343,382)
(344,381)(345,370)(346,369)(347,371)(348,372)(349,376)(350,375)(351,374)
(352,373);
s2 := Sym(384)!( 1,257)( 2,258)( 3,262)( 4,261)( 5,260)( 6,259)( 7,264)
( 8,263)( 9,273)( 10,274)( 11,278)( 12,277)( 13,276)( 14,275)( 15,280)
( 16,279)( 17,265)( 18,266)( 19,270)( 20,269)( 21,268)( 22,267)( 23,272)
( 24,271)( 25,281)( 26,282)( 27,286)( 28,285)( 29,284)( 30,283)( 31,288)
( 32,287)( 33,225)( 34,226)( 35,230)( 36,229)( 37,228)( 38,227)( 39,232)
( 40,231)( 41,241)( 42,242)( 43,246)( 44,245)( 45,244)( 46,243)( 47,248)
( 48,247)( 49,233)( 50,234)( 51,238)( 52,237)( 53,236)( 54,235)( 55,240)
( 56,239)( 57,249)( 58,250)( 59,254)( 60,253)( 61,252)( 62,251)( 63,256)
( 64,255)( 65,193)( 66,194)( 67,198)( 68,197)( 69,196)( 70,195)( 71,200)
( 72,199)( 73,209)( 74,210)( 75,214)( 76,213)( 77,212)( 78,211)( 79,216)
( 80,215)( 81,201)( 82,202)( 83,206)( 84,205)( 85,204)( 86,203)( 87,208)
( 88,207)( 89,217)( 90,218)( 91,222)( 92,221)( 93,220)( 94,219)( 95,224)
( 96,223)( 97,354)( 98,353)( 99,357)(100,358)(101,355)(102,356)(103,359)
(104,360)(105,370)(106,369)(107,373)(108,374)(109,371)(110,372)(111,375)
(112,376)(113,362)(114,361)(115,365)(116,366)(117,363)(118,364)(119,367)
(120,368)(121,378)(122,377)(123,381)(124,382)(125,379)(126,380)(127,383)
(128,384)(129,322)(130,321)(131,325)(132,326)(133,323)(134,324)(135,327)
(136,328)(137,338)(138,337)(139,341)(140,342)(141,339)(142,340)(143,343)
(144,344)(145,330)(146,329)(147,333)(148,334)(149,331)(150,332)(151,335)
(152,336)(153,346)(154,345)(155,349)(156,350)(157,347)(158,348)(159,351)
(160,352)(161,290)(162,289)(163,293)(164,294)(165,291)(166,292)(167,295)
(168,296)(169,306)(170,305)(171,309)(172,310)(173,307)(174,308)(175,311)
(176,312)(177,298)(178,297)(179,301)(180,302)(181,299)(182,300)(183,303)
(184,304)(185,314)(186,313)(187,317)(188,318)(189,315)(190,316)(191,319)
(192,320);
s3 := Sym(384)!( 1,100)( 2, 99)( 3, 97)( 4, 98)( 5,104)( 6,103)( 7,101)
( 8,102)( 9,108)( 10,107)( 11,105)( 12,106)( 13,112)( 14,111)( 15,109)
( 16,110)( 17,116)( 18,115)( 19,113)( 20,114)( 21,120)( 22,119)( 23,117)
( 24,118)( 25,124)( 26,123)( 27,121)( 28,122)( 29,128)( 30,127)( 31,125)
( 32,126)( 33,132)( 34,131)( 35,129)( 36,130)( 37,136)( 38,135)( 39,133)
( 40,134)( 41,140)( 42,139)( 43,137)( 44,138)( 45,144)( 46,143)( 47,141)
( 48,142)( 49,148)( 50,147)( 51,145)( 52,146)( 53,152)( 54,151)( 55,149)
( 56,150)( 57,156)( 58,155)( 59,153)( 60,154)( 61,160)( 62,159)( 63,157)
( 64,158)( 65,164)( 66,163)( 67,161)( 68,162)( 69,168)( 70,167)( 71,165)
( 72,166)( 73,172)( 74,171)( 75,169)( 76,170)( 77,176)( 78,175)( 79,173)
( 80,174)( 81,180)( 82,179)( 83,177)( 84,178)( 85,184)( 86,183)( 87,181)
( 88,182)( 89,188)( 90,187)( 91,185)( 92,186)( 93,192)( 94,191)( 95,189)
( 96,190)(193,292)(194,291)(195,289)(196,290)(197,296)(198,295)(199,293)
(200,294)(201,300)(202,299)(203,297)(204,298)(205,304)(206,303)(207,301)
(208,302)(209,308)(210,307)(211,305)(212,306)(213,312)(214,311)(215,309)
(216,310)(217,316)(218,315)(219,313)(220,314)(221,320)(222,319)(223,317)
(224,318)(225,324)(226,323)(227,321)(228,322)(229,328)(230,327)(231,325)
(232,326)(233,332)(234,331)(235,329)(236,330)(237,336)(238,335)(239,333)
(240,334)(241,340)(242,339)(243,337)(244,338)(245,344)(246,343)(247,341)
(248,342)(249,348)(250,347)(251,345)(252,346)(253,352)(254,351)(255,349)
(256,350)(257,356)(258,355)(259,353)(260,354)(261,360)(262,359)(263,357)
(264,358)(265,364)(266,363)(267,361)(268,362)(269,368)(270,367)(271,365)
(272,366)(273,372)(274,371)(275,369)(276,370)(277,376)(278,375)(279,373)
(280,374)(281,380)(282,379)(283,377)(284,378)(285,384)(286,383)(287,381)
(288,382);
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*s0*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