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