Polytope of Type {8,4,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,4,6}*768a
if this polytope has a name.
Group : SmallGroup(768,141629)
Rank : 4
Schlafli Type : {8,4,6}
Number of vertices, edges, etc : 16, 32, 24, 6
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 : {8,4,6}*384a, {8,4,6}*384b, {4,4,6}*384a
3-fold quotients : {8,4,2}*256a
4-fold quotients : {4,4,6}*192, {8,2,6}*192
6-fold quotients : {8,4,2}*128a, {8,4,2}*128b, {4,4,2}*128
8-fold quotients : {8,2,3}*96, {2,4,6}*96a, {4,2,6}*96
12-fold quotients : {4,4,2}*64, {8,2,2}*64
16-fold quotients : {4,2,3}*48, {2,2,6}*48
24-fold quotients : {2,4,2}*32, {4,2,2}*32
32-fold quotients : {2,2,3}*24
48-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,193)( 2,194)( 3,195)( 4,196)( 5,197)( 6,198)( 7,199)( 8,200)
( 9,201)( 10,202)( 11,203)( 12,204)( 13,208)( 14,209)( 15,210)( 16,205)
( 17,206)( 18,207)( 19,214)( 20,215)( 21,216)( 22,211)( 23,212)( 24,213)
( 25,223)( 26,224)( 27,225)( 28,226)( 29,227)( 30,228)( 31,217)( 32,218)
( 33,219)( 34,220)( 35,221)( 36,222)( 37,238)( 38,239)( 39,240)( 40,235)
( 41,236)( 42,237)( 43,232)( 44,233)( 45,234)( 46,229)( 47,230)( 48,231)
( 49,241)( 50,242)( 51,243)( 52,244)( 53,245)( 54,246)( 55,247)( 56,248)
( 57,249)( 58,250)( 59,251)( 60,252)( 61,256)( 62,257)( 63,258)( 64,253)
( 65,254)( 66,255)( 67,262)( 68,263)( 69,264)( 70,259)( 71,260)( 72,261)
( 73,271)( 74,272)( 75,273)( 76,274)( 77,275)( 78,276)( 79,265)( 80,266)
( 81,267)( 82,268)( 83,269)( 84,270)( 85,286)( 86,287)( 87,288)( 88,283)
( 89,284)( 90,285)( 91,280)( 92,281)( 93,282)( 94,277)( 95,278)( 96,279)
( 97,289)( 98,290)( 99,291)(100,292)(101,293)(102,294)(103,295)(104,296)
(105,297)(106,298)(107,299)(108,300)(109,304)(110,305)(111,306)(112,301)
(113,302)(114,303)(115,310)(116,311)(117,312)(118,307)(119,308)(120,309)
(121,319)(122,320)(123,321)(124,322)(125,323)(126,324)(127,313)(128,314)
(129,315)(130,316)(131,317)(132,318)(133,334)(134,335)(135,336)(136,331)
(137,332)(138,333)(139,328)(140,329)(141,330)(142,325)(143,326)(144,327)
(145,337)(146,338)(147,339)(148,340)(149,341)(150,342)(151,343)(152,344)
(153,345)(154,346)(155,347)(156,348)(157,352)(158,353)(159,354)(160,349)
(161,350)(162,351)(163,358)(164,359)(165,360)(166,355)(167,356)(168,357)
(169,367)(170,368)(171,369)(172,370)(173,371)(174,372)(175,361)(176,362)
(177,363)(178,364)(179,365)(180,366)(181,382)(182,383)(183,384)(184,379)
(185,380)(186,381)(187,376)(188,377)(189,378)(190,373)(191,374)(192,375);;
s1 := ( 25, 31)( 26, 32)( 27, 33)( 28, 34)( 29, 35)( 30, 36)( 37, 43)( 38, 44)
( 39, 45)( 40, 46)( 41, 47)( 42, 48)( 73, 79)( 74, 80)( 75, 81)( 76, 82)
( 77, 83)( 78, 84)( 85, 91)( 86, 92)( 87, 93)( 88, 94)( 89, 95)( 90, 96)
( 97,109)( 98,110)( 99,111)(100,112)(101,113)(102,114)(103,115)(104,116)
(105,117)(106,118)(107,119)(108,120)(121,139)(122,140)(123,141)(124,142)
(125,143)(126,144)(127,133)(128,134)(129,135)(130,136)(131,137)(132,138)
(145,157)(146,158)(147,159)(148,160)(149,161)(150,162)(151,163)(152,164)
(153,165)(154,166)(155,167)(156,168)(169,187)(170,188)(171,189)(172,190)
(173,191)(174,192)(175,181)(176,182)(177,183)(178,184)(179,185)(180,186)
(193,217)(194,218)(195,219)(196,220)(197,221)(198,222)(199,223)(200,224)
(201,225)(202,226)(203,227)(204,228)(205,229)(206,230)(207,231)(208,232)
(209,233)(210,234)(211,235)(212,236)(213,237)(214,238)(215,239)(216,240)
(241,265)(242,266)(243,267)(244,268)(245,269)(246,270)(247,271)(248,272)
(249,273)(250,274)(251,275)(252,276)(253,277)(254,278)(255,279)(256,280)
(257,281)(258,282)(259,283)(260,284)(261,285)(262,286)(263,287)(264,288)
(289,328)(290,329)(291,330)(292,325)(293,326)(294,327)(295,334)(296,335)
(297,336)(298,331)(299,332)(300,333)(301,316)(302,317)(303,318)(304,313)
(305,314)(306,315)(307,322)(308,323)(309,324)(310,319)(311,320)(312,321)
(337,376)(338,377)(339,378)(340,373)(341,374)(342,375)(343,382)(344,383)
(345,384)(346,379)(347,380)(348,381)(349,364)(350,365)(351,366)(352,361)
(353,362)(354,363)(355,370)(356,371)(357,372)(358,367)(359,368)(360,369);;
s2 := ( 1,145)( 2,147)( 3,146)( 4,148)( 5,150)( 6,149)( 7,151)( 8,153)
( 9,152)( 10,154)( 11,156)( 12,155)( 13,157)( 14,159)( 15,158)( 16,160)
( 17,162)( 18,161)( 19,163)( 20,165)( 21,164)( 22,166)( 23,168)( 24,167)
( 25,172)( 26,174)( 27,173)( 28,169)( 29,171)( 30,170)( 31,178)( 32,180)
( 33,179)( 34,175)( 35,177)( 36,176)( 37,184)( 38,186)( 39,185)( 40,181)
( 41,183)( 42,182)( 43,190)( 44,192)( 45,191)( 46,187)( 47,189)( 48,188)
( 49, 97)( 50, 99)( 51, 98)( 52,100)( 53,102)( 54,101)( 55,103)( 56,105)
( 57,104)( 58,106)( 59,108)( 60,107)( 61,109)( 62,111)( 63,110)( 64,112)
( 65,114)( 66,113)( 67,115)( 68,117)( 69,116)( 70,118)( 71,120)( 72,119)
( 73,124)( 74,126)( 75,125)( 76,121)( 77,123)( 78,122)( 79,130)( 80,132)
( 81,131)( 82,127)( 83,129)( 84,128)( 85,136)( 86,138)( 87,137)( 88,133)
( 89,135)( 90,134)( 91,142)( 92,144)( 93,143)( 94,139)( 95,141)( 96,140)
(193,337)(194,339)(195,338)(196,340)(197,342)(198,341)(199,343)(200,345)
(201,344)(202,346)(203,348)(204,347)(205,349)(206,351)(207,350)(208,352)
(209,354)(210,353)(211,355)(212,357)(213,356)(214,358)(215,360)(216,359)
(217,364)(218,366)(219,365)(220,361)(221,363)(222,362)(223,370)(224,372)
(225,371)(226,367)(227,369)(228,368)(229,376)(230,378)(231,377)(232,373)
(233,375)(234,374)(235,382)(236,384)(237,383)(238,379)(239,381)(240,380)
(241,289)(242,291)(243,290)(244,292)(245,294)(246,293)(247,295)(248,297)
(249,296)(250,298)(251,300)(252,299)(253,301)(254,303)(255,302)(256,304)
(257,306)(258,305)(259,307)(260,309)(261,308)(262,310)(263,312)(264,311)
(265,316)(266,318)(267,317)(268,313)(269,315)(270,314)(271,322)(272,324)
(273,323)(274,319)(275,321)(276,320)(277,328)(278,330)(279,329)(280,325)
(281,327)(282,326)(283,334)(284,336)(285,335)(286,331)(287,333)(288,332);;
s3 := ( 1, 51)( 2, 50)( 3, 49)( 4, 54)( 5, 53)( 6, 52)( 7, 57)( 8, 56)
( 9, 55)( 10, 60)( 11, 59)( 12, 58)( 13, 63)( 14, 62)( 15, 61)( 16, 66)
( 17, 65)( 18, 64)( 19, 69)( 20, 68)( 21, 67)( 22, 72)( 23, 71)( 24, 70)
( 25, 75)( 26, 74)( 27, 73)( 28, 78)( 29, 77)( 30, 76)( 31, 81)( 32, 80)
( 33, 79)( 34, 84)( 35, 83)( 36, 82)( 37, 87)( 38, 86)( 39, 85)( 40, 90)
( 41, 89)( 42, 88)( 43, 93)( 44, 92)( 45, 91)( 46, 96)( 47, 95)( 48, 94)
( 97,147)( 98,146)( 99,145)(100,150)(101,149)(102,148)(103,153)(104,152)
(105,151)(106,156)(107,155)(108,154)(109,159)(110,158)(111,157)(112,162)
(113,161)(114,160)(115,165)(116,164)(117,163)(118,168)(119,167)(120,166)
(121,171)(122,170)(123,169)(124,174)(125,173)(126,172)(127,177)(128,176)
(129,175)(130,180)(131,179)(132,178)(133,183)(134,182)(135,181)(136,186)
(137,185)(138,184)(139,189)(140,188)(141,187)(142,192)(143,191)(144,190)
(193,243)(194,242)(195,241)(196,246)(197,245)(198,244)(199,249)(200,248)
(201,247)(202,252)(203,251)(204,250)(205,255)(206,254)(207,253)(208,258)
(209,257)(210,256)(211,261)(212,260)(213,259)(214,264)(215,263)(216,262)
(217,267)(218,266)(219,265)(220,270)(221,269)(222,268)(223,273)(224,272)
(225,271)(226,276)(227,275)(228,274)(229,279)(230,278)(231,277)(232,282)
(233,281)(234,280)(235,285)(236,284)(237,283)(238,288)(239,287)(240,286)
(289,339)(290,338)(291,337)(292,342)(293,341)(294,340)(295,345)(296,344)
(297,343)(298,348)(299,347)(300,346)(301,351)(302,350)(303,349)(304,354)
(305,353)(306,352)(307,357)(308,356)(309,355)(310,360)(311,359)(312,358)
(313,363)(314,362)(315,361)(316,366)(317,365)(318,364)(319,369)(320,368)
(321,367)(322,372)(323,371)(324,370)(325,375)(326,374)(327,373)(328,378)
(329,377)(330,376)(331,381)(332,380)(333,379)(334,384)(335,383)(336,382);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(384)!( 1,193)( 2,194)( 3,195)( 4,196)( 5,197)( 6,198)( 7,199)
( 8,200)( 9,201)( 10,202)( 11,203)( 12,204)( 13,208)( 14,209)( 15,210)
( 16,205)( 17,206)( 18,207)( 19,214)( 20,215)( 21,216)( 22,211)( 23,212)
( 24,213)( 25,223)( 26,224)( 27,225)( 28,226)( 29,227)( 30,228)( 31,217)
( 32,218)( 33,219)( 34,220)( 35,221)( 36,222)( 37,238)( 38,239)( 39,240)
( 40,235)( 41,236)( 42,237)( 43,232)( 44,233)( 45,234)( 46,229)( 47,230)
( 48,231)( 49,241)( 50,242)( 51,243)( 52,244)( 53,245)( 54,246)( 55,247)
( 56,248)( 57,249)( 58,250)( 59,251)( 60,252)( 61,256)( 62,257)( 63,258)
( 64,253)( 65,254)( 66,255)( 67,262)( 68,263)( 69,264)( 70,259)( 71,260)
( 72,261)( 73,271)( 74,272)( 75,273)( 76,274)( 77,275)( 78,276)( 79,265)
( 80,266)( 81,267)( 82,268)( 83,269)( 84,270)( 85,286)( 86,287)( 87,288)
( 88,283)( 89,284)( 90,285)( 91,280)( 92,281)( 93,282)( 94,277)( 95,278)
( 96,279)( 97,289)( 98,290)( 99,291)(100,292)(101,293)(102,294)(103,295)
(104,296)(105,297)(106,298)(107,299)(108,300)(109,304)(110,305)(111,306)
(112,301)(113,302)(114,303)(115,310)(116,311)(117,312)(118,307)(119,308)
(120,309)(121,319)(122,320)(123,321)(124,322)(125,323)(126,324)(127,313)
(128,314)(129,315)(130,316)(131,317)(132,318)(133,334)(134,335)(135,336)
(136,331)(137,332)(138,333)(139,328)(140,329)(141,330)(142,325)(143,326)
(144,327)(145,337)(146,338)(147,339)(148,340)(149,341)(150,342)(151,343)
(152,344)(153,345)(154,346)(155,347)(156,348)(157,352)(158,353)(159,354)
(160,349)(161,350)(162,351)(163,358)(164,359)(165,360)(166,355)(167,356)
(168,357)(169,367)(170,368)(171,369)(172,370)(173,371)(174,372)(175,361)
(176,362)(177,363)(178,364)(179,365)(180,366)(181,382)(182,383)(183,384)
(184,379)(185,380)(186,381)(187,376)(188,377)(189,378)(190,373)(191,374)
(192,375);
s1 := Sym(384)!( 25, 31)( 26, 32)( 27, 33)( 28, 34)( 29, 35)( 30, 36)( 37, 43)
( 38, 44)( 39, 45)( 40, 46)( 41, 47)( 42, 48)( 73, 79)( 74, 80)( 75, 81)
( 76, 82)( 77, 83)( 78, 84)( 85, 91)( 86, 92)( 87, 93)( 88, 94)( 89, 95)
( 90, 96)( 97,109)( 98,110)( 99,111)(100,112)(101,113)(102,114)(103,115)
(104,116)(105,117)(106,118)(107,119)(108,120)(121,139)(122,140)(123,141)
(124,142)(125,143)(126,144)(127,133)(128,134)(129,135)(130,136)(131,137)
(132,138)(145,157)(146,158)(147,159)(148,160)(149,161)(150,162)(151,163)
(152,164)(153,165)(154,166)(155,167)(156,168)(169,187)(170,188)(171,189)
(172,190)(173,191)(174,192)(175,181)(176,182)(177,183)(178,184)(179,185)
(180,186)(193,217)(194,218)(195,219)(196,220)(197,221)(198,222)(199,223)
(200,224)(201,225)(202,226)(203,227)(204,228)(205,229)(206,230)(207,231)
(208,232)(209,233)(210,234)(211,235)(212,236)(213,237)(214,238)(215,239)
(216,240)(241,265)(242,266)(243,267)(244,268)(245,269)(246,270)(247,271)
(248,272)(249,273)(250,274)(251,275)(252,276)(253,277)(254,278)(255,279)
(256,280)(257,281)(258,282)(259,283)(260,284)(261,285)(262,286)(263,287)
(264,288)(289,328)(290,329)(291,330)(292,325)(293,326)(294,327)(295,334)
(296,335)(297,336)(298,331)(299,332)(300,333)(301,316)(302,317)(303,318)
(304,313)(305,314)(306,315)(307,322)(308,323)(309,324)(310,319)(311,320)
(312,321)(337,376)(338,377)(339,378)(340,373)(341,374)(342,375)(343,382)
(344,383)(345,384)(346,379)(347,380)(348,381)(349,364)(350,365)(351,366)
(352,361)(353,362)(354,363)(355,370)(356,371)(357,372)(358,367)(359,368)
(360,369);
s2 := Sym(384)!( 1,145)( 2,147)( 3,146)( 4,148)( 5,150)( 6,149)( 7,151)
( 8,153)( 9,152)( 10,154)( 11,156)( 12,155)( 13,157)( 14,159)( 15,158)
( 16,160)( 17,162)( 18,161)( 19,163)( 20,165)( 21,164)( 22,166)( 23,168)
( 24,167)( 25,172)( 26,174)( 27,173)( 28,169)( 29,171)( 30,170)( 31,178)
( 32,180)( 33,179)( 34,175)( 35,177)( 36,176)( 37,184)( 38,186)( 39,185)
( 40,181)( 41,183)( 42,182)( 43,190)( 44,192)( 45,191)( 46,187)( 47,189)
( 48,188)( 49, 97)( 50, 99)( 51, 98)( 52,100)( 53,102)( 54,101)( 55,103)
( 56,105)( 57,104)( 58,106)( 59,108)( 60,107)( 61,109)( 62,111)( 63,110)
( 64,112)( 65,114)( 66,113)( 67,115)( 68,117)( 69,116)( 70,118)( 71,120)
( 72,119)( 73,124)( 74,126)( 75,125)( 76,121)( 77,123)( 78,122)( 79,130)
( 80,132)( 81,131)( 82,127)( 83,129)( 84,128)( 85,136)( 86,138)( 87,137)
( 88,133)( 89,135)( 90,134)( 91,142)( 92,144)( 93,143)( 94,139)( 95,141)
( 96,140)(193,337)(194,339)(195,338)(196,340)(197,342)(198,341)(199,343)
(200,345)(201,344)(202,346)(203,348)(204,347)(205,349)(206,351)(207,350)
(208,352)(209,354)(210,353)(211,355)(212,357)(213,356)(214,358)(215,360)
(216,359)(217,364)(218,366)(219,365)(220,361)(221,363)(222,362)(223,370)
(224,372)(225,371)(226,367)(227,369)(228,368)(229,376)(230,378)(231,377)
(232,373)(233,375)(234,374)(235,382)(236,384)(237,383)(238,379)(239,381)
(240,380)(241,289)(242,291)(243,290)(244,292)(245,294)(246,293)(247,295)
(248,297)(249,296)(250,298)(251,300)(252,299)(253,301)(254,303)(255,302)
(256,304)(257,306)(258,305)(259,307)(260,309)(261,308)(262,310)(263,312)
(264,311)(265,316)(266,318)(267,317)(268,313)(269,315)(270,314)(271,322)
(272,324)(273,323)(274,319)(275,321)(276,320)(277,328)(278,330)(279,329)
(280,325)(281,327)(282,326)(283,334)(284,336)(285,335)(286,331)(287,333)
(288,332);
s3 := Sym(384)!( 1, 51)( 2, 50)( 3, 49)( 4, 54)( 5, 53)( 6, 52)( 7, 57)
( 8, 56)( 9, 55)( 10, 60)( 11, 59)( 12, 58)( 13, 63)( 14, 62)( 15, 61)
( 16, 66)( 17, 65)( 18, 64)( 19, 69)( 20, 68)( 21, 67)( 22, 72)( 23, 71)
( 24, 70)( 25, 75)( 26, 74)( 27, 73)( 28, 78)( 29, 77)( 30, 76)( 31, 81)
( 32, 80)( 33, 79)( 34, 84)( 35, 83)( 36, 82)( 37, 87)( 38, 86)( 39, 85)
( 40, 90)( 41, 89)( 42, 88)( 43, 93)( 44, 92)( 45, 91)( 46, 96)( 47, 95)
( 48, 94)( 97,147)( 98,146)( 99,145)(100,150)(101,149)(102,148)(103,153)
(104,152)(105,151)(106,156)(107,155)(108,154)(109,159)(110,158)(111,157)
(112,162)(113,161)(114,160)(115,165)(116,164)(117,163)(118,168)(119,167)
(120,166)(121,171)(122,170)(123,169)(124,174)(125,173)(126,172)(127,177)
(128,176)(129,175)(130,180)(131,179)(132,178)(133,183)(134,182)(135,181)
(136,186)(137,185)(138,184)(139,189)(140,188)(141,187)(142,192)(143,191)
(144,190)(193,243)(194,242)(195,241)(196,246)(197,245)(198,244)(199,249)
(200,248)(201,247)(202,252)(203,251)(204,250)(205,255)(206,254)(207,253)
(208,258)(209,257)(210,256)(211,261)(212,260)(213,259)(214,264)(215,263)
(216,262)(217,267)(218,266)(219,265)(220,270)(221,269)(222,268)(223,273)
(224,272)(225,271)(226,276)(227,275)(228,274)(229,279)(230,278)(231,277)
(232,282)(233,281)(234,280)(235,285)(236,284)(237,283)(238,288)(239,287)
(240,286)(289,339)(290,338)(291,337)(292,342)(293,341)(294,340)(295,345)
(296,344)(297,343)(298,348)(299,347)(300,346)(301,351)(302,350)(303,349)
(304,354)(305,353)(306,352)(307,357)(308,356)(309,355)(310,360)(311,359)
(312,358)(313,363)(314,362)(315,361)(316,366)(317,365)(318,364)(319,369)
(320,368)(321,367)(322,372)(323,371)(324,370)(325,375)(326,374)(327,373)
(328,378)(329,377)(330,376)(331,381)(332,380)(333,379)(334,384)(335,383)
(336,382);
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,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope