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Polytope of Type {8,8,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,8,6}*768d
if this polytope has a name.
Group : SmallGroup(768,150681)
Rank : 4
Schlafli Type : {8,8,6}
Number of vertices, edges, etc : 8, 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 : {4,8,6}*384b, {8,4,6}*384b
   3-fold quotients : {8,8,2}*256d
   4-fold quotients : {4,4,6}*192
   6-fold quotients : {4,8,2}*128b, {8,4,2}*128b
   8-fold quotients : {2,4,6}*96a, {4,2,6}*96
   12-fold quotients : {4,4,2}*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 := ( 13, 22)( 14, 23)( 15, 24)( 16, 19)( 17, 20)( 18, 21)( 25, 31)( 26, 32)
( 27, 33)( 28, 34)( 29, 35)( 30, 36)( 37, 40)( 38, 41)( 39, 42)( 43, 46)
( 44, 47)( 45, 48)( 61, 70)( 62, 71)( 63, 72)( 64, 67)( 65, 68)( 66, 69)
( 73, 79)( 74, 80)( 75, 81)( 76, 82)( 77, 83)( 78, 84)( 85, 88)( 86, 89)
( 87, 90)( 91, 94)( 92, 95)( 93, 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)
(205,238)(206,239)(207,240)(208,235)(209,236)(210,237)(211,232)(212,233)
(213,234)(214,229)(215,230)(216,231)(241,265)(242,266)(243,267)(244,268)
(245,269)(246,270)(247,271)(248,272)(249,273)(250,274)(251,275)(252,276)
(253,286)(254,287)(255,288)(256,283)(257,284)(258,285)(259,280)(260,281)
(261,282)(262,277)(263,278)(264,279)(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)( 13,166)( 14,168)( 15,167)( 16,163)
( 17,165)( 18,164)( 19,160)( 20,162)( 21,161)( 22,157)( 23,159)( 24,158)
( 25,172)( 26,174)( 27,173)( 28,169)( 29,171)( 30,170)( 31,178)( 32,180)
( 33,179)( 34,175)( 35,177)( 36,176)( 37,187)( 38,189)( 39,188)( 40,190)
( 41,192)( 42,191)( 43,181)( 44,183)( 45,182)( 46,184)( 47,186)( 48,185)
( 49, 97)( 50, 99)( 51, 98)( 52,100)( 53,102)( 54,101)( 55,103)( 56,105)
( 57,104)( 58,106)( 59,108)( 60,107)( 61,118)( 62,120)( 63,119)( 64,115)
( 65,117)( 66,116)( 67,112)( 68,114)( 69,113)( 70,109)( 71,111)( 72,110)
( 73,124)( 74,126)( 75,125)( 76,121)( 77,123)( 78,122)( 79,130)( 80,132)
( 81,131)( 82,127)( 83,129)( 84,128)( 85,139)( 86,141)( 87,140)( 88,142)
( 89,144)( 90,143)( 91,133)( 92,135)( 93,134)( 94,136)( 95,138)( 96,137)
(193,337)(194,339)(195,338)(196,340)(197,342)(198,341)(199,343)(200,345)
(201,344)(202,346)(203,348)(204,347)(205,358)(206,360)(207,359)(208,355)
(209,357)(210,356)(211,352)(212,354)(213,353)(214,349)(215,351)(216,350)
(217,364)(218,366)(219,365)(220,361)(221,363)(222,362)(223,370)(224,372)
(225,371)(226,367)(227,369)(228,368)(229,379)(230,381)(231,380)(232,382)
(233,384)(234,383)(235,373)(236,375)(237,374)(238,376)(239,378)(240,377)
(241,289)(242,291)(243,290)(244,292)(245,294)(246,293)(247,295)(248,297)
(249,296)(250,298)(251,300)(252,299)(253,310)(254,312)(255,311)(256,307)
(257,309)(258,308)(259,304)(260,306)(261,305)(262,301)(263,303)(264,302)
(265,316)(266,318)(267,317)(268,313)(269,315)(270,314)(271,322)(272,324)
(273,323)(274,319)(275,321)(276,320)(277,331)(278,333)(279,332)(280,334)
(281,336)(282,335)(283,325)(284,327)(285,326)(286,328)(287,330)(288,329);;
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*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s0*s1*s2*s1*s2*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)!( 13, 22)( 14, 23)( 15, 24)( 16, 19)( 17, 20)( 18, 21)( 25, 31)
( 26, 32)( 27, 33)( 28, 34)( 29, 35)( 30, 36)( 37, 40)( 38, 41)( 39, 42)
( 43, 46)( 44, 47)( 45, 48)( 61, 70)( 62, 71)( 63, 72)( 64, 67)( 65, 68)
( 66, 69)( 73, 79)( 74, 80)( 75, 81)( 76, 82)( 77, 83)( 78, 84)( 85, 88)
( 86, 89)( 87, 90)( 91, 94)( 92, 95)( 93, 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)(205,238)(206,239)(207,240)(208,235)(209,236)(210,237)(211,232)
(212,233)(213,234)(214,229)(215,230)(216,231)(241,265)(242,266)(243,267)
(244,268)(245,269)(246,270)(247,271)(248,272)(249,273)(250,274)(251,275)
(252,276)(253,286)(254,287)(255,288)(256,283)(257,284)(258,285)(259,280)
(260,281)(261,282)(262,277)(263,278)(264,279)(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)( 13,166)( 14,168)( 15,167)
( 16,163)( 17,165)( 18,164)( 19,160)( 20,162)( 21,161)( 22,157)( 23,159)
( 24,158)( 25,172)( 26,174)( 27,173)( 28,169)( 29,171)( 30,170)( 31,178)
( 32,180)( 33,179)( 34,175)( 35,177)( 36,176)( 37,187)( 38,189)( 39,188)
( 40,190)( 41,192)( 42,191)( 43,181)( 44,183)( 45,182)( 46,184)( 47,186)
( 48,185)( 49, 97)( 50, 99)( 51, 98)( 52,100)( 53,102)( 54,101)( 55,103)
( 56,105)( 57,104)( 58,106)( 59,108)( 60,107)( 61,118)( 62,120)( 63,119)
( 64,115)( 65,117)( 66,116)( 67,112)( 68,114)( 69,113)( 70,109)( 71,111)
( 72,110)( 73,124)( 74,126)( 75,125)( 76,121)( 77,123)( 78,122)( 79,130)
( 80,132)( 81,131)( 82,127)( 83,129)( 84,128)( 85,139)( 86,141)( 87,140)
( 88,142)( 89,144)( 90,143)( 91,133)( 92,135)( 93,134)( 94,136)( 95,138)
( 96,137)(193,337)(194,339)(195,338)(196,340)(197,342)(198,341)(199,343)
(200,345)(201,344)(202,346)(203,348)(204,347)(205,358)(206,360)(207,359)
(208,355)(209,357)(210,356)(211,352)(212,354)(213,353)(214,349)(215,351)
(216,350)(217,364)(218,366)(219,365)(220,361)(221,363)(222,362)(223,370)
(224,372)(225,371)(226,367)(227,369)(228,368)(229,379)(230,381)(231,380)
(232,382)(233,384)(234,383)(235,373)(236,375)(237,374)(238,376)(239,378)
(240,377)(241,289)(242,291)(243,290)(244,292)(245,294)(246,293)(247,295)
(248,297)(249,296)(250,298)(251,300)(252,299)(253,310)(254,312)(255,311)
(256,307)(257,309)(258,308)(259,304)(260,306)(261,305)(262,301)(263,303)
(264,302)(265,316)(266,318)(267,317)(268,313)(269,315)(270,314)(271,322)
(272,324)(273,323)(274,319)(275,321)(276,320)(277,331)(278,333)(279,332)
(280,334)(281,336)(282,335)(283,325)(284,327)(285,326)(286,328)(287,330)
(288,329);
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*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope