Polytope of Type {8,8,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,8,6}*768b
Also Known As : {{8,8|2},{8,6|2}}. if this polytope has another name.
Group : SmallGroup(768,145169)
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}*384a, {8,4,6}*384a
3-fold quotients : {8,8,2}*256b
4-fold quotients : {4,4,6}*192, {2,8,6}*192, {8,2,6}*192
6-fold quotients : {4,8,2}*128a, {8,4,2}*128a
8-fold quotients : {8,2,3}*96, {2,4,6}*96a, {4,2,6}*96
12-fold quotients : {4,4,2}*64, {2,8,2}*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, 97)( 2, 98)( 3, 99)( 4,100)( 5,101)( 6,102)( 7,103)( 8,104)
( 9,105)( 10,106)( 11,107)( 12,108)( 13,112)( 14,113)( 15,114)( 16,109)
( 17,110)( 18,111)( 19,118)( 20,119)( 21,120)( 22,115)( 23,116)( 24,117)
( 25,124)( 26,125)( 27,126)( 28,121)( 29,122)( 30,123)( 31,130)( 32,131)
( 33,132)( 34,127)( 35,128)( 36,129)( 37,133)( 38,134)( 39,135)( 40,136)
( 41,137)( 42,138)( 43,139)( 44,140)( 45,141)( 46,142)( 47,143)( 48,144)
( 49,145)( 50,146)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)( 56,152)
( 57,153)( 58,154)( 59,155)( 60,156)( 61,160)( 62,161)( 63,162)( 64,157)
( 65,158)( 66,159)( 67,166)( 68,167)( 69,168)( 70,163)( 71,164)( 72,165)
( 73,172)( 74,173)( 75,174)( 76,169)( 77,170)( 78,171)( 79,178)( 80,179)
( 81,180)( 82,175)( 83,176)( 84,177)( 85,181)( 86,182)( 87,183)( 88,184)
( 89,185)( 90,186)( 91,187)( 92,188)( 93,189)( 94,190)( 95,191)( 96,192)
(193,289)(194,290)(195,291)(196,292)(197,293)(198,294)(199,295)(200,296)
(201,297)(202,298)(203,299)(204,300)(205,304)(206,305)(207,306)(208,301)
(209,302)(210,303)(211,310)(212,311)(213,312)(214,307)(215,308)(216,309)
(217,316)(218,317)(219,318)(220,313)(221,314)(222,315)(223,322)(224,323)
(225,324)(226,319)(227,320)(228,321)(229,325)(230,326)(231,327)(232,328)
(233,329)(234,330)(235,331)(236,332)(237,333)(238,334)(239,335)(240,336)
(241,337)(242,338)(243,339)(244,340)(245,341)(246,342)(247,343)(248,344)
(249,345)(250,346)(251,347)(252,348)(253,352)(254,353)(255,354)(256,349)
(257,350)(258,351)(259,358)(260,359)(261,360)(262,355)(263,356)(264,357)
(265,364)(266,365)(267,366)(268,361)(269,362)(270,363)(271,370)(272,371)
(273,372)(274,367)(275,368)(276,369)(277,373)(278,374)(279,375)(280,376)
(281,377)(282,378)(283,379)(284,380)(285,381)(286,382)(287,383)(288,384);;
s1 := ( 13, 16)( 14, 17)( 15, 18)( 19, 22)( 20, 23)( 21, 24)( 25, 31)( 26, 32)
( 27, 33)( 28, 34)( 29, 35)( 30, 36)( 37, 46)( 38, 47)( 39, 48)( 40, 43)
( 41, 44)( 42, 45)( 61, 64)( 62, 65)( 63, 66)( 67, 70)( 68, 71)( 69, 72)
( 73, 79)( 74, 80)( 75, 81)( 76, 82)( 77, 83)( 78, 84)( 85, 94)( 86, 95)
( 87, 96)( 88, 91)( 89, 92)( 90, 93)( 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)
(205,232)(206,233)(207,234)(208,229)(209,230)(210,231)(211,238)(212,239)
(213,240)(214,235)(215,236)(216,237)(241,265)(242,266)(243,267)(244,268)
(245,269)(246,270)(247,271)(248,272)(249,273)(250,274)(251,275)(252,276)
(253,280)(254,281)(255,282)(256,277)(257,278)(258,279)(259,286)(260,287)
(261,288)(262,283)(263,284)(264,285)(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 := ( 1,337)( 2,339)( 3,338)( 4,340)( 5,342)( 6,341)( 7,343)( 8,345)
( 9,344)( 10,346)( 11,348)( 12,347)( 13,349)( 14,351)( 15,350)( 16,352)
( 17,354)( 18,353)( 19,355)( 20,357)( 21,356)( 22,358)( 23,360)( 24,359)
( 25,370)( 26,372)( 27,371)( 28,367)( 29,369)( 30,368)( 31,364)( 32,366)
( 33,365)( 34,361)( 35,363)( 36,362)( 37,382)( 38,384)( 39,383)( 40,379)
( 41,381)( 42,380)( 43,376)( 44,378)( 45,377)( 46,373)( 47,375)( 48,374)
( 49,289)( 50,291)( 51,290)( 52,292)( 53,294)( 54,293)( 55,295)( 56,297)
( 57,296)( 58,298)( 59,300)( 60,299)( 61,301)( 62,303)( 63,302)( 64,304)
( 65,306)( 66,305)( 67,307)( 68,309)( 69,308)( 70,310)( 71,312)( 72,311)
( 73,322)( 74,324)( 75,323)( 76,319)( 77,321)( 78,320)( 79,316)( 80,318)
( 81,317)( 82,313)( 83,315)( 84,314)( 85,334)( 86,336)( 87,335)( 88,331)
( 89,333)( 90,332)( 91,328)( 92,330)( 93,329)( 94,325)( 95,327)( 96,326)
( 97,241)( 98,243)( 99,242)(100,244)(101,246)(102,245)(103,247)(104,249)
(105,248)(106,250)(107,252)(108,251)(109,253)(110,255)(111,254)(112,256)
(113,258)(114,257)(115,259)(116,261)(117,260)(118,262)(119,264)(120,263)
(121,274)(122,276)(123,275)(124,271)(125,273)(126,272)(127,268)(128,270)
(129,269)(130,265)(131,267)(132,266)(133,286)(134,288)(135,287)(136,283)
(137,285)(138,284)(139,280)(140,282)(141,281)(142,277)(143,279)(144,278)
(145,193)(146,195)(147,194)(148,196)(149,198)(150,197)(151,199)(152,201)
(153,200)(154,202)(155,204)(156,203)(157,205)(158,207)(159,206)(160,208)
(161,210)(162,209)(163,211)(164,213)(165,212)(166,214)(167,216)(168,215)
(169,226)(170,228)(171,227)(172,223)(173,225)(174,224)(175,220)(176,222)
(177,221)(178,217)(179,219)(180,218)(181,238)(182,240)(183,239)(184,235)
(185,237)(186,236)(187,232)(188,234)(189,233)(190,229)(191,231)(192,230);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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)( 13,112)( 14,113)( 15,114)
( 16,109)( 17,110)( 18,111)( 19,118)( 20,119)( 21,120)( 22,115)( 23,116)
( 24,117)( 25,124)( 26,125)( 27,126)( 28,121)( 29,122)( 30,123)( 31,130)
( 32,131)( 33,132)( 34,127)( 35,128)( 36,129)( 37,133)( 38,134)( 39,135)
( 40,136)( 41,137)( 42,138)( 43,139)( 44,140)( 45,141)( 46,142)( 47,143)
( 48,144)( 49,145)( 50,146)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)
( 56,152)( 57,153)( 58,154)( 59,155)( 60,156)( 61,160)( 62,161)( 63,162)
( 64,157)( 65,158)( 66,159)( 67,166)( 68,167)( 69,168)( 70,163)( 71,164)
( 72,165)( 73,172)( 74,173)( 75,174)( 76,169)( 77,170)( 78,171)( 79,178)
( 80,179)( 81,180)( 82,175)( 83,176)( 84,177)( 85,181)( 86,182)( 87,183)
( 88,184)( 89,185)( 90,186)( 91,187)( 92,188)( 93,189)( 94,190)( 95,191)
( 96,192)(193,289)(194,290)(195,291)(196,292)(197,293)(198,294)(199,295)
(200,296)(201,297)(202,298)(203,299)(204,300)(205,304)(206,305)(207,306)
(208,301)(209,302)(210,303)(211,310)(212,311)(213,312)(214,307)(215,308)
(216,309)(217,316)(218,317)(219,318)(220,313)(221,314)(222,315)(223,322)
(224,323)(225,324)(226,319)(227,320)(228,321)(229,325)(230,326)(231,327)
(232,328)(233,329)(234,330)(235,331)(236,332)(237,333)(238,334)(239,335)
(240,336)(241,337)(242,338)(243,339)(244,340)(245,341)(246,342)(247,343)
(248,344)(249,345)(250,346)(251,347)(252,348)(253,352)(254,353)(255,354)
(256,349)(257,350)(258,351)(259,358)(260,359)(261,360)(262,355)(263,356)
(264,357)(265,364)(266,365)(267,366)(268,361)(269,362)(270,363)(271,370)
(272,371)(273,372)(274,367)(275,368)(276,369)(277,373)(278,374)(279,375)
(280,376)(281,377)(282,378)(283,379)(284,380)(285,381)(286,382)(287,383)
(288,384);
s1 := Sym(384)!( 13, 16)( 14, 17)( 15, 18)( 19, 22)( 20, 23)( 21, 24)( 25, 31)
( 26, 32)( 27, 33)( 28, 34)( 29, 35)( 30, 36)( 37, 46)( 38, 47)( 39, 48)
( 40, 43)( 41, 44)( 42, 45)( 61, 64)( 62, 65)( 63, 66)( 67, 70)( 68, 71)
( 69, 72)( 73, 79)( 74, 80)( 75, 81)( 76, 82)( 77, 83)( 78, 84)( 85, 94)
( 86, 95)( 87, 96)( 88, 91)( 89, 92)( 90, 93)( 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)(205,232)(206,233)(207,234)(208,229)(209,230)(210,231)(211,238)
(212,239)(213,240)(214,235)(215,236)(216,237)(241,265)(242,266)(243,267)
(244,268)(245,269)(246,270)(247,271)(248,272)(249,273)(250,274)(251,275)
(252,276)(253,280)(254,281)(255,282)(256,277)(257,278)(258,279)(259,286)
(260,287)(261,288)(262,283)(263,284)(264,285)(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)!( 1,337)( 2,339)( 3,338)( 4,340)( 5,342)( 6,341)( 7,343)
( 8,345)( 9,344)( 10,346)( 11,348)( 12,347)( 13,349)( 14,351)( 15,350)
( 16,352)( 17,354)( 18,353)( 19,355)( 20,357)( 21,356)( 22,358)( 23,360)
( 24,359)( 25,370)( 26,372)( 27,371)( 28,367)( 29,369)( 30,368)( 31,364)
( 32,366)( 33,365)( 34,361)( 35,363)( 36,362)( 37,382)( 38,384)( 39,383)
( 40,379)( 41,381)( 42,380)( 43,376)( 44,378)( 45,377)( 46,373)( 47,375)
( 48,374)( 49,289)( 50,291)( 51,290)( 52,292)( 53,294)( 54,293)( 55,295)
( 56,297)( 57,296)( 58,298)( 59,300)( 60,299)( 61,301)( 62,303)( 63,302)
( 64,304)( 65,306)( 66,305)( 67,307)( 68,309)( 69,308)( 70,310)( 71,312)
( 72,311)( 73,322)( 74,324)( 75,323)( 76,319)( 77,321)( 78,320)( 79,316)
( 80,318)( 81,317)( 82,313)( 83,315)( 84,314)( 85,334)( 86,336)( 87,335)
( 88,331)( 89,333)( 90,332)( 91,328)( 92,330)( 93,329)( 94,325)( 95,327)
( 96,326)( 97,241)( 98,243)( 99,242)(100,244)(101,246)(102,245)(103,247)
(104,249)(105,248)(106,250)(107,252)(108,251)(109,253)(110,255)(111,254)
(112,256)(113,258)(114,257)(115,259)(116,261)(117,260)(118,262)(119,264)
(120,263)(121,274)(122,276)(123,275)(124,271)(125,273)(126,272)(127,268)
(128,270)(129,269)(130,265)(131,267)(132,266)(133,286)(134,288)(135,287)
(136,283)(137,285)(138,284)(139,280)(140,282)(141,281)(142,277)(143,279)
(144,278)(145,193)(146,195)(147,194)(148,196)(149,198)(150,197)(151,199)
(152,201)(153,200)(154,202)(155,204)(156,203)(157,205)(158,207)(159,206)
(160,208)(161,210)(162,209)(163,211)(164,213)(165,212)(166,214)(167,216)
(168,215)(169,226)(170,228)(171,227)(172,223)(173,225)(174,224)(175,220)
(176,222)(177,221)(178,217)(179,219)(180,218)(181,238)(182,240)(183,239)
(184,235)(185,237)(186,236)(187,232)(188,234)(189,233)(190,229)(191,231)
(192,230);
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,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope