Polytope of Type {8,6,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6,8}*768
Also Known As : {{8,6|2},{6,8|2}}. if this polytope has another name.
Group : SmallGroup(768,161912)
Rank : 4
Schlafli Type : {8,6,8}
Number of vertices, edges, etc : 8, 24, 24, 8
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Self-Dual
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 : {4,6,8}*384a, {8,6,4}*384a
3-fold quotients : {8,2,8}*256
4-fold quotients : {4,6,4}*192a, {2,6,8}*192, {8,6,2}*192
6-fold quotients : {4,2,8}*128, {8,2,4}*128
8-fold quotients : {2,6,4}*96a, {4,6,2}*96a
12-fold quotients : {4,2,4}*64, {2,2,8}*64, {8,2,2}*64
16-fold quotients : {2,6,2}*48
24-fold quotients : {2,2,4}*32, {4,2,2}*32
32-fold quotients : {2,3,2}*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)( 13,205)( 14,206)( 15,207)( 16,208)
( 17,209)( 18,210)( 19,211)( 20,212)( 21,213)( 22,214)( 23,215)( 24,216)
( 25,223)( 26,224)( 27,225)( 28,226)( 29,227)( 30,228)( 31,217)( 32,218)
( 33,219)( 34,220)( 35,221)( 36,222)( 37,235)( 38,236)( 39,237)( 40,238)
( 41,239)( 42,240)( 43,229)( 44,230)( 45,231)( 46,232)( 47,233)( 48,234)
( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,271)( 56,272)
( 57,273)( 58,274)( 59,275)( 60,276)( 61,277)( 62,278)( 63,279)( 64,280)
( 65,281)( 66,282)( 67,283)( 68,284)( 69,285)( 70,286)( 71,287)( 72,288)
( 73,241)( 74,242)( 75,243)( 76,244)( 77,245)( 78,246)( 79,247)( 80,248)
( 81,249)( 82,250)( 83,251)( 84,252)( 85,253)( 86,254)( 87,255)( 88,256)
( 89,257)( 90,258)( 91,259)( 92,260)( 93,261)( 94,262)( 95,263)( 96,264)
( 97,289)( 98,290)( 99,291)(100,292)(101,293)(102,294)(103,295)(104,296)
(105,297)(106,298)(107,299)(108,300)(109,301)(110,302)(111,303)(112,304)
(113,305)(114,306)(115,307)(116,308)(117,309)(118,310)(119,311)(120,312)
(121,319)(122,320)(123,321)(124,322)(125,323)(126,324)(127,313)(128,314)
(129,315)(130,316)(131,317)(132,318)(133,331)(134,332)(135,333)(136,334)
(137,335)(138,336)(139,325)(140,326)(141,327)(142,328)(143,329)(144,330)
(145,361)(146,362)(147,363)(148,364)(149,365)(150,366)(151,367)(152,368)
(153,369)(154,370)(155,371)(156,372)(157,373)(158,374)(159,375)(160,376)
(161,377)(162,378)(163,379)(164,380)(165,381)(166,382)(167,383)(168,384)
(169,337)(170,338)(171,339)(172,340)(173,341)(174,342)(175,343)(176,344)
(177,345)(178,346)(179,347)(180,348)(181,349)(182,350)(183,351)(184,352)
(185,353)(186,354)(187,355)(188,356)(189,357)(190,358)(191,359)(192,360);;
s1 := ( 1, 49)( 2, 51)( 3, 50)( 4, 52)( 5, 54)( 6, 53)( 7, 55)( 8, 57)
( 9, 56)( 10, 58)( 11, 60)( 12, 59)( 13, 61)( 14, 63)( 15, 62)( 16, 64)
( 17, 66)( 18, 65)( 19, 67)( 20, 69)( 21, 68)( 22, 70)( 23, 72)( 24, 71)
( 25, 79)( 26, 81)( 27, 80)( 28, 82)( 29, 84)( 30, 83)( 31, 73)( 32, 75)
( 33, 74)( 34, 76)( 35, 78)( 36, 77)( 37, 91)( 38, 93)( 39, 92)( 40, 94)
( 41, 96)( 42, 95)( 43, 85)( 44, 87)( 45, 86)( 46, 88)( 47, 90)( 48, 89)
( 97,145)( 98,147)( 99,146)(100,148)(101,150)(102,149)(103,151)(104,153)
(105,152)(106,154)(107,156)(108,155)(109,157)(110,159)(111,158)(112,160)
(113,162)(114,161)(115,163)(116,165)(117,164)(118,166)(119,168)(120,167)
(121,175)(122,177)(123,176)(124,178)(125,180)(126,179)(127,169)(128,171)
(129,170)(130,172)(131,174)(132,173)(133,187)(134,189)(135,188)(136,190)
(137,192)(138,191)(139,181)(140,183)(141,182)(142,184)(143,186)(144,185)
(193,241)(194,243)(195,242)(196,244)(197,246)(198,245)(199,247)(200,249)
(201,248)(202,250)(203,252)(204,251)(205,253)(206,255)(207,254)(208,256)
(209,258)(210,257)(211,259)(212,261)(213,260)(214,262)(215,264)(216,263)
(217,271)(218,273)(219,272)(220,274)(221,276)(222,275)(223,265)(224,267)
(225,266)(226,268)(227,270)(228,269)(229,283)(230,285)(231,284)(232,286)
(233,288)(234,287)(235,277)(236,279)(237,278)(238,280)(239,282)(240,281)
(289,337)(290,339)(291,338)(292,340)(293,342)(294,341)(295,343)(296,345)
(297,344)(298,346)(299,348)(300,347)(301,349)(302,351)(303,350)(304,352)
(305,354)(306,353)(307,355)(308,357)(309,356)(310,358)(311,360)(312,359)
(313,367)(314,369)(315,368)(316,370)(317,372)(318,371)(319,361)(320,363)
(321,362)(322,364)(323,366)(324,365)(325,379)(326,381)(327,380)(328,382)
(329,384)(330,383)(331,373)(332,375)(333,374)(334,376)(335,378)(336,377);;
s2 := ( 1, 81)( 2, 80)( 3, 79)( 4, 84)( 5, 83)( 6, 82)( 7, 75)( 8, 74)
( 9, 73)( 10, 78)( 11, 77)( 12, 76)( 13, 96)( 14, 95)( 15, 94)( 16, 93)
( 17, 92)( 18, 91)( 19, 90)( 20, 89)( 21, 88)( 22, 87)( 23, 86)( 24, 85)
( 25, 51)( 26, 50)( 27, 49)( 28, 54)( 29, 53)( 30, 52)( 31, 57)( 32, 56)
( 33, 55)( 34, 60)( 35, 59)( 36, 58)( 37, 66)( 38, 65)( 39, 64)( 40, 63)
( 41, 62)( 42, 61)( 43, 72)( 44, 71)( 45, 70)( 46, 69)( 47, 68)( 48, 67)
( 97,189)( 98,188)( 99,187)(100,192)(101,191)(102,190)(103,183)(104,182)
(105,181)(106,186)(107,185)(108,184)(109,177)(110,176)(111,175)(112,180)
(113,179)(114,178)(115,171)(116,170)(117,169)(118,174)(119,173)(120,172)
(121,159)(122,158)(123,157)(124,162)(125,161)(126,160)(127,165)(128,164)
(129,163)(130,168)(131,167)(132,166)(133,147)(134,146)(135,145)(136,150)
(137,149)(138,148)(139,153)(140,152)(141,151)(142,156)(143,155)(144,154)
(193,249)(194,248)(195,247)(196,252)(197,251)(198,250)(199,243)(200,242)
(201,241)(202,246)(203,245)(204,244)(205,264)(206,263)(207,262)(208,261)
(209,260)(210,259)(211,258)(212,257)(213,256)(214,255)(215,254)(216,253)
(217,273)(218,272)(219,271)(220,276)(221,275)(222,274)(223,267)(224,266)
(225,265)(226,270)(227,269)(228,268)(229,288)(230,287)(231,286)(232,285)
(233,284)(234,283)(235,282)(236,281)(237,280)(238,279)(239,278)(240,277)
(289,357)(290,356)(291,355)(292,360)(293,359)(294,358)(295,351)(296,350)
(297,349)(298,354)(299,353)(300,352)(301,345)(302,344)(303,343)(304,348)
(305,347)(306,346)(307,339)(308,338)(309,337)(310,342)(311,341)(312,340)
(313,381)(314,380)(315,379)(316,384)(317,383)(318,382)(319,375)(320,374)
(321,373)(322,378)(323,377)(324,376)(325,369)(326,368)(327,367)(328,372)
(329,371)(330,370)(331,363)(332,362)(333,361)(334,366)(335,365)(336,364);;
s3 := ( 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)
( 25,121)( 26,122)( 27,123)( 28,124)( 29,125)( 30,126)( 31,127)( 32,128)
( 33,129)( 34,130)( 35,131)( 36,132)( 37,136)( 38,137)( 39,138)( 40,133)
( 41,134)( 42,135)( 43,142)( 44,143)( 45,144)( 46,139)( 47,140)( 48,141)
( 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)
( 73,169)( 74,170)( 75,171)( 76,172)( 77,173)( 78,174)( 79,175)( 80,176)
( 81,177)( 82,178)( 83,179)( 84,180)( 85,184)( 86,185)( 87,186)( 88,181)
( 89,182)( 90,183)( 91,190)( 92,191)( 93,192)( 94,187)( 95,188)( 96,189)
(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)
(217,313)(218,314)(219,315)(220,316)(221,317)(222,318)(223,319)(224,320)
(225,321)(226,322)(227,323)(228,324)(229,328)(230,329)(231,330)(232,325)
(233,326)(234,327)(235,334)(236,335)(237,336)(238,331)(239,332)(240,333)
(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)
(265,361)(266,362)(267,363)(268,364)(269,365)(270,366)(271,367)(272,368)
(273,369)(274,370)(275,371)(276,372)(277,376)(278,377)(279,378)(280,373)
(281,374)(282,375)(283,382)(284,383)(285,384)(286,379)(287,380)(288,381);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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)( 13,205)( 14,206)( 15,207)
( 16,208)( 17,209)( 18,210)( 19,211)( 20,212)( 21,213)( 22,214)( 23,215)
( 24,216)( 25,223)( 26,224)( 27,225)( 28,226)( 29,227)( 30,228)( 31,217)
( 32,218)( 33,219)( 34,220)( 35,221)( 36,222)( 37,235)( 38,236)( 39,237)
( 40,238)( 41,239)( 42,240)( 43,229)( 44,230)( 45,231)( 46,232)( 47,233)
( 48,234)( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,271)
( 56,272)( 57,273)( 58,274)( 59,275)( 60,276)( 61,277)( 62,278)( 63,279)
( 64,280)( 65,281)( 66,282)( 67,283)( 68,284)( 69,285)( 70,286)( 71,287)
( 72,288)( 73,241)( 74,242)( 75,243)( 76,244)( 77,245)( 78,246)( 79,247)
( 80,248)( 81,249)( 82,250)( 83,251)( 84,252)( 85,253)( 86,254)( 87,255)
( 88,256)( 89,257)( 90,258)( 91,259)( 92,260)( 93,261)( 94,262)( 95,263)
( 96,264)( 97,289)( 98,290)( 99,291)(100,292)(101,293)(102,294)(103,295)
(104,296)(105,297)(106,298)(107,299)(108,300)(109,301)(110,302)(111,303)
(112,304)(113,305)(114,306)(115,307)(116,308)(117,309)(118,310)(119,311)
(120,312)(121,319)(122,320)(123,321)(124,322)(125,323)(126,324)(127,313)
(128,314)(129,315)(130,316)(131,317)(132,318)(133,331)(134,332)(135,333)
(136,334)(137,335)(138,336)(139,325)(140,326)(141,327)(142,328)(143,329)
(144,330)(145,361)(146,362)(147,363)(148,364)(149,365)(150,366)(151,367)
(152,368)(153,369)(154,370)(155,371)(156,372)(157,373)(158,374)(159,375)
(160,376)(161,377)(162,378)(163,379)(164,380)(165,381)(166,382)(167,383)
(168,384)(169,337)(170,338)(171,339)(172,340)(173,341)(174,342)(175,343)
(176,344)(177,345)(178,346)(179,347)(180,348)(181,349)(182,350)(183,351)
(184,352)(185,353)(186,354)(187,355)(188,356)(189,357)(190,358)(191,359)
(192,360);
s1 := Sym(384)!( 1, 49)( 2, 51)( 3, 50)( 4, 52)( 5, 54)( 6, 53)( 7, 55)
( 8, 57)( 9, 56)( 10, 58)( 11, 60)( 12, 59)( 13, 61)( 14, 63)( 15, 62)
( 16, 64)( 17, 66)( 18, 65)( 19, 67)( 20, 69)( 21, 68)( 22, 70)( 23, 72)
( 24, 71)( 25, 79)( 26, 81)( 27, 80)( 28, 82)( 29, 84)( 30, 83)( 31, 73)
( 32, 75)( 33, 74)( 34, 76)( 35, 78)( 36, 77)( 37, 91)( 38, 93)( 39, 92)
( 40, 94)( 41, 96)( 42, 95)( 43, 85)( 44, 87)( 45, 86)( 46, 88)( 47, 90)
( 48, 89)( 97,145)( 98,147)( 99,146)(100,148)(101,150)(102,149)(103,151)
(104,153)(105,152)(106,154)(107,156)(108,155)(109,157)(110,159)(111,158)
(112,160)(113,162)(114,161)(115,163)(116,165)(117,164)(118,166)(119,168)
(120,167)(121,175)(122,177)(123,176)(124,178)(125,180)(126,179)(127,169)
(128,171)(129,170)(130,172)(131,174)(132,173)(133,187)(134,189)(135,188)
(136,190)(137,192)(138,191)(139,181)(140,183)(141,182)(142,184)(143,186)
(144,185)(193,241)(194,243)(195,242)(196,244)(197,246)(198,245)(199,247)
(200,249)(201,248)(202,250)(203,252)(204,251)(205,253)(206,255)(207,254)
(208,256)(209,258)(210,257)(211,259)(212,261)(213,260)(214,262)(215,264)
(216,263)(217,271)(218,273)(219,272)(220,274)(221,276)(222,275)(223,265)
(224,267)(225,266)(226,268)(227,270)(228,269)(229,283)(230,285)(231,284)
(232,286)(233,288)(234,287)(235,277)(236,279)(237,278)(238,280)(239,282)
(240,281)(289,337)(290,339)(291,338)(292,340)(293,342)(294,341)(295,343)
(296,345)(297,344)(298,346)(299,348)(300,347)(301,349)(302,351)(303,350)
(304,352)(305,354)(306,353)(307,355)(308,357)(309,356)(310,358)(311,360)
(312,359)(313,367)(314,369)(315,368)(316,370)(317,372)(318,371)(319,361)
(320,363)(321,362)(322,364)(323,366)(324,365)(325,379)(326,381)(327,380)
(328,382)(329,384)(330,383)(331,373)(332,375)(333,374)(334,376)(335,378)
(336,377);
s2 := Sym(384)!( 1, 81)( 2, 80)( 3, 79)( 4, 84)( 5, 83)( 6, 82)( 7, 75)
( 8, 74)( 9, 73)( 10, 78)( 11, 77)( 12, 76)( 13, 96)( 14, 95)( 15, 94)
( 16, 93)( 17, 92)( 18, 91)( 19, 90)( 20, 89)( 21, 88)( 22, 87)( 23, 86)
( 24, 85)( 25, 51)( 26, 50)( 27, 49)( 28, 54)( 29, 53)( 30, 52)( 31, 57)
( 32, 56)( 33, 55)( 34, 60)( 35, 59)( 36, 58)( 37, 66)( 38, 65)( 39, 64)
( 40, 63)( 41, 62)( 42, 61)( 43, 72)( 44, 71)( 45, 70)( 46, 69)( 47, 68)
( 48, 67)( 97,189)( 98,188)( 99,187)(100,192)(101,191)(102,190)(103,183)
(104,182)(105,181)(106,186)(107,185)(108,184)(109,177)(110,176)(111,175)
(112,180)(113,179)(114,178)(115,171)(116,170)(117,169)(118,174)(119,173)
(120,172)(121,159)(122,158)(123,157)(124,162)(125,161)(126,160)(127,165)
(128,164)(129,163)(130,168)(131,167)(132,166)(133,147)(134,146)(135,145)
(136,150)(137,149)(138,148)(139,153)(140,152)(141,151)(142,156)(143,155)
(144,154)(193,249)(194,248)(195,247)(196,252)(197,251)(198,250)(199,243)
(200,242)(201,241)(202,246)(203,245)(204,244)(205,264)(206,263)(207,262)
(208,261)(209,260)(210,259)(211,258)(212,257)(213,256)(214,255)(215,254)
(216,253)(217,273)(218,272)(219,271)(220,276)(221,275)(222,274)(223,267)
(224,266)(225,265)(226,270)(227,269)(228,268)(229,288)(230,287)(231,286)
(232,285)(233,284)(234,283)(235,282)(236,281)(237,280)(238,279)(239,278)
(240,277)(289,357)(290,356)(291,355)(292,360)(293,359)(294,358)(295,351)
(296,350)(297,349)(298,354)(299,353)(300,352)(301,345)(302,344)(303,343)
(304,348)(305,347)(306,346)(307,339)(308,338)(309,337)(310,342)(311,341)
(312,340)(313,381)(314,380)(315,379)(316,384)(317,383)(318,382)(319,375)
(320,374)(321,373)(322,378)(323,377)(324,376)(325,369)(326,368)(327,367)
(328,372)(329,371)(330,370)(331,363)(332,362)(333,361)(334,366)(335,365)
(336,364);
s3 := 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)( 25,121)( 26,122)( 27,123)( 28,124)( 29,125)( 30,126)( 31,127)
( 32,128)( 33,129)( 34,130)( 35,131)( 36,132)( 37,136)( 38,137)( 39,138)
( 40,133)( 41,134)( 42,135)( 43,142)( 44,143)( 45,144)( 46,139)( 47,140)
( 48,141)( 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)( 73,169)( 74,170)( 75,171)( 76,172)( 77,173)( 78,174)( 79,175)
( 80,176)( 81,177)( 82,178)( 83,179)( 84,180)( 85,184)( 86,185)( 87,186)
( 88,181)( 89,182)( 90,183)( 91,190)( 92,191)( 93,192)( 94,187)( 95,188)
( 96,189)(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)(217,313)(218,314)(219,315)(220,316)(221,317)(222,318)(223,319)
(224,320)(225,321)(226,322)(227,323)(228,324)(229,328)(230,329)(231,330)
(232,325)(233,326)(234,327)(235,334)(236,335)(237,336)(238,331)(239,332)
(240,333)(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)(265,361)(266,362)(267,363)(268,364)(269,365)(270,366)(271,367)
(272,368)(273,369)(274,370)(275,371)(276,372)(277,376)(278,377)(279,378)
(280,373)(281,374)(282,375)(283,382)(284,383)(285,384)(286,379)(287,380)
(288,381);
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope