Polytope of Type {6,6,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,8}*768
if this polytope has a name.
Group : SmallGroup(768,1089270)
Rank : 4
Schlafli Type : {6,6,8}
Number of vertices, edges, etc : 8, 24, 32, 8
Order of s0s1s2s3 : 8
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 : {3,6,8}*384, {6,6,4}*384
4-fold quotients : {3,6,4}*192, {6,6,2}*192
8-fold quotients : {3,6,2}*96, {6,3,2}*96
12-fold quotients : {2,2,8}*64
16-fold quotients : {3,3,2}*48
24-fold quotients : {2,2,4}*32
48-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)( 18, 22)
( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)( 39, 40)
( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 51, 52)( 53, 57)( 54, 58)( 55, 60)
( 56, 59)( 63, 64)( 65, 69)( 66, 70)( 67, 72)( 68, 71)( 75, 76)( 77, 81)
( 78, 82)( 79, 84)( 80, 83)( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)
( 99,100)(101,105)(102,106)(103,108)(104,107)(111,112)(113,117)(114,118)
(115,120)(116,119)(123,124)(125,129)(126,130)(127,132)(128,131)(135,136)
(137,141)(138,142)(139,144)(140,143)(147,148)(149,153)(150,154)(151,156)
(152,155)(159,160)(161,165)(162,166)(163,168)(164,167)(171,172)(173,177)
(174,178)(175,180)(176,179)(183,184)(185,189)(186,190)(187,192)(188,191)
(195,196)(197,201)(198,202)(199,204)(200,203)(207,208)(209,213)(210,214)
(211,216)(212,215)(219,220)(221,225)(222,226)(223,228)(224,227)(231,232)
(233,237)(234,238)(235,240)(236,239)(243,244)(245,249)(246,250)(247,252)
(248,251)(255,256)(257,261)(258,262)(259,264)(260,263)(267,268)(269,273)
(270,274)(271,276)(272,275)(279,280)(281,285)(282,286)(283,288)(284,287)
(291,292)(293,297)(294,298)(295,300)(296,299)(303,304)(305,309)(306,310)
(307,312)(308,311)(315,316)(317,321)(318,322)(319,324)(320,323)(327,328)
(329,333)(330,334)(331,336)(332,335)(339,340)(341,345)(342,346)(343,348)
(344,347)(351,352)(353,357)(354,358)(355,360)(356,359)(363,364)(365,369)
(366,370)(367,372)(368,371)(375,376)(377,381)(378,382)(379,384)(380,383);;
s1 := ( 1,201)( 2,203)( 3,202)( 4,204)( 5,197)( 6,199)( 7,198)( 8,200)
( 9,193)( 10,195)( 11,194)( 12,196)( 13,213)( 14,215)( 15,214)( 16,216)
( 17,209)( 18,211)( 19,210)( 20,212)( 21,205)( 22,207)( 23,206)( 24,208)
( 25,225)( 26,227)( 27,226)( 28,228)( 29,221)( 30,223)( 31,222)( 32,224)
( 33,217)( 34,219)( 35,218)( 36,220)( 37,237)( 38,239)( 39,238)( 40,240)
( 41,233)( 42,235)( 43,234)( 44,236)( 45,229)( 46,231)( 47,230)( 48,232)
( 49,249)( 50,251)( 51,250)( 52,252)( 53,245)( 54,247)( 55,246)( 56,248)
( 57,241)( 58,243)( 59,242)( 60,244)( 61,261)( 62,263)( 63,262)( 64,264)
( 65,257)( 66,259)( 67,258)( 68,260)( 69,253)( 70,255)( 71,254)( 72,256)
( 73,273)( 74,275)( 75,274)( 76,276)( 77,269)( 78,271)( 79,270)( 80,272)
( 81,265)( 82,267)( 83,266)( 84,268)( 85,285)( 86,287)( 87,286)( 88,288)
( 89,281)( 90,283)( 91,282)( 92,284)( 93,277)( 94,279)( 95,278)( 96,280)
( 97,297)( 98,299)( 99,298)(100,300)(101,293)(102,295)(103,294)(104,296)
(105,289)(106,291)(107,290)(108,292)(109,309)(110,311)(111,310)(112,312)
(113,305)(114,307)(115,306)(116,308)(117,301)(118,303)(119,302)(120,304)
(121,321)(122,323)(123,322)(124,324)(125,317)(126,319)(127,318)(128,320)
(129,313)(130,315)(131,314)(132,316)(133,333)(134,335)(135,334)(136,336)
(137,329)(138,331)(139,330)(140,332)(141,325)(142,327)(143,326)(144,328)
(145,345)(146,347)(147,346)(148,348)(149,341)(150,343)(151,342)(152,344)
(153,337)(154,339)(155,338)(156,340)(157,357)(158,359)(159,358)(160,360)
(161,353)(162,355)(163,354)(164,356)(165,349)(166,351)(167,350)(168,352)
(169,369)(170,371)(171,370)(172,372)(173,365)(174,367)(175,366)(176,368)
(177,361)(178,363)(179,362)(180,364)(181,381)(182,383)(183,382)(184,384)
(185,377)(186,379)(187,378)(188,380)(189,373)(190,375)(191,374)(192,376);;
s2 := ( 1, 98)( 2, 97)( 3, 99)( 4,100)( 5,106)( 6,105)( 7,107)( 8,108)
( 9,102)( 10,101)( 11,103)( 12,104)( 13,110)( 14,109)( 15,111)( 16,112)
( 17,118)( 18,117)( 19,119)( 20,120)( 21,114)( 22,113)( 23,115)( 24,116)
( 25,134)( 26,133)( 27,135)( 28,136)( 29,142)( 30,141)( 31,143)( 32,144)
( 33,138)( 34,137)( 35,139)( 36,140)( 37,122)( 38,121)( 39,123)( 40,124)
( 41,130)( 42,129)( 43,131)( 44,132)( 45,126)( 46,125)( 47,127)( 48,128)
( 49,170)( 50,169)( 51,171)( 52,172)( 53,178)( 54,177)( 55,179)( 56,180)
( 57,174)( 58,173)( 59,175)( 60,176)( 61,182)( 62,181)( 63,183)( 64,184)
( 65,190)( 66,189)( 67,191)( 68,192)( 69,186)( 70,185)( 71,187)( 72,188)
( 73,146)( 74,145)( 75,147)( 76,148)( 77,154)( 78,153)( 79,155)( 80,156)
( 81,150)( 82,149)( 83,151)( 84,152)( 85,158)( 86,157)( 87,159)( 88,160)
( 89,166)( 90,165)( 91,167)( 92,168)( 93,162)( 94,161)( 95,163)( 96,164)
(193,290)(194,289)(195,291)(196,292)(197,298)(198,297)(199,299)(200,300)
(201,294)(202,293)(203,295)(204,296)(205,302)(206,301)(207,303)(208,304)
(209,310)(210,309)(211,311)(212,312)(213,306)(214,305)(215,307)(216,308)
(217,326)(218,325)(219,327)(220,328)(221,334)(222,333)(223,335)(224,336)
(225,330)(226,329)(227,331)(228,332)(229,314)(230,313)(231,315)(232,316)
(233,322)(234,321)(235,323)(236,324)(237,318)(238,317)(239,319)(240,320)
(241,362)(242,361)(243,363)(244,364)(245,370)(246,369)(247,371)(248,372)
(249,366)(250,365)(251,367)(252,368)(253,374)(254,373)(255,375)(256,376)
(257,382)(258,381)(259,383)(260,384)(261,378)(262,377)(263,379)(264,380)
(265,338)(266,337)(267,339)(268,340)(269,346)(270,345)(271,347)(272,348)
(273,342)(274,341)(275,343)(276,344)(277,350)(278,349)(279,351)(280,352)
(281,358)(282,357)(283,359)(284,360)(285,354)(286,353)(287,355)(288,356);;
s3 := ( 1, 49)( 2, 50)( 3, 51)( 4, 52)( 5, 53)( 6, 54)( 7, 55)( 8, 56)
( 9, 57)( 10, 58)( 11, 59)( 12, 60)( 13, 61)( 14, 62)( 15, 63)( 16, 64)
( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)( 24, 72)
( 25, 85)( 26, 86)( 27, 87)( 28, 88)( 29, 89)( 30, 90)( 31, 91)( 32, 92)
( 33, 93)( 34, 94)( 35, 95)( 36, 96)( 37, 73)( 38, 74)( 39, 75)( 40, 76)
( 41, 77)( 42, 78)( 43, 79)( 44, 80)( 45, 81)( 46, 82)( 47, 83)( 48, 84)
( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)
(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)
(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)
(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)(128,188)
(129,189)(130,190)(131,191)(132,192)(133,169)(134,170)(135,171)(136,172)
(137,173)(138,174)(139,175)(140,176)(141,177)(142,178)(143,179)(144,180)
(193,241)(194,242)(195,243)(196,244)(197,245)(198,246)(199,247)(200,248)
(201,249)(202,250)(203,251)(204,252)(205,253)(206,254)(207,255)(208,256)
(209,257)(210,258)(211,259)(212,260)(213,261)(214,262)(215,263)(216,264)
(217,277)(218,278)(219,279)(220,280)(221,281)(222,282)(223,283)(224,284)
(225,285)(226,286)(227,287)(228,288)(229,265)(230,266)(231,267)(232,268)
(233,269)(234,270)(235,271)(236,272)(237,273)(238,274)(239,275)(240,276)
(289,337)(290,338)(291,339)(292,340)(293,341)(294,342)(295,343)(296,344)
(297,345)(298,346)(299,347)(300,348)(301,349)(302,350)(303,351)(304,352)
(305,353)(306,354)(307,355)(308,356)(309,357)(310,358)(311,359)(312,360)
(313,373)(314,374)(315,375)(316,376)(317,377)(318,378)(319,379)(320,380)
(321,381)(322,382)(323,383)(324,384)(325,361)(326,362)(327,363)(328,364)
(329,365)(330,366)(331,367)(332,368)(333,369)(334,370)(335,371)(336,372);;
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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*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)!( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)
( 18, 22)( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)
( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 51, 52)( 53, 57)( 54, 58)
( 55, 60)( 56, 59)( 63, 64)( 65, 69)( 66, 70)( 67, 72)( 68, 71)( 75, 76)
( 77, 81)( 78, 82)( 79, 84)( 80, 83)( 87, 88)( 89, 93)( 90, 94)( 91, 96)
( 92, 95)( 99,100)(101,105)(102,106)(103,108)(104,107)(111,112)(113,117)
(114,118)(115,120)(116,119)(123,124)(125,129)(126,130)(127,132)(128,131)
(135,136)(137,141)(138,142)(139,144)(140,143)(147,148)(149,153)(150,154)
(151,156)(152,155)(159,160)(161,165)(162,166)(163,168)(164,167)(171,172)
(173,177)(174,178)(175,180)(176,179)(183,184)(185,189)(186,190)(187,192)
(188,191)(195,196)(197,201)(198,202)(199,204)(200,203)(207,208)(209,213)
(210,214)(211,216)(212,215)(219,220)(221,225)(222,226)(223,228)(224,227)
(231,232)(233,237)(234,238)(235,240)(236,239)(243,244)(245,249)(246,250)
(247,252)(248,251)(255,256)(257,261)(258,262)(259,264)(260,263)(267,268)
(269,273)(270,274)(271,276)(272,275)(279,280)(281,285)(282,286)(283,288)
(284,287)(291,292)(293,297)(294,298)(295,300)(296,299)(303,304)(305,309)
(306,310)(307,312)(308,311)(315,316)(317,321)(318,322)(319,324)(320,323)
(327,328)(329,333)(330,334)(331,336)(332,335)(339,340)(341,345)(342,346)
(343,348)(344,347)(351,352)(353,357)(354,358)(355,360)(356,359)(363,364)
(365,369)(366,370)(367,372)(368,371)(375,376)(377,381)(378,382)(379,384)
(380,383);
s1 := Sym(384)!( 1,201)( 2,203)( 3,202)( 4,204)( 5,197)( 6,199)( 7,198)
( 8,200)( 9,193)( 10,195)( 11,194)( 12,196)( 13,213)( 14,215)( 15,214)
( 16,216)( 17,209)( 18,211)( 19,210)( 20,212)( 21,205)( 22,207)( 23,206)
( 24,208)( 25,225)( 26,227)( 27,226)( 28,228)( 29,221)( 30,223)( 31,222)
( 32,224)( 33,217)( 34,219)( 35,218)( 36,220)( 37,237)( 38,239)( 39,238)
( 40,240)( 41,233)( 42,235)( 43,234)( 44,236)( 45,229)( 46,231)( 47,230)
( 48,232)( 49,249)( 50,251)( 51,250)( 52,252)( 53,245)( 54,247)( 55,246)
( 56,248)( 57,241)( 58,243)( 59,242)( 60,244)( 61,261)( 62,263)( 63,262)
( 64,264)( 65,257)( 66,259)( 67,258)( 68,260)( 69,253)( 70,255)( 71,254)
( 72,256)( 73,273)( 74,275)( 75,274)( 76,276)( 77,269)( 78,271)( 79,270)
( 80,272)( 81,265)( 82,267)( 83,266)( 84,268)( 85,285)( 86,287)( 87,286)
( 88,288)( 89,281)( 90,283)( 91,282)( 92,284)( 93,277)( 94,279)( 95,278)
( 96,280)( 97,297)( 98,299)( 99,298)(100,300)(101,293)(102,295)(103,294)
(104,296)(105,289)(106,291)(107,290)(108,292)(109,309)(110,311)(111,310)
(112,312)(113,305)(114,307)(115,306)(116,308)(117,301)(118,303)(119,302)
(120,304)(121,321)(122,323)(123,322)(124,324)(125,317)(126,319)(127,318)
(128,320)(129,313)(130,315)(131,314)(132,316)(133,333)(134,335)(135,334)
(136,336)(137,329)(138,331)(139,330)(140,332)(141,325)(142,327)(143,326)
(144,328)(145,345)(146,347)(147,346)(148,348)(149,341)(150,343)(151,342)
(152,344)(153,337)(154,339)(155,338)(156,340)(157,357)(158,359)(159,358)
(160,360)(161,353)(162,355)(163,354)(164,356)(165,349)(166,351)(167,350)
(168,352)(169,369)(170,371)(171,370)(172,372)(173,365)(174,367)(175,366)
(176,368)(177,361)(178,363)(179,362)(180,364)(181,381)(182,383)(183,382)
(184,384)(185,377)(186,379)(187,378)(188,380)(189,373)(190,375)(191,374)
(192,376);
s2 := Sym(384)!( 1, 98)( 2, 97)( 3, 99)( 4,100)( 5,106)( 6,105)( 7,107)
( 8,108)( 9,102)( 10,101)( 11,103)( 12,104)( 13,110)( 14,109)( 15,111)
( 16,112)( 17,118)( 18,117)( 19,119)( 20,120)( 21,114)( 22,113)( 23,115)
( 24,116)( 25,134)( 26,133)( 27,135)( 28,136)( 29,142)( 30,141)( 31,143)
( 32,144)( 33,138)( 34,137)( 35,139)( 36,140)( 37,122)( 38,121)( 39,123)
( 40,124)( 41,130)( 42,129)( 43,131)( 44,132)( 45,126)( 46,125)( 47,127)
( 48,128)( 49,170)( 50,169)( 51,171)( 52,172)( 53,178)( 54,177)( 55,179)
( 56,180)( 57,174)( 58,173)( 59,175)( 60,176)( 61,182)( 62,181)( 63,183)
( 64,184)( 65,190)( 66,189)( 67,191)( 68,192)( 69,186)( 70,185)( 71,187)
( 72,188)( 73,146)( 74,145)( 75,147)( 76,148)( 77,154)( 78,153)( 79,155)
( 80,156)( 81,150)( 82,149)( 83,151)( 84,152)( 85,158)( 86,157)( 87,159)
( 88,160)( 89,166)( 90,165)( 91,167)( 92,168)( 93,162)( 94,161)( 95,163)
( 96,164)(193,290)(194,289)(195,291)(196,292)(197,298)(198,297)(199,299)
(200,300)(201,294)(202,293)(203,295)(204,296)(205,302)(206,301)(207,303)
(208,304)(209,310)(210,309)(211,311)(212,312)(213,306)(214,305)(215,307)
(216,308)(217,326)(218,325)(219,327)(220,328)(221,334)(222,333)(223,335)
(224,336)(225,330)(226,329)(227,331)(228,332)(229,314)(230,313)(231,315)
(232,316)(233,322)(234,321)(235,323)(236,324)(237,318)(238,317)(239,319)
(240,320)(241,362)(242,361)(243,363)(244,364)(245,370)(246,369)(247,371)
(248,372)(249,366)(250,365)(251,367)(252,368)(253,374)(254,373)(255,375)
(256,376)(257,382)(258,381)(259,383)(260,384)(261,378)(262,377)(263,379)
(264,380)(265,338)(266,337)(267,339)(268,340)(269,346)(270,345)(271,347)
(272,348)(273,342)(274,341)(275,343)(276,344)(277,350)(278,349)(279,351)
(280,352)(281,358)(282,357)(283,359)(284,360)(285,354)(286,353)(287,355)
(288,356);
s3 := Sym(384)!( 1, 49)( 2, 50)( 3, 51)( 4, 52)( 5, 53)( 6, 54)( 7, 55)
( 8, 56)( 9, 57)( 10, 58)( 11, 59)( 12, 60)( 13, 61)( 14, 62)( 15, 63)
( 16, 64)( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)
( 24, 72)( 25, 85)( 26, 86)( 27, 87)( 28, 88)( 29, 89)( 30, 90)( 31, 91)
( 32, 92)( 33, 93)( 34, 94)( 35, 95)( 36, 96)( 37, 73)( 38, 74)( 39, 75)
( 40, 76)( 41, 77)( 42, 78)( 43, 79)( 44, 80)( 45, 81)( 46, 82)( 47, 83)
( 48, 84)( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)
(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)
(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)
(120,168)(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)
(128,188)(129,189)(130,190)(131,191)(132,192)(133,169)(134,170)(135,171)
(136,172)(137,173)(138,174)(139,175)(140,176)(141,177)(142,178)(143,179)
(144,180)(193,241)(194,242)(195,243)(196,244)(197,245)(198,246)(199,247)
(200,248)(201,249)(202,250)(203,251)(204,252)(205,253)(206,254)(207,255)
(208,256)(209,257)(210,258)(211,259)(212,260)(213,261)(214,262)(215,263)
(216,264)(217,277)(218,278)(219,279)(220,280)(221,281)(222,282)(223,283)
(224,284)(225,285)(226,286)(227,287)(228,288)(229,265)(230,266)(231,267)
(232,268)(233,269)(234,270)(235,271)(236,272)(237,273)(238,274)(239,275)
(240,276)(289,337)(290,338)(291,339)(292,340)(293,341)(294,342)(295,343)
(296,344)(297,345)(298,346)(299,347)(300,348)(301,349)(302,350)(303,351)
(304,352)(305,353)(306,354)(307,355)(308,356)(309,357)(310,358)(311,359)
(312,360)(313,373)(314,374)(315,375)(316,376)(317,377)(318,378)(319,379)
(320,380)(321,381)(322,382)(323,383)(324,384)(325,361)(326,362)(327,363)
(328,364)(329,365)(330,366)(331,367)(332,368)(333,369)(334,370)(335,371)
(336,372);
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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope