Polytope of Type {6,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,4}*768a
if this polytope has a name.
Group : SmallGroup(768,1088539)
Rank : 4
Schlafli Type : {6,6,4}
Number of vertices, edges, etc : 8, 48, 32, 8
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 4
Special Properties :
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 : {6,3,4}*384a, {3,6,4}*384b
4-fold quotients : {3,3,4}*192
8-fold quotients : {3,6,2}*96
16-fold quotients : {3,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 5, 7)( 6, 8)( 13, 15)( 14, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)
( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)( 34, 66)( 35, 67)( 36, 68)
( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)( 42, 74)( 43, 75)( 44, 76)
( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)( 50, 90)( 51, 91)( 52, 92)
( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)( 58, 82)( 59, 83)( 60, 84)
( 61, 87)( 62, 88)( 63, 85)( 64, 86)(101,103)(102,104)(109,111)(110,112)
(113,121)(114,122)(115,123)(116,124)(117,127)(118,128)(119,125)(120,126)
(129,161)(130,162)(131,163)(132,164)(133,167)(134,168)(135,165)(136,166)
(137,169)(138,170)(139,171)(140,172)(141,175)(142,176)(143,173)(144,174)
(145,185)(146,186)(147,187)(148,188)(149,191)(150,192)(151,189)(152,190)
(153,177)(154,178)(155,179)(156,180)(157,183)(158,184)(159,181)(160,182)
(197,199)(198,200)(205,207)(206,208)(209,217)(210,218)(211,219)(212,220)
(213,223)(214,224)(215,221)(216,222)(225,257)(226,258)(227,259)(228,260)
(229,263)(230,264)(231,261)(232,262)(233,265)(234,266)(235,267)(236,268)
(237,271)(238,272)(239,269)(240,270)(241,281)(242,282)(243,283)(244,284)
(245,287)(246,288)(247,285)(248,286)(249,273)(250,274)(251,275)(252,276)
(253,279)(254,280)(255,277)(256,278)(293,295)(294,296)(301,303)(302,304)
(305,313)(306,314)(307,315)(308,316)(309,319)(310,320)(311,317)(312,318)
(321,353)(322,354)(323,355)(324,356)(325,359)(326,360)(327,357)(328,358)
(329,361)(330,362)(331,363)(332,364)(333,367)(334,368)(335,365)(336,366)
(337,377)(338,378)(339,379)(340,380)(341,383)(342,384)(343,381)(344,382)
(345,369)(346,370)(347,371)(348,372)(349,375)(350,376)(351,373)(352,374);;
s1 := ( 1, 72)( 2, 71)( 3, 68)( 4, 67)( 5, 70)( 6, 69)( 7, 66)( 8, 65)
( 9, 88)( 10, 87)( 11, 84)( 12, 83)( 13, 86)( 14, 85)( 15, 82)( 16, 81)
( 17, 80)( 18, 79)( 19, 76)( 20, 75)( 21, 78)( 22, 77)( 23, 74)( 24, 73)
( 25, 96)( 26, 95)( 27, 92)( 28, 91)( 29, 94)( 30, 93)( 31, 90)( 32, 89)
( 33, 40)( 34, 39)( 35, 36)( 37, 38)( 41, 56)( 42, 55)( 43, 52)( 44, 51)
( 45, 54)( 46, 53)( 47, 50)( 48, 49)( 57, 64)( 58, 63)( 59, 60)( 61, 62)
( 97,168)( 98,167)( 99,164)(100,163)(101,166)(102,165)(103,162)(104,161)
(105,184)(106,183)(107,180)(108,179)(109,182)(110,181)(111,178)(112,177)
(113,176)(114,175)(115,172)(116,171)(117,174)(118,173)(119,170)(120,169)
(121,192)(122,191)(123,188)(124,187)(125,190)(126,189)(127,186)(128,185)
(129,136)(130,135)(131,132)(133,134)(137,152)(138,151)(139,148)(140,147)
(141,150)(142,149)(143,146)(144,145)(153,160)(154,159)(155,156)(157,158)
(193,264)(194,263)(195,260)(196,259)(197,262)(198,261)(199,258)(200,257)
(201,280)(202,279)(203,276)(204,275)(205,278)(206,277)(207,274)(208,273)
(209,272)(210,271)(211,268)(212,267)(213,270)(214,269)(215,266)(216,265)
(217,288)(218,287)(219,284)(220,283)(221,286)(222,285)(223,282)(224,281)
(225,232)(226,231)(227,228)(229,230)(233,248)(234,247)(235,244)(236,243)
(237,246)(238,245)(239,242)(240,241)(249,256)(250,255)(251,252)(253,254)
(289,360)(290,359)(291,356)(292,355)(293,358)(294,357)(295,354)(296,353)
(297,376)(298,375)(299,372)(300,371)(301,374)(302,373)(303,370)(304,369)
(305,368)(306,367)(307,364)(308,363)(309,366)(310,365)(311,362)(312,361)
(313,384)(314,383)(315,380)(316,379)(317,382)(318,381)(319,378)(320,377)
(321,328)(322,327)(323,324)(325,326)(329,344)(330,343)(331,340)(332,339)
(333,342)(334,341)(335,338)(336,337)(345,352)(346,351)(347,348)(349,350);;
s2 := ( 1,201)( 2,202)( 3,203)( 4,204)( 5,208)( 6,207)( 7,206)( 8,205)
( 9,193)( 10,194)( 11,195)( 12,196)( 13,200)( 14,199)( 15,198)( 16,197)
( 17,209)( 18,210)( 19,211)( 20,212)( 21,216)( 22,215)( 23,214)( 24,213)
( 25,217)( 26,218)( 27,219)( 28,220)( 29,224)( 30,223)( 31,222)( 32,221)
( 33,265)( 34,266)( 35,267)( 36,268)( 37,272)( 38,271)( 39,270)( 40,269)
( 41,257)( 42,258)( 43,259)( 44,260)( 45,264)( 46,263)( 47,262)( 48,261)
( 49,273)( 50,274)( 51,275)( 52,276)( 53,280)( 54,279)( 55,278)( 56,277)
( 57,281)( 58,282)( 59,283)( 60,284)( 61,288)( 62,287)( 63,286)( 64,285)
( 65,233)( 66,234)( 67,235)( 68,236)( 69,240)( 70,239)( 71,238)( 72,237)
( 73,225)( 74,226)( 75,227)( 76,228)( 77,232)( 78,231)( 79,230)( 80,229)
( 81,241)( 82,242)( 83,243)( 84,244)( 85,248)( 86,247)( 87,246)( 88,245)
( 89,249)( 90,250)( 91,251)( 92,252)( 93,256)( 94,255)( 95,254)( 96,253)
( 97,297)( 98,298)( 99,299)(100,300)(101,304)(102,303)(103,302)(104,301)
(105,289)(106,290)(107,291)(108,292)(109,296)(110,295)(111,294)(112,293)
(113,305)(114,306)(115,307)(116,308)(117,312)(118,311)(119,310)(120,309)
(121,313)(122,314)(123,315)(124,316)(125,320)(126,319)(127,318)(128,317)
(129,361)(130,362)(131,363)(132,364)(133,368)(134,367)(135,366)(136,365)
(137,353)(138,354)(139,355)(140,356)(141,360)(142,359)(143,358)(144,357)
(145,369)(146,370)(147,371)(148,372)(149,376)(150,375)(151,374)(152,373)
(153,377)(154,378)(155,379)(156,380)(157,384)(158,383)(159,382)(160,381)
(161,329)(162,330)(163,331)(164,332)(165,336)(166,335)(167,334)(168,333)
(169,321)(170,322)(171,323)(172,324)(173,328)(174,327)(175,326)(176,325)
(177,337)(178,338)(179,339)(180,340)(181,344)(182,343)(183,342)(184,341)
(185,345)(186,346)(187,347)(188,348)(189,352)(190,351)(191,350)(192,349);;
s3 := ( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,101)( 6,102)( 7,103)( 8,104)
( 9,108)( 10,107)( 11,106)( 12,105)( 13,112)( 14,111)( 15,110)( 16,109)
( 17,118)( 18,117)( 19,120)( 20,119)( 21,114)( 22,113)( 23,116)( 24,115)
( 25,128)( 26,127)( 27,126)( 28,125)( 29,124)( 30,123)( 31,122)( 32,121)
( 33,129)( 34,130)( 35,131)( 36,132)( 37,133)( 38,134)( 39,135)( 40,136)
( 41,140)( 42,139)( 43,138)( 44,137)( 45,144)( 46,143)( 47,142)( 48,141)
( 49,150)( 50,149)( 51,152)( 52,151)( 53,146)( 54,145)( 55,148)( 56,147)
( 57,160)( 58,159)( 59,158)( 60,157)( 61,156)( 62,155)( 63,154)( 64,153)
( 65,161)( 66,162)( 67,163)( 68,164)( 69,165)( 70,166)( 71,167)( 72,168)
( 73,172)( 74,171)( 75,170)( 76,169)( 77,176)( 78,175)( 79,174)( 80,173)
( 81,182)( 82,181)( 83,184)( 84,183)( 85,178)( 86,177)( 87,180)( 88,179)
( 89,192)( 90,191)( 91,190)( 92,189)( 93,188)( 94,187)( 95,186)( 96,185)
(193,289)(194,290)(195,291)(196,292)(197,293)(198,294)(199,295)(200,296)
(201,300)(202,299)(203,298)(204,297)(205,304)(206,303)(207,302)(208,301)
(209,310)(210,309)(211,312)(212,311)(213,306)(214,305)(215,308)(216,307)
(217,320)(218,319)(219,318)(220,317)(221,316)(222,315)(223,314)(224,313)
(225,321)(226,322)(227,323)(228,324)(229,325)(230,326)(231,327)(232,328)
(233,332)(234,331)(235,330)(236,329)(237,336)(238,335)(239,334)(240,333)
(241,342)(242,341)(243,344)(244,343)(245,338)(246,337)(247,340)(248,339)
(249,352)(250,351)(251,350)(252,349)(253,348)(254,347)(255,346)(256,345)
(257,353)(258,354)(259,355)(260,356)(261,357)(262,358)(263,359)(264,360)
(265,364)(266,363)(267,362)(268,361)(269,368)(270,367)(271,366)(272,365)
(273,374)(274,373)(275,376)(276,375)(277,370)(278,369)(279,372)(280,371)
(281,384)(282,383)(283,382)(284,381)(285,380)(286,379)(287,378)(288,377);;
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, s2*s3*s2*s3*s2*s3*s2*s3,
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,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s0*s2*s1*s3*s2*s1*s3*s0*s1*s2*s3*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(384)!( 5, 7)( 6, 8)( 13, 15)( 14, 16)( 17, 25)( 18, 26)( 19, 27)
( 20, 28)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)( 34, 66)( 35, 67)
( 36, 68)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)( 42, 74)( 43, 75)
( 44, 76)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)( 50, 90)( 51, 91)
( 52, 92)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)( 58, 82)( 59, 83)
( 60, 84)( 61, 87)( 62, 88)( 63, 85)( 64, 86)(101,103)(102,104)(109,111)
(110,112)(113,121)(114,122)(115,123)(116,124)(117,127)(118,128)(119,125)
(120,126)(129,161)(130,162)(131,163)(132,164)(133,167)(134,168)(135,165)
(136,166)(137,169)(138,170)(139,171)(140,172)(141,175)(142,176)(143,173)
(144,174)(145,185)(146,186)(147,187)(148,188)(149,191)(150,192)(151,189)
(152,190)(153,177)(154,178)(155,179)(156,180)(157,183)(158,184)(159,181)
(160,182)(197,199)(198,200)(205,207)(206,208)(209,217)(210,218)(211,219)
(212,220)(213,223)(214,224)(215,221)(216,222)(225,257)(226,258)(227,259)
(228,260)(229,263)(230,264)(231,261)(232,262)(233,265)(234,266)(235,267)
(236,268)(237,271)(238,272)(239,269)(240,270)(241,281)(242,282)(243,283)
(244,284)(245,287)(246,288)(247,285)(248,286)(249,273)(250,274)(251,275)
(252,276)(253,279)(254,280)(255,277)(256,278)(293,295)(294,296)(301,303)
(302,304)(305,313)(306,314)(307,315)(308,316)(309,319)(310,320)(311,317)
(312,318)(321,353)(322,354)(323,355)(324,356)(325,359)(326,360)(327,357)
(328,358)(329,361)(330,362)(331,363)(332,364)(333,367)(334,368)(335,365)
(336,366)(337,377)(338,378)(339,379)(340,380)(341,383)(342,384)(343,381)
(344,382)(345,369)(346,370)(347,371)(348,372)(349,375)(350,376)(351,373)
(352,374);
s1 := Sym(384)!( 1, 72)( 2, 71)( 3, 68)( 4, 67)( 5, 70)( 6, 69)( 7, 66)
( 8, 65)( 9, 88)( 10, 87)( 11, 84)( 12, 83)( 13, 86)( 14, 85)( 15, 82)
( 16, 81)( 17, 80)( 18, 79)( 19, 76)( 20, 75)( 21, 78)( 22, 77)( 23, 74)
( 24, 73)( 25, 96)( 26, 95)( 27, 92)( 28, 91)( 29, 94)( 30, 93)( 31, 90)
( 32, 89)( 33, 40)( 34, 39)( 35, 36)( 37, 38)( 41, 56)( 42, 55)( 43, 52)
( 44, 51)( 45, 54)( 46, 53)( 47, 50)( 48, 49)( 57, 64)( 58, 63)( 59, 60)
( 61, 62)( 97,168)( 98,167)( 99,164)(100,163)(101,166)(102,165)(103,162)
(104,161)(105,184)(106,183)(107,180)(108,179)(109,182)(110,181)(111,178)
(112,177)(113,176)(114,175)(115,172)(116,171)(117,174)(118,173)(119,170)
(120,169)(121,192)(122,191)(123,188)(124,187)(125,190)(126,189)(127,186)
(128,185)(129,136)(130,135)(131,132)(133,134)(137,152)(138,151)(139,148)
(140,147)(141,150)(142,149)(143,146)(144,145)(153,160)(154,159)(155,156)
(157,158)(193,264)(194,263)(195,260)(196,259)(197,262)(198,261)(199,258)
(200,257)(201,280)(202,279)(203,276)(204,275)(205,278)(206,277)(207,274)
(208,273)(209,272)(210,271)(211,268)(212,267)(213,270)(214,269)(215,266)
(216,265)(217,288)(218,287)(219,284)(220,283)(221,286)(222,285)(223,282)
(224,281)(225,232)(226,231)(227,228)(229,230)(233,248)(234,247)(235,244)
(236,243)(237,246)(238,245)(239,242)(240,241)(249,256)(250,255)(251,252)
(253,254)(289,360)(290,359)(291,356)(292,355)(293,358)(294,357)(295,354)
(296,353)(297,376)(298,375)(299,372)(300,371)(301,374)(302,373)(303,370)
(304,369)(305,368)(306,367)(307,364)(308,363)(309,366)(310,365)(311,362)
(312,361)(313,384)(314,383)(315,380)(316,379)(317,382)(318,381)(319,378)
(320,377)(321,328)(322,327)(323,324)(325,326)(329,344)(330,343)(331,340)
(332,339)(333,342)(334,341)(335,338)(336,337)(345,352)(346,351)(347,348)
(349,350);
s2 := Sym(384)!( 1,201)( 2,202)( 3,203)( 4,204)( 5,208)( 6,207)( 7,206)
( 8,205)( 9,193)( 10,194)( 11,195)( 12,196)( 13,200)( 14,199)( 15,198)
( 16,197)( 17,209)( 18,210)( 19,211)( 20,212)( 21,216)( 22,215)( 23,214)
( 24,213)( 25,217)( 26,218)( 27,219)( 28,220)( 29,224)( 30,223)( 31,222)
( 32,221)( 33,265)( 34,266)( 35,267)( 36,268)( 37,272)( 38,271)( 39,270)
( 40,269)( 41,257)( 42,258)( 43,259)( 44,260)( 45,264)( 46,263)( 47,262)
( 48,261)( 49,273)( 50,274)( 51,275)( 52,276)( 53,280)( 54,279)( 55,278)
( 56,277)( 57,281)( 58,282)( 59,283)( 60,284)( 61,288)( 62,287)( 63,286)
( 64,285)( 65,233)( 66,234)( 67,235)( 68,236)( 69,240)( 70,239)( 71,238)
( 72,237)( 73,225)( 74,226)( 75,227)( 76,228)( 77,232)( 78,231)( 79,230)
( 80,229)( 81,241)( 82,242)( 83,243)( 84,244)( 85,248)( 86,247)( 87,246)
( 88,245)( 89,249)( 90,250)( 91,251)( 92,252)( 93,256)( 94,255)( 95,254)
( 96,253)( 97,297)( 98,298)( 99,299)(100,300)(101,304)(102,303)(103,302)
(104,301)(105,289)(106,290)(107,291)(108,292)(109,296)(110,295)(111,294)
(112,293)(113,305)(114,306)(115,307)(116,308)(117,312)(118,311)(119,310)
(120,309)(121,313)(122,314)(123,315)(124,316)(125,320)(126,319)(127,318)
(128,317)(129,361)(130,362)(131,363)(132,364)(133,368)(134,367)(135,366)
(136,365)(137,353)(138,354)(139,355)(140,356)(141,360)(142,359)(143,358)
(144,357)(145,369)(146,370)(147,371)(148,372)(149,376)(150,375)(151,374)
(152,373)(153,377)(154,378)(155,379)(156,380)(157,384)(158,383)(159,382)
(160,381)(161,329)(162,330)(163,331)(164,332)(165,336)(166,335)(167,334)
(168,333)(169,321)(170,322)(171,323)(172,324)(173,328)(174,327)(175,326)
(176,325)(177,337)(178,338)(179,339)(180,340)(181,344)(182,343)(183,342)
(184,341)(185,345)(186,346)(187,347)(188,348)(189,352)(190,351)(191,350)
(192,349);
s3 := Sym(384)!( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,101)( 6,102)( 7,103)
( 8,104)( 9,108)( 10,107)( 11,106)( 12,105)( 13,112)( 14,111)( 15,110)
( 16,109)( 17,118)( 18,117)( 19,120)( 20,119)( 21,114)( 22,113)( 23,116)
( 24,115)( 25,128)( 26,127)( 27,126)( 28,125)( 29,124)( 30,123)( 31,122)
( 32,121)( 33,129)( 34,130)( 35,131)( 36,132)( 37,133)( 38,134)( 39,135)
( 40,136)( 41,140)( 42,139)( 43,138)( 44,137)( 45,144)( 46,143)( 47,142)
( 48,141)( 49,150)( 50,149)( 51,152)( 52,151)( 53,146)( 54,145)( 55,148)
( 56,147)( 57,160)( 58,159)( 59,158)( 60,157)( 61,156)( 62,155)( 63,154)
( 64,153)( 65,161)( 66,162)( 67,163)( 68,164)( 69,165)( 70,166)( 71,167)
( 72,168)( 73,172)( 74,171)( 75,170)( 76,169)( 77,176)( 78,175)( 79,174)
( 80,173)( 81,182)( 82,181)( 83,184)( 84,183)( 85,178)( 86,177)( 87,180)
( 88,179)( 89,192)( 90,191)( 91,190)( 92,189)( 93,188)( 94,187)( 95,186)
( 96,185)(193,289)(194,290)(195,291)(196,292)(197,293)(198,294)(199,295)
(200,296)(201,300)(202,299)(203,298)(204,297)(205,304)(206,303)(207,302)
(208,301)(209,310)(210,309)(211,312)(212,311)(213,306)(214,305)(215,308)
(216,307)(217,320)(218,319)(219,318)(220,317)(221,316)(222,315)(223,314)
(224,313)(225,321)(226,322)(227,323)(228,324)(229,325)(230,326)(231,327)
(232,328)(233,332)(234,331)(235,330)(236,329)(237,336)(238,335)(239,334)
(240,333)(241,342)(242,341)(243,344)(244,343)(245,338)(246,337)(247,340)
(248,339)(249,352)(250,351)(251,350)(252,349)(253,348)(254,347)(255,346)
(256,345)(257,353)(258,354)(259,355)(260,356)(261,357)(262,358)(263,359)
(264,360)(265,364)(266,363)(267,362)(268,361)(269,368)(270,367)(271,366)
(272,365)(273,374)(274,373)(275,376)(276,375)(277,370)(278,369)(279,372)
(280,371)(281,384)(282,383)(283,382)(284,381)(285,380)(286,379)(287,378)
(288,377);
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,
s2*s3*s2*s3*s2*s3*s2*s3, 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,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s0*s2*s1*s3*s2*s1*s3*s0*s1*s2*s3*s1*s2*s0*s1 >;
References : None.
to this polytope