Polytope of Type {6,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,4}*768d
if this polytope has a name.
Group : SmallGroup(768,1089108)
Rank : 4
Schlafli Type : {6,6,4}
Number of vertices, edges, etc : 8, 48, 32, 8
Order of s0s1s2s3 : 4
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 : {3,6,4}*384b, {6,3,4}*384b
4-fold quotients : {3,3,4}*192, {6,6,2}*192
8-fold quotients : {3,6,2}*96, {6,3,2}*96
16-fold quotients : {3,3,2}*48
48-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 55, 56)
( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)
( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)( 76, 95)
( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 99,100)(103,104)(105,109)(106,110)
(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,133)(118,134)
(119,136)(120,135)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)
(127,140)(128,139)(147,148)(151,152)(153,157)(154,158)(155,160)(156,159)
(161,177)(162,178)(163,180)(164,179)(165,181)(166,182)(167,184)(168,183)
(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)(176,187)
(195,196)(199,200)(201,205)(202,206)(203,208)(204,207)(209,225)(210,226)
(211,228)(212,227)(213,229)(214,230)(215,232)(216,231)(217,237)(218,238)
(219,240)(220,239)(221,233)(222,234)(223,236)(224,235)(243,244)(247,248)
(249,253)(250,254)(251,256)(252,255)(257,273)(258,274)(259,276)(260,275)
(261,277)(262,278)(263,280)(264,279)(265,285)(266,286)(267,288)(268,287)
(269,281)(270,282)(271,284)(272,283)(291,292)(295,296)(297,301)(298,302)
(299,304)(300,303)(305,321)(306,322)(307,324)(308,323)(309,325)(310,326)
(311,328)(312,327)(313,333)(314,334)(315,336)(316,335)(317,329)(318,330)
(319,332)(320,331)(339,340)(343,344)(345,349)(346,350)(347,352)(348,351)
(353,369)(354,370)(355,372)(356,371)(357,373)(358,374)(359,376)(360,375)
(361,381)(362,382)(363,384)(364,383)(365,377)(366,378)(367,380)(368,379);;
s1 := ( 1,225)( 2,227)( 3,226)( 4,228)( 5,233)( 6,235)( 7,234)( 8,236)
( 9,229)( 10,231)( 11,230)( 12,232)( 13,237)( 14,239)( 15,238)( 16,240)
( 17,209)( 18,211)( 19,210)( 20,212)( 21,217)( 22,219)( 23,218)( 24,220)
( 25,213)( 26,215)( 27,214)( 28,216)( 29,221)( 30,223)( 31,222)( 32,224)
( 33,193)( 34,195)( 35,194)( 36,196)( 37,201)( 38,203)( 39,202)( 40,204)
( 41,197)( 42,199)( 43,198)( 44,200)( 45,205)( 46,207)( 47,206)( 48,208)
( 49,273)( 50,275)( 51,274)( 52,276)( 53,281)( 54,283)( 55,282)( 56,284)
( 57,277)( 58,279)( 59,278)( 60,280)( 61,285)( 62,287)( 63,286)( 64,288)
( 65,257)( 66,259)( 67,258)( 68,260)( 69,265)( 70,267)( 71,266)( 72,268)
( 73,261)( 74,263)( 75,262)( 76,264)( 77,269)( 78,271)( 79,270)( 80,272)
( 81,241)( 82,243)( 83,242)( 84,244)( 85,249)( 86,251)( 87,250)( 88,252)
( 89,245)( 90,247)( 91,246)( 92,248)( 93,253)( 94,255)( 95,254)( 96,256)
( 97,321)( 98,323)( 99,322)(100,324)(101,329)(102,331)(103,330)(104,332)
(105,325)(106,327)(107,326)(108,328)(109,333)(110,335)(111,334)(112,336)
(113,305)(114,307)(115,306)(116,308)(117,313)(118,315)(119,314)(120,316)
(121,309)(122,311)(123,310)(124,312)(125,317)(126,319)(127,318)(128,320)
(129,289)(130,291)(131,290)(132,292)(133,297)(134,299)(135,298)(136,300)
(137,293)(138,295)(139,294)(140,296)(141,301)(142,303)(143,302)(144,304)
(145,369)(146,371)(147,370)(148,372)(149,377)(150,379)(151,378)(152,380)
(153,373)(154,375)(155,374)(156,376)(157,381)(158,383)(159,382)(160,384)
(161,353)(162,355)(163,354)(164,356)(165,361)(166,363)(167,362)(168,364)
(169,357)(170,359)(171,358)(172,360)(173,365)(174,367)(175,366)(176,368)
(177,337)(178,339)(179,338)(180,340)(181,345)(182,347)(183,346)(184,348)
(185,341)(186,343)(187,342)(188,344)(189,349)(190,351)(191,350)(192,352);;
s2 := ( 1,101)( 2,102)( 3,104)( 4,103)( 5, 97)( 6, 98)( 7,100)( 8, 99)
( 9,105)( 10,106)( 11,108)( 12,107)( 13,109)( 14,110)( 15,112)( 16,111)
( 17,133)( 18,134)( 19,136)( 20,135)( 21,129)( 22,130)( 23,132)( 24,131)
( 25,137)( 26,138)( 27,140)( 28,139)( 29,141)( 30,142)( 31,144)( 32,143)
( 33,117)( 34,118)( 35,120)( 36,119)( 37,113)( 38,114)( 39,116)( 40,115)
( 41,121)( 42,122)( 43,124)( 44,123)( 45,125)( 46,126)( 47,128)( 48,127)
( 49,149)( 50,150)( 51,152)( 52,151)( 53,145)( 54,146)( 55,148)( 56,147)
( 57,153)( 58,154)( 59,156)( 60,155)( 61,157)( 62,158)( 63,160)( 64,159)
( 65,181)( 66,182)( 67,184)( 68,183)( 69,177)( 70,178)( 71,180)( 72,179)
( 73,185)( 74,186)( 75,188)( 76,187)( 77,189)( 78,190)( 79,192)( 80,191)
( 81,165)( 82,166)( 83,168)( 84,167)( 85,161)( 86,162)( 87,164)( 88,163)
( 89,169)( 90,170)( 91,172)( 92,171)( 93,173)( 94,174)( 95,176)( 96,175)
(193,293)(194,294)(195,296)(196,295)(197,289)(198,290)(199,292)(200,291)
(201,297)(202,298)(203,300)(204,299)(205,301)(206,302)(207,304)(208,303)
(209,325)(210,326)(211,328)(212,327)(213,321)(214,322)(215,324)(216,323)
(217,329)(218,330)(219,332)(220,331)(221,333)(222,334)(223,336)(224,335)
(225,309)(226,310)(227,312)(228,311)(229,305)(230,306)(231,308)(232,307)
(233,313)(234,314)(235,316)(236,315)(237,317)(238,318)(239,320)(240,319)
(241,341)(242,342)(243,344)(244,343)(245,337)(246,338)(247,340)(248,339)
(249,345)(250,346)(251,348)(252,347)(253,349)(254,350)(255,352)(256,351)
(257,373)(258,374)(259,376)(260,375)(261,369)(262,370)(263,372)(264,371)
(265,377)(266,378)(267,380)(268,379)(269,381)(270,382)(271,384)(272,383)
(273,357)(274,358)(275,360)(276,359)(277,353)(278,354)(279,356)(280,355)
(281,361)(282,362)(283,364)(284,363)(285,365)(286,366)(287,368)(288,367);;
s3 := ( 1, 49)( 2, 50)( 3, 51)( 4, 52)( 5, 54)( 6, 53)( 7, 56)( 8, 55)
( 9, 59)( 10, 60)( 11, 57)( 12, 58)( 13, 64)( 14, 63)( 15, 62)( 16, 61)
( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 70)( 22, 69)( 23, 72)( 24, 71)
( 25, 75)( 26, 76)( 27, 73)( 28, 74)( 29, 80)( 30, 79)( 31, 78)( 32, 77)
( 33, 81)( 34, 82)( 35, 83)( 36, 84)( 37, 86)( 38, 85)( 39, 88)( 40, 87)
( 41, 91)( 42, 92)( 43, 89)( 44, 90)( 45, 96)( 46, 95)( 47, 94)( 48, 93)
( 97,145)( 98,146)( 99,147)(100,148)(101,150)(102,149)(103,152)(104,151)
(105,155)(106,156)(107,153)(108,154)(109,160)(110,159)(111,158)(112,157)
(113,161)(114,162)(115,163)(116,164)(117,166)(118,165)(119,168)(120,167)
(121,171)(122,172)(123,169)(124,170)(125,176)(126,175)(127,174)(128,173)
(129,177)(130,178)(131,179)(132,180)(133,182)(134,181)(135,184)(136,183)
(137,187)(138,188)(139,185)(140,186)(141,192)(142,191)(143,190)(144,189)
(193,241)(194,242)(195,243)(196,244)(197,246)(198,245)(199,248)(200,247)
(201,251)(202,252)(203,249)(204,250)(205,256)(206,255)(207,254)(208,253)
(209,257)(210,258)(211,259)(212,260)(213,262)(214,261)(215,264)(216,263)
(217,267)(218,268)(219,265)(220,266)(221,272)(222,271)(223,270)(224,269)
(225,273)(226,274)(227,275)(228,276)(229,278)(230,277)(231,280)(232,279)
(233,283)(234,284)(235,281)(236,282)(237,288)(238,287)(239,286)(240,285)
(289,337)(290,338)(291,339)(292,340)(293,342)(294,341)(295,344)(296,343)
(297,347)(298,348)(299,345)(300,346)(301,352)(302,351)(303,350)(304,349)
(305,353)(306,354)(307,355)(308,356)(309,358)(310,357)(311,360)(312,359)
(313,363)(314,364)(315,361)(316,362)(317,368)(318,367)(319,366)(320,365)
(321,369)(322,370)(323,371)(324,372)(325,374)(326,373)(327,376)(328,375)
(329,379)(330,380)(331,377)(332,378)(333,384)(334,383)(335,382)(336,381);;
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, 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*s0*s2*s3*s1*s2*s3*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(384)!( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)
( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)
( 55, 56)( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)
( 68, 83)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)
( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 99,100)(103,104)(105,109)
(106,110)(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,133)
(118,134)(119,136)(120,135)(121,141)(122,142)(123,144)(124,143)(125,137)
(126,138)(127,140)(128,139)(147,148)(151,152)(153,157)(154,158)(155,160)
(156,159)(161,177)(162,178)(163,180)(164,179)(165,181)(166,182)(167,184)
(168,183)(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)
(176,187)(195,196)(199,200)(201,205)(202,206)(203,208)(204,207)(209,225)
(210,226)(211,228)(212,227)(213,229)(214,230)(215,232)(216,231)(217,237)
(218,238)(219,240)(220,239)(221,233)(222,234)(223,236)(224,235)(243,244)
(247,248)(249,253)(250,254)(251,256)(252,255)(257,273)(258,274)(259,276)
(260,275)(261,277)(262,278)(263,280)(264,279)(265,285)(266,286)(267,288)
(268,287)(269,281)(270,282)(271,284)(272,283)(291,292)(295,296)(297,301)
(298,302)(299,304)(300,303)(305,321)(306,322)(307,324)(308,323)(309,325)
(310,326)(311,328)(312,327)(313,333)(314,334)(315,336)(316,335)(317,329)
(318,330)(319,332)(320,331)(339,340)(343,344)(345,349)(346,350)(347,352)
(348,351)(353,369)(354,370)(355,372)(356,371)(357,373)(358,374)(359,376)
(360,375)(361,381)(362,382)(363,384)(364,383)(365,377)(366,378)(367,380)
(368,379);
s1 := Sym(384)!( 1,225)( 2,227)( 3,226)( 4,228)( 5,233)( 6,235)( 7,234)
( 8,236)( 9,229)( 10,231)( 11,230)( 12,232)( 13,237)( 14,239)( 15,238)
( 16,240)( 17,209)( 18,211)( 19,210)( 20,212)( 21,217)( 22,219)( 23,218)
( 24,220)( 25,213)( 26,215)( 27,214)( 28,216)( 29,221)( 30,223)( 31,222)
( 32,224)( 33,193)( 34,195)( 35,194)( 36,196)( 37,201)( 38,203)( 39,202)
( 40,204)( 41,197)( 42,199)( 43,198)( 44,200)( 45,205)( 46,207)( 47,206)
( 48,208)( 49,273)( 50,275)( 51,274)( 52,276)( 53,281)( 54,283)( 55,282)
( 56,284)( 57,277)( 58,279)( 59,278)( 60,280)( 61,285)( 62,287)( 63,286)
( 64,288)( 65,257)( 66,259)( 67,258)( 68,260)( 69,265)( 70,267)( 71,266)
( 72,268)( 73,261)( 74,263)( 75,262)( 76,264)( 77,269)( 78,271)( 79,270)
( 80,272)( 81,241)( 82,243)( 83,242)( 84,244)( 85,249)( 86,251)( 87,250)
( 88,252)( 89,245)( 90,247)( 91,246)( 92,248)( 93,253)( 94,255)( 95,254)
( 96,256)( 97,321)( 98,323)( 99,322)(100,324)(101,329)(102,331)(103,330)
(104,332)(105,325)(106,327)(107,326)(108,328)(109,333)(110,335)(111,334)
(112,336)(113,305)(114,307)(115,306)(116,308)(117,313)(118,315)(119,314)
(120,316)(121,309)(122,311)(123,310)(124,312)(125,317)(126,319)(127,318)
(128,320)(129,289)(130,291)(131,290)(132,292)(133,297)(134,299)(135,298)
(136,300)(137,293)(138,295)(139,294)(140,296)(141,301)(142,303)(143,302)
(144,304)(145,369)(146,371)(147,370)(148,372)(149,377)(150,379)(151,378)
(152,380)(153,373)(154,375)(155,374)(156,376)(157,381)(158,383)(159,382)
(160,384)(161,353)(162,355)(163,354)(164,356)(165,361)(166,363)(167,362)
(168,364)(169,357)(170,359)(171,358)(172,360)(173,365)(174,367)(175,366)
(176,368)(177,337)(178,339)(179,338)(180,340)(181,345)(182,347)(183,346)
(184,348)(185,341)(186,343)(187,342)(188,344)(189,349)(190,351)(191,350)
(192,352);
s2 := Sym(384)!( 1,101)( 2,102)( 3,104)( 4,103)( 5, 97)( 6, 98)( 7,100)
( 8, 99)( 9,105)( 10,106)( 11,108)( 12,107)( 13,109)( 14,110)( 15,112)
( 16,111)( 17,133)( 18,134)( 19,136)( 20,135)( 21,129)( 22,130)( 23,132)
( 24,131)( 25,137)( 26,138)( 27,140)( 28,139)( 29,141)( 30,142)( 31,144)
( 32,143)( 33,117)( 34,118)( 35,120)( 36,119)( 37,113)( 38,114)( 39,116)
( 40,115)( 41,121)( 42,122)( 43,124)( 44,123)( 45,125)( 46,126)( 47,128)
( 48,127)( 49,149)( 50,150)( 51,152)( 52,151)( 53,145)( 54,146)( 55,148)
( 56,147)( 57,153)( 58,154)( 59,156)( 60,155)( 61,157)( 62,158)( 63,160)
( 64,159)( 65,181)( 66,182)( 67,184)( 68,183)( 69,177)( 70,178)( 71,180)
( 72,179)( 73,185)( 74,186)( 75,188)( 76,187)( 77,189)( 78,190)( 79,192)
( 80,191)( 81,165)( 82,166)( 83,168)( 84,167)( 85,161)( 86,162)( 87,164)
( 88,163)( 89,169)( 90,170)( 91,172)( 92,171)( 93,173)( 94,174)( 95,176)
( 96,175)(193,293)(194,294)(195,296)(196,295)(197,289)(198,290)(199,292)
(200,291)(201,297)(202,298)(203,300)(204,299)(205,301)(206,302)(207,304)
(208,303)(209,325)(210,326)(211,328)(212,327)(213,321)(214,322)(215,324)
(216,323)(217,329)(218,330)(219,332)(220,331)(221,333)(222,334)(223,336)
(224,335)(225,309)(226,310)(227,312)(228,311)(229,305)(230,306)(231,308)
(232,307)(233,313)(234,314)(235,316)(236,315)(237,317)(238,318)(239,320)
(240,319)(241,341)(242,342)(243,344)(244,343)(245,337)(246,338)(247,340)
(248,339)(249,345)(250,346)(251,348)(252,347)(253,349)(254,350)(255,352)
(256,351)(257,373)(258,374)(259,376)(260,375)(261,369)(262,370)(263,372)
(264,371)(265,377)(266,378)(267,380)(268,379)(269,381)(270,382)(271,384)
(272,383)(273,357)(274,358)(275,360)(276,359)(277,353)(278,354)(279,356)
(280,355)(281,361)(282,362)(283,364)(284,363)(285,365)(286,366)(287,368)
(288,367);
s3 := Sym(384)!( 1, 49)( 2, 50)( 3, 51)( 4, 52)( 5, 54)( 6, 53)( 7, 56)
( 8, 55)( 9, 59)( 10, 60)( 11, 57)( 12, 58)( 13, 64)( 14, 63)( 15, 62)
( 16, 61)( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 70)( 22, 69)( 23, 72)
( 24, 71)( 25, 75)( 26, 76)( 27, 73)( 28, 74)( 29, 80)( 30, 79)( 31, 78)
( 32, 77)( 33, 81)( 34, 82)( 35, 83)( 36, 84)( 37, 86)( 38, 85)( 39, 88)
( 40, 87)( 41, 91)( 42, 92)( 43, 89)( 44, 90)( 45, 96)( 46, 95)( 47, 94)
( 48, 93)( 97,145)( 98,146)( 99,147)(100,148)(101,150)(102,149)(103,152)
(104,151)(105,155)(106,156)(107,153)(108,154)(109,160)(110,159)(111,158)
(112,157)(113,161)(114,162)(115,163)(116,164)(117,166)(118,165)(119,168)
(120,167)(121,171)(122,172)(123,169)(124,170)(125,176)(126,175)(127,174)
(128,173)(129,177)(130,178)(131,179)(132,180)(133,182)(134,181)(135,184)
(136,183)(137,187)(138,188)(139,185)(140,186)(141,192)(142,191)(143,190)
(144,189)(193,241)(194,242)(195,243)(196,244)(197,246)(198,245)(199,248)
(200,247)(201,251)(202,252)(203,249)(204,250)(205,256)(206,255)(207,254)
(208,253)(209,257)(210,258)(211,259)(212,260)(213,262)(214,261)(215,264)
(216,263)(217,267)(218,268)(219,265)(220,266)(221,272)(222,271)(223,270)
(224,269)(225,273)(226,274)(227,275)(228,276)(229,278)(230,277)(231,280)
(232,279)(233,283)(234,284)(235,281)(236,282)(237,288)(238,287)(239,286)
(240,285)(289,337)(290,338)(291,339)(292,340)(293,342)(294,341)(295,344)
(296,343)(297,347)(298,348)(299,345)(300,346)(301,352)(302,351)(303,350)
(304,349)(305,353)(306,354)(307,355)(308,356)(309,358)(310,357)(311,360)
(312,359)(313,363)(314,364)(315,361)(316,362)(317,368)(318,367)(319,366)
(320,365)(321,369)(322,370)(323,371)(324,372)(325,374)(326,373)(327,376)
(328,375)(329,379)(330,380)(331,377)(332,378)(333,384)(334,383)(335,382)
(336,381);
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,
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*s0*s2*s3*s1*s2*s3*s0*s1*s2*s0*s1 >;
References : None.
to this polytope