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