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