Polytope of Type {6,8,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,8,4}*768a
if this polytope has a name.
Group : SmallGroup(768,141629)
Rank : 4
Schlafli Type : {6,8,4}
Number of vertices, edges, etc : 6, 48, 32, 8
Order of s0s1s2s3 : 24
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 : {6,8,4}*384a, {6,8,4}*384b, {6,4,4}*384a
3-fold quotients : {2,8,4}*256a
4-fold quotients : {6,4,4}*192, {6,8,2}*192
6-fold quotients : {2,8,4}*128a, {2,8,4}*128b, {2,4,4}*128
8-fold quotients : {6,2,4}*96, {6,4,2}*96a
12-fold quotients : {2,4,4}*64, {2,8,2}*64
16-fold quotients : {3,2,4}*48, {6,2,2}*48
24-fold quotients : {2,2,4}*32, {2,4,2}*32
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 := ( 1, 49)( 2, 51)( 3, 50)( 4, 52)( 5, 54)( 6, 53)( 7, 55)( 8, 57)
( 9, 56)( 10, 58)( 11, 60)( 12, 59)( 13, 61)( 14, 63)( 15, 62)( 16, 64)
( 17, 66)( 18, 65)( 19, 67)( 20, 69)( 21, 68)( 22, 70)( 23, 72)( 24, 71)
( 25, 73)( 26, 75)( 27, 74)( 28, 76)( 29, 78)( 30, 77)( 31, 79)( 32, 81)
( 33, 80)( 34, 82)( 35, 84)( 36, 83)( 37, 85)( 38, 87)( 39, 86)( 40, 88)
( 41, 90)( 42, 89)( 43, 91)( 44, 93)( 45, 92)( 46, 94)( 47, 96)( 48, 95)
( 97,145)( 98,147)( 99,146)(100,148)(101,150)(102,149)(103,151)(104,153)
(105,152)(106,154)(107,156)(108,155)(109,157)(110,159)(111,158)(112,160)
(113,162)(114,161)(115,163)(116,165)(117,164)(118,166)(119,168)(120,167)
(121,169)(122,171)(123,170)(124,172)(125,174)(126,173)(127,175)(128,177)
(129,176)(130,178)(131,180)(132,179)(133,181)(134,183)(135,182)(136,184)
(137,186)(138,185)(139,187)(140,189)(141,188)(142,190)(143,192)(144,191)
(193,241)(194,243)(195,242)(196,244)(197,246)(198,245)(199,247)(200,249)
(201,248)(202,250)(203,252)(204,251)(205,253)(206,255)(207,254)(208,256)
(209,258)(210,257)(211,259)(212,261)(213,260)(214,262)(215,264)(216,263)
(217,265)(218,267)(219,266)(220,268)(221,270)(222,269)(223,271)(224,273)
(225,272)(226,274)(227,276)(228,275)(229,277)(230,279)(231,278)(232,280)
(233,282)(234,281)(235,283)(236,285)(237,284)(238,286)(239,288)(240,287)
(289,337)(290,339)(291,338)(292,340)(293,342)(294,341)(295,343)(296,345)
(297,344)(298,346)(299,348)(300,347)(301,349)(302,351)(303,350)(304,352)
(305,354)(306,353)(307,355)(308,357)(309,356)(310,358)(311,360)(312,359)
(313,361)(314,363)(315,362)(316,364)(317,366)(318,365)(319,367)(320,369)
(321,368)(322,370)(323,372)(324,371)(325,373)(326,375)(327,374)(328,376)
(329,378)(330,377)(331,379)(332,381)(333,380)(334,382)(335,384)(336,383);;
s1 := ( 1,243)( 2,242)( 3,241)( 4,246)( 5,245)( 6,244)( 7,249)( 8,248)
( 9,247)( 10,252)( 11,251)( 12,250)( 13,258)( 14,257)( 15,256)( 16,255)
( 17,254)( 18,253)( 19,264)( 20,263)( 21,262)( 22,261)( 23,260)( 24,259)
( 25,273)( 26,272)( 27,271)( 28,276)( 29,275)( 30,274)( 31,267)( 32,266)
( 33,265)( 34,270)( 35,269)( 36,268)( 37,288)( 38,287)( 39,286)( 40,285)
( 41,284)( 42,283)( 43,282)( 44,281)( 45,280)( 46,279)( 47,278)( 48,277)
( 49,195)( 50,194)( 51,193)( 52,198)( 53,197)( 54,196)( 55,201)( 56,200)
( 57,199)( 58,204)( 59,203)( 60,202)( 61,210)( 62,209)( 63,208)( 64,207)
( 65,206)( 66,205)( 67,216)( 68,215)( 69,214)( 70,213)( 71,212)( 72,211)
( 73,225)( 74,224)( 75,223)( 76,228)( 77,227)( 78,226)( 79,219)( 80,218)
( 81,217)( 82,222)( 83,221)( 84,220)( 85,240)( 86,239)( 87,238)( 88,237)
( 89,236)( 90,235)( 91,234)( 92,233)( 93,232)( 94,231)( 95,230)( 96,229)
( 97,339)( 98,338)( 99,337)(100,342)(101,341)(102,340)(103,345)(104,344)
(105,343)(106,348)(107,347)(108,346)(109,354)(110,353)(111,352)(112,351)
(113,350)(114,349)(115,360)(116,359)(117,358)(118,357)(119,356)(120,355)
(121,369)(122,368)(123,367)(124,372)(125,371)(126,370)(127,363)(128,362)
(129,361)(130,366)(131,365)(132,364)(133,384)(134,383)(135,382)(136,381)
(137,380)(138,379)(139,378)(140,377)(141,376)(142,375)(143,374)(144,373)
(145,291)(146,290)(147,289)(148,294)(149,293)(150,292)(151,297)(152,296)
(153,295)(154,300)(155,299)(156,298)(157,306)(158,305)(159,304)(160,303)
(161,302)(162,301)(163,312)(164,311)(165,310)(166,309)(167,308)(168,307)
(169,321)(170,320)(171,319)(172,324)(173,323)(174,322)(175,315)(176,314)
(177,313)(178,318)(179,317)(180,316)(181,336)(182,335)(183,334)(184,333)
(185,332)(186,331)(187,330)(188,329)(189,328)(190,327)(191,326)(192,325);;
s2 := ( 25, 31)( 26, 32)( 27, 33)( 28, 34)( 29, 35)( 30, 36)( 37, 43)( 38, 44)
( 39, 45)( 40, 46)( 41, 47)( 42, 48)( 73, 79)( 74, 80)( 75, 81)( 76, 82)
( 77, 83)( 78, 84)( 85, 91)( 86, 92)( 87, 93)( 88, 94)( 89, 95)( 90, 96)
( 97,109)( 98,110)( 99,111)(100,112)(101,113)(102,114)(103,115)(104,116)
(105,117)(106,118)(107,119)(108,120)(121,139)(122,140)(123,141)(124,142)
(125,143)(126,144)(127,133)(128,134)(129,135)(130,136)(131,137)(132,138)
(145,157)(146,158)(147,159)(148,160)(149,161)(150,162)(151,163)(152,164)
(153,165)(154,166)(155,167)(156,168)(169,187)(170,188)(171,189)(172,190)
(173,191)(174,192)(175,181)(176,182)(177,183)(178,184)(179,185)(180,186)
(193,217)(194,218)(195,219)(196,220)(197,221)(198,222)(199,223)(200,224)
(201,225)(202,226)(203,227)(204,228)(205,229)(206,230)(207,231)(208,232)
(209,233)(210,234)(211,235)(212,236)(213,237)(214,238)(215,239)(216,240)
(241,265)(242,266)(243,267)(244,268)(245,269)(246,270)(247,271)(248,272)
(249,273)(250,274)(251,275)(252,276)(253,277)(254,278)(255,279)(256,280)
(257,281)(258,282)(259,283)(260,284)(261,285)(262,286)(263,287)(264,288)
(289,328)(290,329)(291,330)(292,325)(293,326)(294,327)(295,334)(296,335)
(297,336)(298,331)(299,332)(300,333)(301,316)(302,317)(303,318)(304,313)
(305,314)(306,315)(307,322)(308,323)(309,324)(310,319)(311,320)(312,321)
(337,376)(338,377)(339,378)(340,373)(341,374)(342,375)(343,382)(344,383)
(345,384)(346,379)(347,380)(348,381)(349,364)(350,365)(351,366)(352,361)
(353,362)(354,363)(355,370)(356,371)(357,372)(358,367)(359,368)(360,369);;
s3 := ( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,101)( 6,102)( 7,103)( 8,104)
( 9,105)( 10,106)( 11,107)( 12,108)( 13,109)( 14,110)( 15,111)( 16,112)
( 17,113)( 18,114)( 19,115)( 20,116)( 21,117)( 22,118)( 23,119)( 24,120)
( 25,124)( 26,125)( 27,126)( 28,121)( 29,122)( 30,123)( 31,130)( 32,131)
( 33,132)( 34,127)( 35,128)( 36,129)( 37,136)( 38,137)( 39,138)( 40,133)
( 41,134)( 42,135)( 43,142)( 44,143)( 45,144)( 46,139)( 47,140)( 48,141)
( 49,145)( 50,146)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)( 56,152)
( 57,153)( 58,154)( 59,155)( 60,156)( 61,157)( 62,158)( 63,159)( 64,160)
( 65,161)( 66,162)( 67,163)( 68,164)( 69,165)( 70,166)( 71,167)( 72,168)
( 73,172)( 74,173)( 75,174)( 76,169)( 77,170)( 78,171)( 79,178)( 80,179)
( 81,180)( 82,175)( 83,176)( 84,177)( 85,184)( 86,185)( 87,186)( 88,181)
( 89,182)( 90,183)( 91,190)( 92,191)( 93,192)( 94,187)( 95,188)( 96,189)
(193,289)(194,290)(195,291)(196,292)(197,293)(198,294)(199,295)(200,296)
(201,297)(202,298)(203,299)(204,300)(205,301)(206,302)(207,303)(208,304)
(209,305)(210,306)(211,307)(212,308)(213,309)(214,310)(215,311)(216,312)
(217,316)(218,317)(219,318)(220,313)(221,314)(222,315)(223,322)(224,323)
(225,324)(226,319)(227,320)(228,321)(229,328)(230,329)(231,330)(232,325)
(233,326)(234,327)(235,334)(236,335)(237,336)(238,331)(239,332)(240,333)
(241,337)(242,338)(243,339)(244,340)(245,341)(246,342)(247,343)(248,344)
(249,345)(250,346)(251,347)(252,348)(253,349)(254,350)(255,351)(256,352)
(257,353)(258,354)(259,355)(260,356)(261,357)(262,358)(263,359)(264,360)
(265,364)(266,365)(267,366)(268,361)(269,362)(270,363)(271,370)(272,371)
(273,372)(274,367)(275,368)(276,369)(277,376)(278,377)(279,378)(280,373)
(281,374)(282,375)(283,382)(284,383)(285,384)(286,379)(287,380)(288,381);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s3*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(384)!( 1, 49)( 2, 51)( 3, 50)( 4, 52)( 5, 54)( 6, 53)( 7, 55)
( 8, 57)( 9, 56)( 10, 58)( 11, 60)( 12, 59)( 13, 61)( 14, 63)( 15, 62)
( 16, 64)( 17, 66)( 18, 65)( 19, 67)( 20, 69)( 21, 68)( 22, 70)( 23, 72)
( 24, 71)( 25, 73)( 26, 75)( 27, 74)( 28, 76)( 29, 78)( 30, 77)( 31, 79)
( 32, 81)( 33, 80)( 34, 82)( 35, 84)( 36, 83)( 37, 85)( 38, 87)( 39, 86)
( 40, 88)( 41, 90)( 42, 89)( 43, 91)( 44, 93)( 45, 92)( 46, 94)( 47, 96)
( 48, 95)( 97,145)( 98,147)( 99,146)(100,148)(101,150)(102,149)(103,151)
(104,153)(105,152)(106,154)(107,156)(108,155)(109,157)(110,159)(111,158)
(112,160)(113,162)(114,161)(115,163)(116,165)(117,164)(118,166)(119,168)
(120,167)(121,169)(122,171)(123,170)(124,172)(125,174)(126,173)(127,175)
(128,177)(129,176)(130,178)(131,180)(132,179)(133,181)(134,183)(135,182)
(136,184)(137,186)(138,185)(139,187)(140,189)(141,188)(142,190)(143,192)
(144,191)(193,241)(194,243)(195,242)(196,244)(197,246)(198,245)(199,247)
(200,249)(201,248)(202,250)(203,252)(204,251)(205,253)(206,255)(207,254)
(208,256)(209,258)(210,257)(211,259)(212,261)(213,260)(214,262)(215,264)
(216,263)(217,265)(218,267)(219,266)(220,268)(221,270)(222,269)(223,271)
(224,273)(225,272)(226,274)(227,276)(228,275)(229,277)(230,279)(231,278)
(232,280)(233,282)(234,281)(235,283)(236,285)(237,284)(238,286)(239,288)
(240,287)(289,337)(290,339)(291,338)(292,340)(293,342)(294,341)(295,343)
(296,345)(297,344)(298,346)(299,348)(300,347)(301,349)(302,351)(303,350)
(304,352)(305,354)(306,353)(307,355)(308,357)(309,356)(310,358)(311,360)
(312,359)(313,361)(314,363)(315,362)(316,364)(317,366)(318,365)(319,367)
(320,369)(321,368)(322,370)(323,372)(324,371)(325,373)(326,375)(327,374)
(328,376)(329,378)(330,377)(331,379)(332,381)(333,380)(334,382)(335,384)
(336,383);
s1 := Sym(384)!( 1,243)( 2,242)( 3,241)( 4,246)( 5,245)( 6,244)( 7,249)
( 8,248)( 9,247)( 10,252)( 11,251)( 12,250)( 13,258)( 14,257)( 15,256)
( 16,255)( 17,254)( 18,253)( 19,264)( 20,263)( 21,262)( 22,261)( 23,260)
( 24,259)( 25,273)( 26,272)( 27,271)( 28,276)( 29,275)( 30,274)( 31,267)
( 32,266)( 33,265)( 34,270)( 35,269)( 36,268)( 37,288)( 38,287)( 39,286)
( 40,285)( 41,284)( 42,283)( 43,282)( 44,281)( 45,280)( 46,279)( 47,278)
( 48,277)( 49,195)( 50,194)( 51,193)( 52,198)( 53,197)( 54,196)( 55,201)
( 56,200)( 57,199)( 58,204)( 59,203)( 60,202)( 61,210)( 62,209)( 63,208)
( 64,207)( 65,206)( 66,205)( 67,216)( 68,215)( 69,214)( 70,213)( 71,212)
( 72,211)( 73,225)( 74,224)( 75,223)( 76,228)( 77,227)( 78,226)( 79,219)
( 80,218)( 81,217)( 82,222)( 83,221)( 84,220)( 85,240)( 86,239)( 87,238)
( 88,237)( 89,236)( 90,235)( 91,234)( 92,233)( 93,232)( 94,231)( 95,230)
( 96,229)( 97,339)( 98,338)( 99,337)(100,342)(101,341)(102,340)(103,345)
(104,344)(105,343)(106,348)(107,347)(108,346)(109,354)(110,353)(111,352)
(112,351)(113,350)(114,349)(115,360)(116,359)(117,358)(118,357)(119,356)
(120,355)(121,369)(122,368)(123,367)(124,372)(125,371)(126,370)(127,363)
(128,362)(129,361)(130,366)(131,365)(132,364)(133,384)(134,383)(135,382)
(136,381)(137,380)(138,379)(139,378)(140,377)(141,376)(142,375)(143,374)
(144,373)(145,291)(146,290)(147,289)(148,294)(149,293)(150,292)(151,297)
(152,296)(153,295)(154,300)(155,299)(156,298)(157,306)(158,305)(159,304)
(160,303)(161,302)(162,301)(163,312)(164,311)(165,310)(166,309)(167,308)
(168,307)(169,321)(170,320)(171,319)(172,324)(173,323)(174,322)(175,315)
(176,314)(177,313)(178,318)(179,317)(180,316)(181,336)(182,335)(183,334)
(184,333)(185,332)(186,331)(187,330)(188,329)(189,328)(190,327)(191,326)
(192,325);
s2 := Sym(384)!( 25, 31)( 26, 32)( 27, 33)( 28, 34)( 29, 35)( 30, 36)( 37, 43)
( 38, 44)( 39, 45)( 40, 46)( 41, 47)( 42, 48)( 73, 79)( 74, 80)( 75, 81)
( 76, 82)( 77, 83)( 78, 84)( 85, 91)( 86, 92)( 87, 93)( 88, 94)( 89, 95)
( 90, 96)( 97,109)( 98,110)( 99,111)(100,112)(101,113)(102,114)(103,115)
(104,116)(105,117)(106,118)(107,119)(108,120)(121,139)(122,140)(123,141)
(124,142)(125,143)(126,144)(127,133)(128,134)(129,135)(130,136)(131,137)
(132,138)(145,157)(146,158)(147,159)(148,160)(149,161)(150,162)(151,163)
(152,164)(153,165)(154,166)(155,167)(156,168)(169,187)(170,188)(171,189)
(172,190)(173,191)(174,192)(175,181)(176,182)(177,183)(178,184)(179,185)
(180,186)(193,217)(194,218)(195,219)(196,220)(197,221)(198,222)(199,223)
(200,224)(201,225)(202,226)(203,227)(204,228)(205,229)(206,230)(207,231)
(208,232)(209,233)(210,234)(211,235)(212,236)(213,237)(214,238)(215,239)
(216,240)(241,265)(242,266)(243,267)(244,268)(245,269)(246,270)(247,271)
(248,272)(249,273)(250,274)(251,275)(252,276)(253,277)(254,278)(255,279)
(256,280)(257,281)(258,282)(259,283)(260,284)(261,285)(262,286)(263,287)
(264,288)(289,328)(290,329)(291,330)(292,325)(293,326)(294,327)(295,334)
(296,335)(297,336)(298,331)(299,332)(300,333)(301,316)(302,317)(303,318)
(304,313)(305,314)(306,315)(307,322)(308,323)(309,324)(310,319)(311,320)
(312,321)(337,376)(338,377)(339,378)(340,373)(341,374)(342,375)(343,382)
(344,383)(345,384)(346,379)(347,380)(348,381)(349,364)(350,365)(351,366)
(352,361)(353,362)(354,363)(355,370)(356,371)(357,372)(358,367)(359,368)
(360,369);
s3 := Sym(384)!( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,101)( 6,102)( 7,103)
( 8,104)( 9,105)( 10,106)( 11,107)( 12,108)( 13,109)( 14,110)( 15,111)
( 16,112)( 17,113)( 18,114)( 19,115)( 20,116)( 21,117)( 22,118)( 23,119)
( 24,120)( 25,124)( 26,125)( 27,126)( 28,121)( 29,122)( 30,123)( 31,130)
( 32,131)( 33,132)( 34,127)( 35,128)( 36,129)( 37,136)( 38,137)( 39,138)
( 40,133)( 41,134)( 42,135)( 43,142)( 44,143)( 45,144)( 46,139)( 47,140)
( 48,141)( 49,145)( 50,146)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)
( 56,152)( 57,153)( 58,154)( 59,155)( 60,156)( 61,157)( 62,158)( 63,159)
( 64,160)( 65,161)( 66,162)( 67,163)( 68,164)( 69,165)( 70,166)( 71,167)
( 72,168)( 73,172)( 74,173)( 75,174)( 76,169)( 77,170)( 78,171)( 79,178)
( 80,179)( 81,180)( 82,175)( 83,176)( 84,177)( 85,184)( 86,185)( 87,186)
( 88,181)( 89,182)( 90,183)( 91,190)( 92,191)( 93,192)( 94,187)( 95,188)
( 96,189)(193,289)(194,290)(195,291)(196,292)(197,293)(198,294)(199,295)
(200,296)(201,297)(202,298)(203,299)(204,300)(205,301)(206,302)(207,303)
(208,304)(209,305)(210,306)(211,307)(212,308)(213,309)(214,310)(215,311)
(216,312)(217,316)(218,317)(219,318)(220,313)(221,314)(222,315)(223,322)
(224,323)(225,324)(226,319)(227,320)(228,321)(229,328)(230,329)(231,330)
(232,325)(233,326)(234,327)(235,334)(236,335)(237,336)(238,331)(239,332)
(240,333)(241,337)(242,338)(243,339)(244,340)(245,341)(246,342)(247,343)
(248,344)(249,345)(250,346)(251,347)(252,348)(253,349)(254,350)(255,351)
(256,352)(257,353)(258,354)(259,355)(260,356)(261,357)(262,358)(263,359)
(264,360)(265,364)(266,365)(267,366)(268,361)(269,362)(270,363)(271,370)
(272,371)(273,372)(274,367)(275,368)(276,369)(277,376)(278,377)(279,378)
(280,373)(281,374)(282,375)(283,382)(284,383)(285,384)(286,379)(287,380)
(288,381);
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,
s0*s1*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s3*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope