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