Polytope of Type {8,12,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,12,4}*768b
if this polytope has a name.
Group : SmallGroup(768,200907)
Rank : 4
Schlafli Type : {8,12,4}
Number of vertices, edges, etc : 8, 48, 24, 4
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 : {4,12,4}*384a, {8,12,2}*384b
3-fold quotients : {8,4,4}*256b
4-fold quotients : {2,12,4}*192a, {4,12,2}*192a, {4,6,4}*192a
6-fold quotients : {4,4,4}*128, {8,4,2}*128b
8-fold quotients : {2,12,2}*96, {2,6,4}*96a, {4,6,2}*96a
12-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
16-fold quotients : {2,6,2}*48
24-fold quotients : {2,2,4}*32, {2,4,2}*32, {4,2,2}*32
32-fold quotients : {2,3,2}*24
48-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 49)( 2, 50)( 3, 51)( 4, 52)( 5, 53)( 6, 54)( 7, 55)( 8, 56)
( 9, 57)( 10, 58)( 11, 59)( 12, 60)( 13, 61)( 14, 62)( 15, 63)( 16, 64)
( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)( 24, 72)
( 25, 76)( 26, 77)( 27, 78)( 28, 73)( 29, 74)( 30, 75)( 31, 82)( 32, 83)
( 33, 84)( 34, 79)( 35, 80)( 36, 81)( 37, 88)( 38, 89)( 39, 90)( 40, 85)
( 41, 86)( 42, 87)( 43, 94)( 44, 95)( 45, 96)( 46, 91)( 47, 92)( 48, 93)
( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)
(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)
(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)
(121,172)(122,173)(123,174)(124,169)(125,170)(126,171)(127,178)(128,179)
(129,180)(130,175)(131,176)(132,177)(133,184)(134,185)(135,186)(136,181)
(137,182)(138,183)(139,190)(140,191)(141,192)(142,187)(143,188)(144,189)
(193,241)(194,242)(195,243)(196,244)(197,245)(198,246)(199,247)(200,248)
(201,249)(202,250)(203,251)(204,252)(205,253)(206,254)(207,255)(208,256)
(209,257)(210,258)(211,259)(212,260)(213,261)(214,262)(215,263)(216,264)
(217,268)(218,269)(219,270)(220,265)(221,266)(222,267)(223,274)(224,275)
(225,276)(226,271)(227,272)(228,273)(229,280)(230,281)(231,282)(232,277)
(233,278)(234,279)(235,286)(236,287)(237,288)(238,283)(239,284)(240,285)
(289,337)(290,338)(291,339)(292,340)(293,341)(294,342)(295,343)(296,344)
(297,345)(298,346)(299,347)(300,348)(301,349)(302,350)(303,351)(304,352)
(305,353)(306,354)(307,355)(308,356)(309,357)(310,358)(311,359)(312,360)
(313,364)(314,365)(315,366)(316,361)(317,362)(318,363)(319,370)(320,371)
(321,372)(322,367)(323,368)(324,369)(325,376)(326,377)(327,378)(328,373)
(329,374)(330,375)(331,382)(332,383)(333,384)(334,379)(335,380)(336,381);;
s1 := ( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 19, 22)
( 20, 24)( 21, 23)( 25, 28)( 26, 30)( 27, 29)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 41, 42)( 44, 45)( 47, 48)( 49, 73)( 50, 75)( 51, 74)( 52, 76)
( 53, 78)( 54, 77)( 55, 79)( 56, 81)( 57, 80)( 58, 82)( 59, 84)( 60, 83)
( 61, 88)( 62, 90)( 63, 89)( 64, 85)( 65, 87)( 66, 86)( 67, 94)( 68, 96)
( 69, 95)( 70, 91)( 71, 93)( 72, 92)( 97,109)( 98,111)( 99,110)(100,112)
(101,114)(102,113)(103,115)(104,117)(105,116)(106,118)(107,120)(108,119)
(121,136)(122,138)(123,137)(124,133)(125,135)(126,134)(127,142)(128,144)
(129,143)(130,139)(131,141)(132,140)(145,181)(146,183)(147,182)(148,184)
(149,186)(150,185)(151,187)(152,189)(153,188)(154,190)(155,192)(156,191)
(157,169)(158,171)(159,170)(160,172)(161,174)(162,173)(163,175)(164,177)
(165,176)(166,178)(167,180)(168,179)(193,217)(194,219)(195,218)(196,220)
(197,222)(198,221)(199,223)(200,225)(201,224)(202,226)(203,228)(204,227)
(205,232)(206,234)(207,233)(208,229)(209,231)(210,230)(211,238)(212,240)
(213,239)(214,235)(215,237)(216,236)(242,243)(245,246)(248,249)(251,252)
(253,256)(254,258)(255,257)(259,262)(260,264)(261,263)(265,268)(266,270)
(267,269)(271,274)(272,276)(273,275)(278,279)(281,282)(284,285)(287,288)
(289,325)(290,327)(291,326)(292,328)(293,330)(294,329)(295,331)(296,333)
(297,332)(298,334)(299,336)(300,335)(301,313)(302,315)(303,314)(304,316)
(305,318)(306,317)(307,319)(308,321)(309,320)(310,322)(311,324)(312,323)
(337,349)(338,351)(339,350)(340,352)(341,354)(342,353)(343,355)(344,357)
(345,356)(346,358)(347,360)(348,359)(361,376)(362,378)(363,377)(364,373)
(365,375)(366,374)(367,382)(368,384)(369,383)(370,379)(371,381)(372,380);;
s2 := ( 1,147)( 2,146)( 3,145)( 4,150)( 5,149)( 6,148)( 7,153)( 8,152)
( 9,151)( 10,156)( 11,155)( 12,154)( 13,162)( 14,161)( 15,160)( 16,159)
( 17,158)( 18,157)( 19,168)( 20,167)( 21,166)( 22,165)( 23,164)( 24,163)
( 25,174)( 26,173)( 27,172)( 28,171)( 29,170)( 30,169)( 31,180)( 32,179)
( 33,178)( 34,177)( 35,176)( 36,175)( 37,183)( 38,182)( 39,181)( 40,186)
( 41,185)( 42,184)( 43,189)( 44,188)( 45,187)( 46,192)( 47,191)( 48,190)
( 49, 99)( 50, 98)( 51, 97)( 52,102)( 53,101)( 54,100)( 55,105)( 56,104)
( 57,103)( 58,108)( 59,107)( 60,106)( 61,114)( 62,113)( 63,112)( 64,111)
( 65,110)( 66,109)( 67,120)( 68,119)( 69,118)( 70,117)( 71,116)( 72,115)
( 73,126)( 74,125)( 75,124)( 76,123)( 77,122)( 78,121)( 79,132)( 80,131)
( 81,130)( 82,129)( 83,128)( 84,127)( 85,135)( 86,134)( 87,133)( 88,138)
( 89,137)( 90,136)( 91,141)( 92,140)( 93,139)( 94,144)( 95,143)( 96,142)
(193,339)(194,338)(195,337)(196,342)(197,341)(198,340)(199,345)(200,344)
(201,343)(202,348)(203,347)(204,346)(205,354)(206,353)(207,352)(208,351)
(209,350)(210,349)(211,360)(212,359)(213,358)(214,357)(215,356)(216,355)
(217,366)(218,365)(219,364)(220,363)(221,362)(222,361)(223,372)(224,371)
(225,370)(226,369)(227,368)(228,367)(229,375)(230,374)(231,373)(232,378)
(233,377)(234,376)(235,381)(236,380)(237,379)(238,384)(239,383)(240,382)
(241,291)(242,290)(243,289)(244,294)(245,293)(246,292)(247,297)(248,296)
(249,295)(250,300)(251,299)(252,298)(253,306)(254,305)(255,304)(256,303)
(257,302)(258,301)(259,312)(260,311)(261,310)(262,309)(263,308)(264,307)
(265,318)(266,317)(267,316)(268,315)(269,314)(270,313)(271,324)(272,323)
(273,322)(274,321)(275,320)(276,319)(277,327)(278,326)(279,325)(280,330)
(281,329)(282,328)(283,333)(284,332)(285,331)(286,336)(287,335)(288,334);;
s3 := ( 1,241)( 2,242)( 3,243)( 4,244)( 5,245)( 6,246)( 7,247)( 8,248)
( 9,249)( 10,250)( 11,251)( 12,252)( 13,253)( 14,254)( 15,255)( 16,256)
( 17,257)( 18,258)( 19,259)( 20,260)( 21,261)( 22,262)( 23,263)( 24,264)
( 25,265)( 26,266)( 27,267)( 28,268)( 29,269)( 30,270)( 31,271)( 32,272)
( 33,273)( 34,274)( 35,275)( 36,276)( 37,277)( 38,278)( 39,279)( 40,280)
( 41,281)( 42,282)( 43,283)( 44,284)( 45,285)( 46,286)( 47,287)( 48,288)
( 49,193)( 50,194)( 51,195)( 52,196)( 53,197)( 54,198)( 55,199)( 56,200)
( 57,201)( 58,202)( 59,203)( 60,204)( 61,205)( 62,206)( 63,207)( 64,208)
( 65,209)( 66,210)( 67,211)( 68,212)( 69,213)( 70,214)( 71,215)( 72,216)
( 73,217)( 74,218)( 75,219)( 76,220)( 77,221)( 78,222)( 79,223)( 80,224)
( 81,225)( 82,226)( 83,227)( 84,228)( 85,229)( 86,230)( 87,231)( 88,232)
( 89,233)( 90,234)( 91,235)( 92,236)( 93,237)( 94,238)( 95,239)( 96,240)
( 97,343)( 98,344)( 99,345)(100,346)(101,347)(102,348)(103,337)(104,338)
(105,339)(106,340)(107,341)(108,342)(109,355)(110,356)(111,357)(112,358)
(113,359)(114,360)(115,349)(116,350)(117,351)(118,352)(119,353)(120,354)
(121,367)(122,368)(123,369)(124,370)(125,371)(126,372)(127,361)(128,362)
(129,363)(130,364)(131,365)(132,366)(133,379)(134,380)(135,381)(136,382)
(137,383)(138,384)(139,373)(140,374)(141,375)(142,376)(143,377)(144,378)
(145,295)(146,296)(147,297)(148,298)(149,299)(150,300)(151,289)(152,290)
(153,291)(154,292)(155,293)(156,294)(157,307)(158,308)(159,309)(160,310)
(161,311)(162,312)(163,301)(164,302)(165,303)(166,304)(167,305)(168,306)
(169,319)(170,320)(171,321)(172,322)(173,323)(174,324)(175,313)(176,314)
(177,315)(178,316)(179,317)(180,318)(181,331)(182,332)(183,333)(184,334)
(185,335)(186,336)(187,325)(188,326)(189,327)(190,328)(191,329)(192,330);;
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*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*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, 50)( 3, 51)( 4, 52)( 5, 53)( 6, 54)( 7, 55)
( 8, 56)( 9, 57)( 10, 58)( 11, 59)( 12, 60)( 13, 61)( 14, 62)( 15, 63)
( 16, 64)( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)
( 24, 72)( 25, 76)( 26, 77)( 27, 78)( 28, 73)( 29, 74)( 30, 75)( 31, 82)
( 32, 83)( 33, 84)( 34, 79)( 35, 80)( 36, 81)( 37, 88)( 38, 89)( 39, 90)
( 40, 85)( 41, 86)( 42, 87)( 43, 94)( 44, 95)( 45, 96)( 46, 91)( 47, 92)
( 48, 93)( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)
(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)
(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)
(120,168)(121,172)(122,173)(123,174)(124,169)(125,170)(126,171)(127,178)
(128,179)(129,180)(130,175)(131,176)(132,177)(133,184)(134,185)(135,186)
(136,181)(137,182)(138,183)(139,190)(140,191)(141,192)(142,187)(143,188)
(144,189)(193,241)(194,242)(195,243)(196,244)(197,245)(198,246)(199,247)
(200,248)(201,249)(202,250)(203,251)(204,252)(205,253)(206,254)(207,255)
(208,256)(209,257)(210,258)(211,259)(212,260)(213,261)(214,262)(215,263)
(216,264)(217,268)(218,269)(219,270)(220,265)(221,266)(222,267)(223,274)
(224,275)(225,276)(226,271)(227,272)(228,273)(229,280)(230,281)(231,282)
(232,277)(233,278)(234,279)(235,286)(236,287)(237,288)(238,283)(239,284)
(240,285)(289,337)(290,338)(291,339)(292,340)(293,341)(294,342)(295,343)
(296,344)(297,345)(298,346)(299,347)(300,348)(301,349)(302,350)(303,351)
(304,352)(305,353)(306,354)(307,355)(308,356)(309,357)(310,358)(311,359)
(312,360)(313,364)(314,365)(315,366)(316,361)(317,362)(318,363)(319,370)
(320,371)(321,372)(322,367)(323,368)(324,369)(325,376)(326,377)(327,378)
(328,373)(329,374)(330,375)(331,382)(332,383)(333,384)(334,379)(335,380)
(336,381);
s1 := Sym(384)!( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 19, 22)( 20, 24)( 21, 23)( 25, 28)( 26, 30)( 27, 29)( 31, 34)( 32, 36)
( 33, 35)( 38, 39)( 41, 42)( 44, 45)( 47, 48)( 49, 73)( 50, 75)( 51, 74)
( 52, 76)( 53, 78)( 54, 77)( 55, 79)( 56, 81)( 57, 80)( 58, 82)( 59, 84)
( 60, 83)( 61, 88)( 62, 90)( 63, 89)( 64, 85)( 65, 87)( 66, 86)( 67, 94)
( 68, 96)( 69, 95)( 70, 91)( 71, 93)( 72, 92)( 97,109)( 98,111)( 99,110)
(100,112)(101,114)(102,113)(103,115)(104,117)(105,116)(106,118)(107,120)
(108,119)(121,136)(122,138)(123,137)(124,133)(125,135)(126,134)(127,142)
(128,144)(129,143)(130,139)(131,141)(132,140)(145,181)(146,183)(147,182)
(148,184)(149,186)(150,185)(151,187)(152,189)(153,188)(154,190)(155,192)
(156,191)(157,169)(158,171)(159,170)(160,172)(161,174)(162,173)(163,175)
(164,177)(165,176)(166,178)(167,180)(168,179)(193,217)(194,219)(195,218)
(196,220)(197,222)(198,221)(199,223)(200,225)(201,224)(202,226)(203,228)
(204,227)(205,232)(206,234)(207,233)(208,229)(209,231)(210,230)(211,238)
(212,240)(213,239)(214,235)(215,237)(216,236)(242,243)(245,246)(248,249)
(251,252)(253,256)(254,258)(255,257)(259,262)(260,264)(261,263)(265,268)
(266,270)(267,269)(271,274)(272,276)(273,275)(278,279)(281,282)(284,285)
(287,288)(289,325)(290,327)(291,326)(292,328)(293,330)(294,329)(295,331)
(296,333)(297,332)(298,334)(299,336)(300,335)(301,313)(302,315)(303,314)
(304,316)(305,318)(306,317)(307,319)(308,321)(309,320)(310,322)(311,324)
(312,323)(337,349)(338,351)(339,350)(340,352)(341,354)(342,353)(343,355)
(344,357)(345,356)(346,358)(347,360)(348,359)(361,376)(362,378)(363,377)
(364,373)(365,375)(366,374)(367,382)(368,384)(369,383)(370,379)(371,381)
(372,380);
s2 := 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)( 13,162)( 14,161)( 15,160)
( 16,159)( 17,158)( 18,157)( 19,168)( 20,167)( 21,166)( 22,165)( 23,164)
( 24,163)( 25,174)( 26,173)( 27,172)( 28,171)( 29,170)( 30,169)( 31,180)
( 32,179)( 33,178)( 34,177)( 35,176)( 36,175)( 37,183)( 38,182)( 39,181)
( 40,186)( 41,185)( 42,184)( 43,189)( 44,188)( 45,187)( 46,192)( 47,191)
( 48,190)( 49, 99)( 50, 98)( 51, 97)( 52,102)( 53,101)( 54,100)( 55,105)
( 56,104)( 57,103)( 58,108)( 59,107)( 60,106)( 61,114)( 62,113)( 63,112)
( 64,111)( 65,110)( 66,109)( 67,120)( 68,119)( 69,118)( 70,117)( 71,116)
( 72,115)( 73,126)( 74,125)( 75,124)( 76,123)( 77,122)( 78,121)( 79,132)
( 80,131)( 81,130)( 82,129)( 83,128)( 84,127)( 85,135)( 86,134)( 87,133)
( 88,138)( 89,137)( 90,136)( 91,141)( 92,140)( 93,139)( 94,144)( 95,143)
( 96,142)(193,339)(194,338)(195,337)(196,342)(197,341)(198,340)(199,345)
(200,344)(201,343)(202,348)(203,347)(204,346)(205,354)(206,353)(207,352)
(208,351)(209,350)(210,349)(211,360)(212,359)(213,358)(214,357)(215,356)
(216,355)(217,366)(218,365)(219,364)(220,363)(221,362)(222,361)(223,372)
(224,371)(225,370)(226,369)(227,368)(228,367)(229,375)(230,374)(231,373)
(232,378)(233,377)(234,376)(235,381)(236,380)(237,379)(238,384)(239,383)
(240,382)(241,291)(242,290)(243,289)(244,294)(245,293)(246,292)(247,297)
(248,296)(249,295)(250,300)(251,299)(252,298)(253,306)(254,305)(255,304)
(256,303)(257,302)(258,301)(259,312)(260,311)(261,310)(262,309)(263,308)
(264,307)(265,318)(266,317)(267,316)(268,315)(269,314)(270,313)(271,324)
(272,323)(273,322)(274,321)(275,320)(276,319)(277,327)(278,326)(279,325)
(280,330)(281,329)(282,328)(283,333)(284,332)(285,331)(286,336)(287,335)
(288,334);
s3 := Sym(384)!( 1,241)( 2,242)( 3,243)( 4,244)( 5,245)( 6,246)( 7,247)
( 8,248)( 9,249)( 10,250)( 11,251)( 12,252)( 13,253)( 14,254)( 15,255)
( 16,256)( 17,257)( 18,258)( 19,259)( 20,260)( 21,261)( 22,262)( 23,263)
( 24,264)( 25,265)( 26,266)( 27,267)( 28,268)( 29,269)( 30,270)( 31,271)
( 32,272)( 33,273)( 34,274)( 35,275)( 36,276)( 37,277)( 38,278)( 39,279)
( 40,280)( 41,281)( 42,282)( 43,283)( 44,284)( 45,285)( 46,286)( 47,287)
( 48,288)( 49,193)( 50,194)( 51,195)( 52,196)( 53,197)( 54,198)( 55,199)
( 56,200)( 57,201)( 58,202)( 59,203)( 60,204)( 61,205)( 62,206)( 63,207)
( 64,208)( 65,209)( 66,210)( 67,211)( 68,212)( 69,213)( 70,214)( 71,215)
( 72,216)( 73,217)( 74,218)( 75,219)( 76,220)( 77,221)( 78,222)( 79,223)
( 80,224)( 81,225)( 82,226)( 83,227)( 84,228)( 85,229)( 86,230)( 87,231)
( 88,232)( 89,233)( 90,234)( 91,235)( 92,236)( 93,237)( 94,238)( 95,239)
( 96,240)( 97,343)( 98,344)( 99,345)(100,346)(101,347)(102,348)(103,337)
(104,338)(105,339)(106,340)(107,341)(108,342)(109,355)(110,356)(111,357)
(112,358)(113,359)(114,360)(115,349)(116,350)(117,351)(118,352)(119,353)
(120,354)(121,367)(122,368)(123,369)(124,370)(125,371)(126,372)(127,361)
(128,362)(129,363)(130,364)(131,365)(132,366)(133,379)(134,380)(135,381)
(136,382)(137,383)(138,384)(139,373)(140,374)(141,375)(142,376)(143,377)
(144,378)(145,295)(146,296)(147,297)(148,298)(149,299)(150,300)(151,289)
(152,290)(153,291)(154,292)(155,293)(156,294)(157,307)(158,308)(159,309)
(160,310)(161,311)(162,312)(163,301)(164,302)(165,303)(166,304)(167,305)
(168,306)(169,319)(170,320)(171,321)(172,322)(173,323)(174,324)(175,313)
(176,314)(177,315)(178,316)(179,317)(180,318)(181,331)(182,332)(183,333)
(184,334)(185,335)(186,336)(187,325)(188,326)(189,327)(190,328)(191,329)
(192,330);
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*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope