Polytope of Type {12,8,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,8,4}*768d
Also Known As : {{12,8|2},{8,4|2}}. if this polytope has another name.
Group : SmallGroup(768,201309)
Rank : 4
Schlafli Type : {12,8,4}
Number of vertices, edges, etc : 12, 48, 16, 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 : {12,4,4}*384, {12,8,2}*384a, {6,8,4}*384a
3-fold quotients : {4,8,4}*256d
4-fold quotients : {12,4,2}*192a, {12,2,4}*192, {6,4,4}*192, {6,8,2}*192
6-fold quotients : {4,4,4}*128, {2,8,4}*128a, {4,8,2}*128a
8-fold quotients : {12,2,2}*96, {6,2,4}*96, {6,4,2}*96a
12-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*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, {4,2,2}*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 := ( 1,241)( 2,243)( 3,242)( 4,244)( 5,246)( 6,245)( 7,247)( 8,249)
( 9,248)( 10,250)( 11,252)( 12,251)( 13,253)( 14,255)( 15,254)( 16,256)
( 17,258)( 18,257)( 19,259)( 20,261)( 21,260)( 22,262)( 23,264)( 24,263)
( 25,265)( 26,267)( 27,266)( 28,268)( 29,270)( 30,269)( 31,271)( 32,273)
( 33,272)( 34,274)( 35,276)( 36,275)( 37,277)( 38,279)( 39,278)( 40,280)
( 41,282)( 42,281)( 43,283)( 44,285)( 45,284)( 46,286)( 47,288)( 48,287)
( 49,193)( 50,195)( 51,194)( 52,196)( 53,198)( 54,197)( 55,199)( 56,201)
( 57,200)( 58,202)( 59,204)( 60,203)( 61,205)( 62,207)( 63,206)( 64,208)
( 65,210)( 66,209)( 67,211)( 68,213)( 69,212)( 70,214)( 71,216)( 72,215)
( 73,217)( 74,219)( 75,218)( 76,220)( 77,222)( 78,221)( 79,223)( 80,225)
( 81,224)( 82,226)( 83,228)( 84,227)( 85,229)( 86,231)( 87,230)( 88,232)
( 89,234)( 90,233)( 91,235)( 92,237)( 93,236)( 94,238)( 95,240)( 96,239)
( 97,343)( 98,345)( 99,344)(100,346)(101,348)(102,347)(103,337)(104,339)
(105,338)(106,340)(107,342)(108,341)(109,355)(110,357)(111,356)(112,358)
(113,360)(114,359)(115,349)(116,351)(117,350)(118,352)(119,354)(120,353)
(121,367)(122,369)(123,368)(124,370)(125,372)(126,371)(127,361)(128,363)
(129,362)(130,364)(131,366)(132,365)(133,379)(134,381)(135,380)(136,382)
(137,384)(138,383)(139,373)(140,375)(141,374)(142,376)(143,378)(144,377)
(145,295)(146,297)(147,296)(148,298)(149,300)(150,299)(151,289)(152,291)
(153,290)(154,292)(155,294)(156,293)(157,307)(158,309)(159,308)(160,310)
(161,312)(162,311)(163,301)(164,303)(165,302)(166,304)(167,306)(168,305)
(169,319)(170,321)(171,320)(172,322)(173,324)(174,323)(175,313)(176,315)
(177,314)(178,316)(179,318)(180,317)(181,331)(182,333)(183,332)(184,334)
(185,336)(186,335)(187,325)(188,327)(189,326)(190,328)(191,330)(192,329);;
s1 := ( 1, 99)( 2, 98)( 3, 97)( 4,102)( 5,101)( 6,100)( 7,105)( 8,104)
( 9,103)( 10,108)( 11,107)( 12,106)( 13,114)( 14,113)( 15,112)( 16,111)
( 17,110)( 18,109)( 19,120)( 20,119)( 21,118)( 22,117)( 23,116)( 24,115)
( 25,123)( 26,122)( 27,121)( 28,126)( 29,125)( 30,124)( 31,129)( 32,128)
( 33,127)( 34,132)( 35,131)( 36,130)( 37,138)( 38,137)( 39,136)( 40,135)
( 41,134)( 42,133)( 43,144)( 44,143)( 45,142)( 46,141)( 47,140)( 48,139)
( 49,147)( 50,146)( 51,145)( 52,150)( 53,149)( 54,148)( 55,153)( 56,152)
( 57,151)( 58,156)( 59,155)( 60,154)( 61,162)( 62,161)( 63,160)( 64,159)
( 65,158)( 66,157)( 67,168)( 68,167)( 69,166)( 70,165)( 71,164)( 72,163)
( 73,171)( 74,170)( 75,169)( 76,174)( 77,173)( 78,172)( 79,177)( 80,176)
( 81,175)( 82,180)( 83,179)( 84,178)( 85,186)( 86,185)( 87,184)( 88,183)
( 89,182)( 90,181)( 91,192)( 92,191)( 93,190)( 94,189)( 95,188)( 96,187)
(193,291)(194,290)(195,289)(196,294)(197,293)(198,292)(199,297)(200,296)
(201,295)(202,300)(203,299)(204,298)(205,306)(206,305)(207,304)(208,303)
(209,302)(210,301)(211,312)(212,311)(213,310)(214,309)(215,308)(216,307)
(217,315)(218,314)(219,313)(220,318)(221,317)(222,316)(223,321)(224,320)
(225,319)(226,324)(227,323)(228,322)(229,330)(230,329)(231,328)(232,327)
(233,326)(234,325)(235,336)(236,335)(237,334)(238,333)(239,332)(240,331)
(241,339)(242,338)(243,337)(244,342)(245,341)(246,340)(247,345)(248,344)
(249,343)(250,348)(251,347)(252,346)(253,354)(254,353)(255,352)(256,351)
(257,350)(258,349)(259,360)(260,359)(261,358)(262,357)(263,356)(264,355)
(265,363)(266,362)(267,361)(268,366)(269,365)(270,364)(271,369)(272,368)
(273,367)(274,372)(275,371)(276,370)(277,378)(278,377)(279,376)(280,375)
(281,374)(282,373)(283,384)(284,383)(285,382)(286,381)(287,380)(288,379);;
s2 := ( 13, 16)( 14, 17)( 15, 18)( 19, 22)( 20, 23)( 21, 24)( 37, 40)( 38, 41)
( 39, 42)( 43, 46)( 44, 47)( 45, 48)( 49, 73)( 50, 74)( 51, 75)( 52, 76)
( 53, 77)( 54, 78)( 55, 79)( 56, 80)( 57, 81)( 58, 82)( 59, 83)( 60, 84)
( 61, 88)( 62, 89)( 63, 90)( 64, 85)( 65, 86)( 66, 87)( 67, 94)( 68, 95)
( 69, 96)( 70, 91)( 71, 92)( 72, 93)( 97,109)( 98,110)( 99,111)(100,112)
(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)(108,120)
(121,133)(122,134)(123,135)(124,136)(125,137)(126,138)(127,139)(128,140)
(129,141)(130,142)(131,143)(132,144)(145,181)(146,182)(147,183)(148,184)
(149,185)(150,186)(151,187)(152,188)(153,189)(154,190)(155,191)(156,192)
(157,169)(158,170)(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)
(165,177)(166,178)(167,179)(168,180)(193,217)(194,218)(195,219)(196,220)
(197,221)(198,222)(199,223)(200,224)(201,225)(202,226)(203,227)(204,228)
(205,232)(206,233)(207,234)(208,229)(209,230)(210,231)(211,238)(212,239)
(213,240)(214,235)(215,236)(216,237)(253,256)(254,257)(255,258)(259,262)
(260,263)(261,264)(277,280)(278,281)(279,282)(283,286)(284,287)(285,288)
(289,325)(290,326)(291,327)(292,328)(293,329)(294,330)(295,331)(296,332)
(297,333)(298,334)(299,335)(300,336)(301,313)(302,314)(303,315)(304,316)
(305,317)(306,318)(307,319)(308,320)(309,321)(310,322)(311,323)(312,324)
(337,349)(338,350)(339,351)(340,352)(341,353)(342,354)(343,355)(344,356)
(345,357)(346,358)(347,359)(348,360)(361,373)(362,374)(363,375)(364,376)
(365,377)(366,378)(367,379)(368,380)(369,381)(370,382)(371,383)(372,384);;
s3 := ( 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, 73)( 26, 74)( 27, 75)( 28, 76)( 29, 77)( 30, 78)( 31, 79)( 32, 80)
( 33, 81)( 34, 82)( 35, 83)( 36, 84)( 37, 85)( 38, 86)( 39, 87)( 40, 88)
( 41, 89)( 42, 90)( 43, 91)( 44, 92)( 45, 93)( 46, 94)( 47, 95)( 48, 96)
( 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)
(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)
(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)(136,184)
(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)(144,192)
(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)
(217,265)(218,266)(219,267)(220,268)(221,269)(222,270)(223,271)(224,272)
(225,273)(226,274)(227,275)(228,276)(229,277)(230,278)(231,279)(232,280)
(233,281)(234,282)(235,283)(236,284)(237,285)(238,286)(239,287)(240,288)
(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)
(313,361)(314,362)(315,363)(316,364)(317,365)(318,366)(319,367)(320,368)
(321,369)(322,370)(323,371)(324,372)(325,373)(326,374)(327,375)(328,376)
(329,377)(330,378)(331,379)(332,380)(333,381)(334,382)(335,383)(336,384);;
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*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(384)!( 1,241)( 2,243)( 3,242)( 4,244)( 5,246)( 6,245)( 7,247)
( 8,249)( 9,248)( 10,250)( 11,252)( 12,251)( 13,253)( 14,255)( 15,254)
( 16,256)( 17,258)( 18,257)( 19,259)( 20,261)( 21,260)( 22,262)( 23,264)
( 24,263)( 25,265)( 26,267)( 27,266)( 28,268)( 29,270)( 30,269)( 31,271)
( 32,273)( 33,272)( 34,274)( 35,276)( 36,275)( 37,277)( 38,279)( 39,278)
( 40,280)( 41,282)( 42,281)( 43,283)( 44,285)( 45,284)( 46,286)( 47,288)
( 48,287)( 49,193)( 50,195)( 51,194)( 52,196)( 53,198)( 54,197)( 55,199)
( 56,201)( 57,200)( 58,202)( 59,204)( 60,203)( 61,205)( 62,207)( 63,206)
( 64,208)( 65,210)( 66,209)( 67,211)( 68,213)( 69,212)( 70,214)( 71,216)
( 72,215)( 73,217)( 74,219)( 75,218)( 76,220)( 77,222)( 78,221)( 79,223)
( 80,225)( 81,224)( 82,226)( 83,228)( 84,227)( 85,229)( 86,231)( 87,230)
( 88,232)( 89,234)( 90,233)( 91,235)( 92,237)( 93,236)( 94,238)( 95,240)
( 96,239)( 97,343)( 98,345)( 99,344)(100,346)(101,348)(102,347)(103,337)
(104,339)(105,338)(106,340)(107,342)(108,341)(109,355)(110,357)(111,356)
(112,358)(113,360)(114,359)(115,349)(116,351)(117,350)(118,352)(119,354)
(120,353)(121,367)(122,369)(123,368)(124,370)(125,372)(126,371)(127,361)
(128,363)(129,362)(130,364)(131,366)(132,365)(133,379)(134,381)(135,380)
(136,382)(137,384)(138,383)(139,373)(140,375)(141,374)(142,376)(143,378)
(144,377)(145,295)(146,297)(147,296)(148,298)(149,300)(150,299)(151,289)
(152,291)(153,290)(154,292)(155,294)(156,293)(157,307)(158,309)(159,308)
(160,310)(161,312)(162,311)(163,301)(164,303)(165,302)(166,304)(167,306)
(168,305)(169,319)(170,321)(171,320)(172,322)(173,324)(174,323)(175,313)
(176,315)(177,314)(178,316)(179,318)(180,317)(181,331)(182,333)(183,332)
(184,334)(185,336)(186,335)(187,325)(188,327)(189,326)(190,328)(191,330)
(192,329);
s1 := Sym(384)!( 1, 99)( 2, 98)( 3, 97)( 4,102)( 5,101)( 6,100)( 7,105)
( 8,104)( 9,103)( 10,108)( 11,107)( 12,106)( 13,114)( 14,113)( 15,112)
( 16,111)( 17,110)( 18,109)( 19,120)( 20,119)( 21,118)( 22,117)( 23,116)
( 24,115)( 25,123)( 26,122)( 27,121)( 28,126)( 29,125)( 30,124)( 31,129)
( 32,128)( 33,127)( 34,132)( 35,131)( 36,130)( 37,138)( 38,137)( 39,136)
( 40,135)( 41,134)( 42,133)( 43,144)( 44,143)( 45,142)( 46,141)( 47,140)
( 48,139)( 49,147)( 50,146)( 51,145)( 52,150)( 53,149)( 54,148)( 55,153)
( 56,152)( 57,151)( 58,156)( 59,155)( 60,154)( 61,162)( 62,161)( 63,160)
( 64,159)( 65,158)( 66,157)( 67,168)( 68,167)( 69,166)( 70,165)( 71,164)
( 72,163)( 73,171)( 74,170)( 75,169)( 76,174)( 77,173)( 78,172)( 79,177)
( 80,176)( 81,175)( 82,180)( 83,179)( 84,178)( 85,186)( 86,185)( 87,184)
( 88,183)( 89,182)( 90,181)( 91,192)( 92,191)( 93,190)( 94,189)( 95,188)
( 96,187)(193,291)(194,290)(195,289)(196,294)(197,293)(198,292)(199,297)
(200,296)(201,295)(202,300)(203,299)(204,298)(205,306)(206,305)(207,304)
(208,303)(209,302)(210,301)(211,312)(212,311)(213,310)(214,309)(215,308)
(216,307)(217,315)(218,314)(219,313)(220,318)(221,317)(222,316)(223,321)
(224,320)(225,319)(226,324)(227,323)(228,322)(229,330)(230,329)(231,328)
(232,327)(233,326)(234,325)(235,336)(236,335)(237,334)(238,333)(239,332)
(240,331)(241,339)(242,338)(243,337)(244,342)(245,341)(246,340)(247,345)
(248,344)(249,343)(250,348)(251,347)(252,346)(253,354)(254,353)(255,352)
(256,351)(257,350)(258,349)(259,360)(260,359)(261,358)(262,357)(263,356)
(264,355)(265,363)(266,362)(267,361)(268,366)(269,365)(270,364)(271,369)
(272,368)(273,367)(274,372)(275,371)(276,370)(277,378)(278,377)(279,376)
(280,375)(281,374)(282,373)(283,384)(284,383)(285,382)(286,381)(287,380)
(288,379);
s2 := Sym(384)!( 13, 16)( 14, 17)( 15, 18)( 19, 22)( 20, 23)( 21, 24)( 37, 40)
( 38, 41)( 39, 42)( 43, 46)( 44, 47)( 45, 48)( 49, 73)( 50, 74)( 51, 75)
( 52, 76)( 53, 77)( 54, 78)( 55, 79)( 56, 80)( 57, 81)( 58, 82)( 59, 83)
( 60, 84)( 61, 88)( 62, 89)( 63, 90)( 64, 85)( 65, 86)( 66, 87)( 67, 94)
( 68, 95)( 69, 96)( 70, 91)( 71, 92)( 72, 93)( 97,109)( 98,110)( 99,111)
(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)
(108,120)(121,133)(122,134)(123,135)(124,136)(125,137)(126,138)(127,139)
(128,140)(129,141)(130,142)(131,143)(132,144)(145,181)(146,182)(147,183)
(148,184)(149,185)(150,186)(151,187)(152,188)(153,189)(154,190)(155,191)
(156,192)(157,169)(158,170)(159,171)(160,172)(161,173)(162,174)(163,175)
(164,176)(165,177)(166,178)(167,179)(168,180)(193,217)(194,218)(195,219)
(196,220)(197,221)(198,222)(199,223)(200,224)(201,225)(202,226)(203,227)
(204,228)(205,232)(206,233)(207,234)(208,229)(209,230)(210,231)(211,238)
(212,239)(213,240)(214,235)(215,236)(216,237)(253,256)(254,257)(255,258)
(259,262)(260,263)(261,264)(277,280)(278,281)(279,282)(283,286)(284,287)
(285,288)(289,325)(290,326)(291,327)(292,328)(293,329)(294,330)(295,331)
(296,332)(297,333)(298,334)(299,335)(300,336)(301,313)(302,314)(303,315)
(304,316)(305,317)(306,318)(307,319)(308,320)(309,321)(310,322)(311,323)
(312,324)(337,349)(338,350)(339,351)(340,352)(341,353)(342,354)(343,355)
(344,356)(345,357)(346,358)(347,359)(348,360)(361,373)(362,374)(363,375)
(364,376)(365,377)(366,378)(367,379)(368,380)(369,381)(370,382)(371,383)
(372,384);
s3 := 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, 73)( 26, 74)( 27, 75)( 28, 76)( 29, 77)( 30, 78)( 31, 79)
( 32, 80)( 33, 81)( 34, 82)( 35, 83)( 36, 84)( 37, 85)( 38, 86)( 39, 87)
( 40, 88)( 41, 89)( 42, 90)( 43, 91)( 44, 92)( 45, 93)( 46, 94)( 47, 95)
( 48, 96)( 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)(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)
(128,176)(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)
(136,184)(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)
(144,192)(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)(217,265)(218,266)(219,267)(220,268)(221,269)(222,270)(223,271)
(224,272)(225,273)(226,274)(227,275)(228,276)(229,277)(230,278)(231,279)
(232,280)(233,281)(234,282)(235,283)(236,284)(237,285)(238,286)(239,287)
(240,288)(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)(313,361)(314,362)(315,363)(316,364)(317,365)(318,366)(319,367)
(320,368)(321,369)(322,370)(323,371)(324,372)(325,373)(326,374)(327,375)
(328,376)(329,377)(330,378)(331,379)(332,380)(333,381)(334,382)(335,383)
(336,384);
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*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope