Polytope of Type {12,4,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4,8}*768a
Also Known As : {{12,4|2},{4,8|2}}. if this polytope has another name.
Group : SmallGroup(768,200564)
Rank : 4
Schlafli Type : {12,4,8}
Number of vertices, edges, etc : 12, 24, 16, 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 : {12,4,4}*384, {12,2,8}*384, {6,4,8}*384a
3-fold quotients : {4,4,8}*256a
4-fold quotients : {12,4,2}*192a, {12,2,4}*192, {6,4,4}*192, {6,2,8}*192
6-fold quotients : {4,4,4}*128, {2,4,8}*128a, {4,2,8}*128
8-fold quotients : {3,2,8}*96, {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, {2,2,8}*64
16-fold quotients : {3,2,4}*48, {6,2,2}*48
24-fold quotients : {2,2,4}*32, {2,4,2}*32, {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)( 13,111)( 14,110)( 15,109)( 16,114)
( 17,113)( 18,112)( 19,117)( 20,116)( 21,115)( 22,120)( 23,119)( 24,118)
( 25,123)( 26,122)( 27,121)( 28,126)( 29,125)( 30,124)( 31,129)( 32,128)
( 33,127)( 34,132)( 35,131)( 36,130)( 37,135)( 38,134)( 39,133)( 40,138)
( 41,137)( 42,136)( 43,141)( 44,140)( 45,139)( 46,144)( 47,143)( 48,142)
( 49,147)( 50,146)( 51,145)( 52,150)( 53,149)( 54,148)( 55,153)( 56,152)
( 57,151)( 58,156)( 59,155)( 60,154)( 61,159)( 62,158)( 63,157)( 64,162)
( 65,161)( 66,160)( 67,165)( 68,164)( 69,163)( 70,168)( 71,167)( 72,166)
( 73,171)( 74,170)( 75,169)( 76,174)( 77,173)( 78,172)( 79,177)( 80,176)
( 81,175)( 82,180)( 83,179)( 84,178)( 85,183)( 86,182)( 87,181)( 88,186)
( 89,185)( 90,184)( 91,189)( 92,188)( 93,187)( 94,192)( 95,191)( 96,190)
(193,291)(194,290)(195,289)(196,294)(197,293)(198,292)(199,297)(200,296)
(201,295)(202,300)(203,299)(204,298)(205,303)(206,302)(207,301)(208,306)
(209,305)(210,304)(211,309)(212,308)(213,307)(214,312)(215,311)(216,310)
(217,315)(218,314)(219,313)(220,318)(221,317)(222,316)(223,321)(224,320)
(225,319)(226,324)(227,323)(228,322)(229,327)(230,326)(231,325)(232,330)
(233,329)(234,328)(235,333)(236,332)(237,331)(238,336)(239,335)(240,334)
(241,339)(242,338)(243,337)(244,342)(245,341)(246,340)(247,345)(248,344)
(249,343)(250,348)(251,347)(252,346)(253,351)(254,350)(255,349)(256,354)
(257,353)(258,352)(259,357)(260,356)(261,355)(262,360)(263,359)(264,358)
(265,363)(266,362)(267,361)(268,366)(269,365)(270,364)(271,369)(272,368)
(273,367)(274,372)(275,371)(276,370)(277,375)(278,374)(279,373)(280,378)
(281,377)(282,376)(283,381)(284,380)(285,379)(286,384)(287,383)(288,382);;
s2 := ( 25, 28)( 26, 29)( 27, 30)( 31, 34)( 32, 35)( 33, 36)( 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, 85)( 62, 86)( 63, 87)( 64, 88)( 65, 89)( 66, 90)( 67, 91)( 68, 92)
( 69, 93)( 70, 94)( 71, 95)( 72, 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)
(121,136)(122,137)(123,138)(124,133)(125,134)(126,135)(127,142)(128,143)
(129,144)(130,139)(131,140)(132,141)(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)
(205,229)(206,230)(207,231)(208,232)(209,233)(210,234)(211,235)(212,236)
(213,237)(214,238)(215,239)(216,240)(265,268)(266,269)(267,270)(271,274)
(272,275)(273,276)(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)(361,376)(362,377)(363,378)(364,373)
(365,374)(366,375)(367,382)(368,383)(369,384)(370,379)(371,380)(372,381);;
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, 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);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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)( 13,111)( 14,110)( 15,109)
( 16,114)( 17,113)( 18,112)( 19,117)( 20,116)( 21,115)( 22,120)( 23,119)
( 24,118)( 25,123)( 26,122)( 27,121)( 28,126)( 29,125)( 30,124)( 31,129)
( 32,128)( 33,127)( 34,132)( 35,131)( 36,130)( 37,135)( 38,134)( 39,133)
( 40,138)( 41,137)( 42,136)( 43,141)( 44,140)( 45,139)( 46,144)( 47,143)
( 48,142)( 49,147)( 50,146)( 51,145)( 52,150)( 53,149)( 54,148)( 55,153)
( 56,152)( 57,151)( 58,156)( 59,155)( 60,154)( 61,159)( 62,158)( 63,157)
( 64,162)( 65,161)( 66,160)( 67,165)( 68,164)( 69,163)( 70,168)( 71,167)
( 72,166)( 73,171)( 74,170)( 75,169)( 76,174)( 77,173)( 78,172)( 79,177)
( 80,176)( 81,175)( 82,180)( 83,179)( 84,178)( 85,183)( 86,182)( 87,181)
( 88,186)( 89,185)( 90,184)( 91,189)( 92,188)( 93,187)( 94,192)( 95,191)
( 96,190)(193,291)(194,290)(195,289)(196,294)(197,293)(198,292)(199,297)
(200,296)(201,295)(202,300)(203,299)(204,298)(205,303)(206,302)(207,301)
(208,306)(209,305)(210,304)(211,309)(212,308)(213,307)(214,312)(215,311)
(216,310)(217,315)(218,314)(219,313)(220,318)(221,317)(222,316)(223,321)
(224,320)(225,319)(226,324)(227,323)(228,322)(229,327)(230,326)(231,325)
(232,330)(233,329)(234,328)(235,333)(236,332)(237,331)(238,336)(239,335)
(240,334)(241,339)(242,338)(243,337)(244,342)(245,341)(246,340)(247,345)
(248,344)(249,343)(250,348)(251,347)(252,346)(253,351)(254,350)(255,349)
(256,354)(257,353)(258,352)(259,357)(260,356)(261,355)(262,360)(263,359)
(264,358)(265,363)(266,362)(267,361)(268,366)(269,365)(270,364)(271,369)
(272,368)(273,367)(274,372)(275,371)(276,370)(277,375)(278,374)(279,373)
(280,378)(281,377)(282,376)(283,381)(284,380)(285,379)(286,384)(287,383)
(288,382);
s2 := Sym(384)!( 25, 28)( 26, 29)( 27, 30)( 31, 34)( 32, 35)( 33, 36)( 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, 85)( 62, 86)( 63, 87)( 64, 88)( 65, 89)( 66, 90)( 67, 91)
( 68, 92)( 69, 93)( 70, 94)( 71, 95)( 72, 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)(121,136)(122,137)(123,138)(124,133)(125,134)(126,135)(127,142)
(128,143)(129,144)(130,139)(131,140)(132,141)(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)(205,229)(206,230)(207,231)(208,232)(209,233)(210,234)(211,235)
(212,236)(213,237)(214,238)(215,239)(216,240)(265,268)(266,269)(267,270)
(271,274)(272,275)(273,276)(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)(361,376)(362,377)(363,378)
(364,373)(365,374)(366,375)(367,382)(368,383)(369,384)(370,379)(371,380)
(372,381);
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, 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);
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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