Polytope of Type {8,4,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,4,12}*768a
Also Known As : {{8,4|2},{4,12|2}}. if this polytope has another name.
Group : SmallGroup(768,200564)
Rank : 4
Schlafli Type : {8,4,12}
Number of vertices, edges, etc : 8, 16, 24, 12
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 : {4,4,12}*384, {8,2,12}*384, {8,4,6}*384a
3-fold quotients : {8,4,4}*256a
4-fold quotients : {2,4,12}*192a, {4,2,12}*192, {4,4,6}*192, {8,2,6}*192
6-fold quotients : {4,4,4}*128, {8,4,2}*128a, {8,2,4}*128
8-fold quotients : {8,2,3}*96, {2,2,12}*96, {2,4,6}*96a, {4,2,6}*96
12-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64, {8,2,2}*64
16-fold quotients : {4,2,3}*48, {2,2,6}*48
24-fold quotients : {2,2,4}*32, {2,4,2}*32, {4,2,2}*32
32-fold quotients : {2,2,3}*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 := ( 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);;
s2 := ( 1, 97)( 2, 99)( 3, 98)( 4,100)( 5,102)( 6,101)( 7,103)( 8,105)
( 9,104)( 10,106)( 11,108)( 12,107)( 13,109)( 14,111)( 15,110)( 16,112)
( 17,114)( 18,113)( 19,115)( 20,117)( 21,116)( 22,118)( 23,120)( 24,119)
( 25,121)( 26,123)( 27,122)( 28,124)( 29,126)( 30,125)( 31,127)( 32,129)
( 33,128)( 34,130)( 35,132)( 36,131)( 37,133)( 38,135)( 39,134)( 40,136)
( 41,138)( 42,137)( 43,139)( 44,141)( 45,140)( 46,142)( 47,144)( 48,143)
( 49,145)( 50,147)( 51,146)( 52,148)( 53,150)( 54,149)( 55,151)( 56,153)
( 57,152)( 58,154)( 59,156)( 60,155)( 61,157)( 62,159)( 63,158)( 64,160)
( 65,162)( 66,161)( 67,163)( 68,165)( 69,164)( 70,166)( 71,168)( 72,167)
( 73,169)( 74,171)( 75,170)( 76,172)( 77,174)( 78,173)( 79,175)( 80,177)
( 81,176)( 82,178)( 83,180)( 84,179)( 85,181)( 86,183)( 87,182)( 88,184)
( 89,186)( 90,185)( 91,187)( 92,189)( 93,188)( 94,190)( 95,192)( 96,191)
(193,289)(194,291)(195,290)(196,292)(197,294)(198,293)(199,295)(200,297)
(201,296)(202,298)(203,300)(204,299)(205,301)(206,303)(207,302)(208,304)
(209,306)(210,305)(211,307)(212,309)(213,308)(214,310)(215,312)(216,311)
(217,313)(218,315)(219,314)(220,316)(221,318)(222,317)(223,319)(224,321)
(225,320)(226,322)(227,324)(228,323)(229,325)(230,327)(231,326)(232,328)
(233,330)(234,329)(235,331)(236,333)(237,332)(238,334)(239,336)(240,335)
(241,337)(242,339)(243,338)(244,340)(245,342)(246,341)(247,343)(248,345)
(249,344)(250,346)(251,348)(252,347)(253,349)(254,351)(255,350)(256,352)
(257,354)(258,353)(259,355)(260,357)(261,356)(262,358)(263,360)(264,359)
(265,361)(266,363)(267,362)(268,364)(269,366)(270,365)(271,367)(272,369)
(273,368)(274,370)(275,372)(276,371)(277,373)(278,375)(279,374)(280,376)
(281,378)(282,377)(283,379)(284,381)(285,380)(286,382)(287,384)(288,383);;
s3 := ( 1,243)( 2,242)( 3,241)( 4,246)( 5,245)( 6,244)( 7,249)( 8,248)
( 9,247)( 10,252)( 11,251)( 12,250)( 13,255)( 14,254)( 15,253)( 16,258)
( 17,257)( 18,256)( 19,261)( 20,260)( 21,259)( 22,264)( 23,263)( 24,262)
( 25,267)( 26,266)( 27,265)( 28,270)( 29,269)( 30,268)( 31,273)( 32,272)
( 33,271)( 34,276)( 35,275)( 36,274)( 37,279)( 38,278)( 39,277)( 40,282)
( 41,281)( 42,280)( 43,285)( 44,284)( 45,283)( 46,288)( 47,287)( 48,286)
( 49,195)( 50,194)( 51,193)( 52,198)( 53,197)( 54,196)( 55,201)( 56,200)
( 57,199)( 58,204)( 59,203)( 60,202)( 61,207)( 62,206)( 63,205)( 64,210)
( 65,209)( 66,208)( 67,213)( 68,212)( 69,211)( 70,216)( 71,215)( 72,214)
( 73,219)( 74,218)( 75,217)( 76,222)( 77,221)( 78,220)( 79,225)( 80,224)
( 81,223)( 82,228)( 83,227)( 84,226)( 85,231)( 86,230)( 87,229)( 88,234)
( 89,233)( 90,232)( 91,237)( 92,236)( 93,235)( 94,240)( 95,239)( 96,238)
( 97,345)( 98,344)( 99,343)(100,348)(101,347)(102,346)(103,339)(104,338)
(105,337)(106,342)(107,341)(108,340)(109,357)(110,356)(111,355)(112,360)
(113,359)(114,358)(115,351)(116,350)(117,349)(118,354)(119,353)(120,352)
(121,369)(122,368)(123,367)(124,372)(125,371)(126,370)(127,363)(128,362)
(129,361)(130,366)(131,365)(132,364)(133,381)(134,380)(135,379)(136,384)
(137,383)(138,382)(139,375)(140,374)(141,373)(142,378)(143,377)(144,376)
(145,297)(146,296)(147,295)(148,300)(149,299)(150,298)(151,291)(152,290)
(153,289)(154,294)(155,293)(156,292)(157,309)(158,308)(159,307)(160,312)
(161,311)(162,310)(163,303)(164,302)(165,301)(166,306)(167,305)(168,304)
(169,321)(170,320)(171,319)(172,324)(173,323)(174,322)(175,315)(176,314)
(177,313)(178,318)(179,317)(180,316)(181,333)(182,332)(183,331)(184,336)
(185,335)(186,334)(187,327)(188,326)(189,325)(190,330)(191,329)(192,328);;
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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*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, 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)!( 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);
s2 := Sym(384)!( 1, 97)( 2, 99)( 3, 98)( 4,100)( 5,102)( 6,101)( 7,103)
( 8,105)( 9,104)( 10,106)( 11,108)( 12,107)( 13,109)( 14,111)( 15,110)
( 16,112)( 17,114)( 18,113)( 19,115)( 20,117)( 21,116)( 22,118)( 23,120)
( 24,119)( 25,121)( 26,123)( 27,122)( 28,124)( 29,126)( 30,125)( 31,127)
( 32,129)( 33,128)( 34,130)( 35,132)( 36,131)( 37,133)( 38,135)( 39,134)
( 40,136)( 41,138)( 42,137)( 43,139)( 44,141)( 45,140)( 46,142)( 47,144)
( 48,143)( 49,145)( 50,147)( 51,146)( 52,148)( 53,150)( 54,149)( 55,151)
( 56,153)( 57,152)( 58,154)( 59,156)( 60,155)( 61,157)( 62,159)( 63,158)
( 64,160)( 65,162)( 66,161)( 67,163)( 68,165)( 69,164)( 70,166)( 71,168)
( 72,167)( 73,169)( 74,171)( 75,170)( 76,172)( 77,174)( 78,173)( 79,175)
( 80,177)( 81,176)( 82,178)( 83,180)( 84,179)( 85,181)( 86,183)( 87,182)
( 88,184)( 89,186)( 90,185)( 91,187)( 92,189)( 93,188)( 94,190)( 95,192)
( 96,191)(193,289)(194,291)(195,290)(196,292)(197,294)(198,293)(199,295)
(200,297)(201,296)(202,298)(203,300)(204,299)(205,301)(206,303)(207,302)
(208,304)(209,306)(210,305)(211,307)(212,309)(213,308)(214,310)(215,312)
(216,311)(217,313)(218,315)(219,314)(220,316)(221,318)(222,317)(223,319)
(224,321)(225,320)(226,322)(227,324)(228,323)(229,325)(230,327)(231,326)
(232,328)(233,330)(234,329)(235,331)(236,333)(237,332)(238,334)(239,336)
(240,335)(241,337)(242,339)(243,338)(244,340)(245,342)(246,341)(247,343)
(248,345)(249,344)(250,346)(251,348)(252,347)(253,349)(254,351)(255,350)
(256,352)(257,354)(258,353)(259,355)(260,357)(261,356)(262,358)(263,360)
(264,359)(265,361)(266,363)(267,362)(268,364)(269,366)(270,365)(271,367)
(272,369)(273,368)(274,370)(275,372)(276,371)(277,373)(278,375)(279,374)
(280,376)(281,378)(282,377)(283,379)(284,381)(285,380)(286,382)(287,384)
(288,383);
s3 := 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)( 13,255)( 14,254)( 15,253)
( 16,258)( 17,257)( 18,256)( 19,261)( 20,260)( 21,259)( 22,264)( 23,263)
( 24,262)( 25,267)( 26,266)( 27,265)( 28,270)( 29,269)( 30,268)( 31,273)
( 32,272)( 33,271)( 34,276)( 35,275)( 36,274)( 37,279)( 38,278)( 39,277)
( 40,282)( 41,281)( 42,280)( 43,285)( 44,284)( 45,283)( 46,288)( 47,287)
( 48,286)( 49,195)( 50,194)( 51,193)( 52,198)( 53,197)( 54,196)( 55,201)
( 56,200)( 57,199)( 58,204)( 59,203)( 60,202)( 61,207)( 62,206)( 63,205)
( 64,210)( 65,209)( 66,208)( 67,213)( 68,212)( 69,211)( 70,216)( 71,215)
( 72,214)( 73,219)( 74,218)( 75,217)( 76,222)( 77,221)( 78,220)( 79,225)
( 80,224)( 81,223)( 82,228)( 83,227)( 84,226)( 85,231)( 86,230)( 87,229)
( 88,234)( 89,233)( 90,232)( 91,237)( 92,236)( 93,235)( 94,240)( 95,239)
( 96,238)( 97,345)( 98,344)( 99,343)(100,348)(101,347)(102,346)(103,339)
(104,338)(105,337)(106,342)(107,341)(108,340)(109,357)(110,356)(111,355)
(112,360)(113,359)(114,358)(115,351)(116,350)(117,349)(118,354)(119,353)
(120,352)(121,369)(122,368)(123,367)(124,372)(125,371)(126,370)(127,363)
(128,362)(129,361)(130,366)(131,365)(132,364)(133,381)(134,380)(135,379)
(136,384)(137,383)(138,382)(139,375)(140,374)(141,373)(142,378)(143,377)
(144,376)(145,297)(146,296)(147,295)(148,300)(149,299)(150,298)(151,291)
(152,290)(153,289)(154,294)(155,293)(156,292)(157,309)(158,308)(159,307)
(160,312)(161,311)(162,310)(163,303)(164,302)(165,301)(166,306)(167,305)
(168,304)(169,321)(170,320)(171,319)(172,324)(173,323)(174,322)(175,315)
(176,314)(177,313)(178,318)(179,317)(180,316)(181,333)(182,332)(183,331)
(184,336)(185,335)(186,334)(187,327)(188,326)(189,325)(190,330)(191,329)
(192,328);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope