Polytope of Type {4,8,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,8,12}*768d
Also Known As : {{4,8|2},{8,12|2}}. if this polytope has another name.
Group : SmallGroup(768,201309)
Rank : 4
Schlafli Type : {4,8,12}
Number of vertices, edges, etc : 4, 16, 48, 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, {2,8,12}*384a, {4,8,6}*384a
3-fold quotients : {4,8,4}*256d
4-fold quotients : {2,4,12}*192a, {4,2,12}*192, {4,4,6}*192, {2,8,6}*192
6-fold quotients : {4,4,4}*128, {2,8,4}*128a, {4,8,2}*128a
8-fold quotients : {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, {2,8,2}*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, 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);;
s1 := ( 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);;
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)( 13,112)( 14,114)( 15,113)( 16,109)
( 17,111)( 18,110)( 19,118)( 20,120)( 21,119)( 22,115)( 23,117)( 24,116)
( 25,121)( 26,123)( 27,122)( 28,124)( 29,126)( 30,125)( 31,127)( 32,129)
( 33,128)( 34,130)( 35,132)( 36,131)( 37,136)( 38,138)( 39,137)( 40,133)
( 41,135)( 42,134)( 43,142)( 44,144)( 45,143)( 46,139)( 47,141)( 48,140)
( 49,145)( 50,147)( 51,146)( 52,148)( 53,150)( 54,149)( 55,151)( 56,153)
( 57,152)( 58,154)( 59,156)( 60,155)( 61,160)( 62,162)( 63,161)( 64,157)
( 65,159)( 66,158)( 67,166)( 68,168)( 69,167)( 70,163)( 71,165)( 72,164)
( 73,169)( 74,171)( 75,170)( 76,172)( 77,174)( 78,173)( 79,175)( 80,177)
( 81,176)( 82,178)( 83,180)( 84,179)( 85,184)( 86,186)( 87,185)( 88,181)
( 89,183)( 90,182)( 91,190)( 92,192)( 93,191)( 94,187)( 95,189)( 96,188)
(193,289)(194,291)(195,290)(196,292)(197,294)(198,293)(199,295)(200,297)
(201,296)(202,298)(203,300)(204,299)(205,304)(206,306)(207,305)(208,301)
(209,303)(210,302)(211,310)(212,312)(213,311)(214,307)(215,309)(216,308)
(217,313)(218,315)(219,314)(220,316)(221,318)(222,317)(223,319)(224,321)
(225,320)(226,322)(227,324)(228,323)(229,328)(230,330)(231,329)(232,325)
(233,327)(234,326)(235,334)(236,336)(237,335)(238,331)(239,333)(240,332)
(241,337)(242,339)(243,338)(244,340)(245,342)(246,341)(247,343)(248,345)
(249,344)(250,346)(251,348)(252,347)(253,352)(254,354)(255,353)(256,349)
(257,351)(258,350)(259,358)(260,360)(261,359)(262,355)(263,357)(264,356)
(265,361)(266,363)(267,362)(268,364)(269,366)(270,365)(271,367)(272,369)
(273,368)(274,370)(275,372)(276,371)(277,376)(278,378)(279,377)(280,373)
(281,375)(282,374)(283,382)(284,384)(285,383)(286,379)(287,381)(288,380);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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, 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);
s1 := 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);
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)( 13,112)( 14,114)( 15,113)
( 16,109)( 17,111)( 18,110)( 19,118)( 20,120)( 21,119)( 22,115)( 23,117)
( 24,116)( 25,121)( 26,123)( 27,122)( 28,124)( 29,126)( 30,125)( 31,127)
( 32,129)( 33,128)( 34,130)( 35,132)( 36,131)( 37,136)( 38,138)( 39,137)
( 40,133)( 41,135)( 42,134)( 43,142)( 44,144)( 45,143)( 46,139)( 47,141)
( 48,140)( 49,145)( 50,147)( 51,146)( 52,148)( 53,150)( 54,149)( 55,151)
( 56,153)( 57,152)( 58,154)( 59,156)( 60,155)( 61,160)( 62,162)( 63,161)
( 64,157)( 65,159)( 66,158)( 67,166)( 68,168)( 69,167)( 70,163)( 71,165)
( 72,164)( 73,169)( 74,171)( 75,170)( 76,172)( 77,174)( 78,173)( 79,175)
( 80,177)( 81,176)( 82,178)( 83,180)( 84,179)( 85,184)( 86,186)( 87,185)
( 88,181)( 89,183)( 90,182)( 91,190)( 92,192)( 93,191)( 94,187)( 95,189)
( 96,188)(193,289)(194,291)(195,290)(196,292)(197,294)(198,293)(199,295)
(200,297)(201,296)(202,298)(203,300)(204,299)(205,304)(206,306)(207,305)
(208,301)(209,303)(210,302)(211,310)(212,312)(213,311)(214,307)(215,309)
(216,308)(217,313)(218,315)(219,314)(220,316)(221,318)(222,317)(223,319)
(224,321)(225,320)(226,322)(227,324)(228,323)(229,328)(230,330)(231,329)
(232,325)(233,327)(234,326)(235,334)(236,336)(237,335)(238,331)(239,333)
(240,332)(241,337)(242,339)(243,338)(244,340)(245,342)(246,341)(247,343)
(248,345)(249,344)(250,346)(251,348)(252,347)(253,352)(254,354)(255,353)
(256,349)(257,351)(258,350)(259,358)(260,360)(261,359)(262,355)(263,357)
(264,356)(265,361)(266,363)(267,362)(268,364)(269,366)(270,365)(271,367)
(272,369)(273,368)(274,370)(275,372)(276,371)(277,376)(278,378)(279,377)
(280,373)(281,375)(282,374)(283,382)(284,384)(285,383)(286,379)(287,381)
(288,380);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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