Polytope of Type {3,8,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,8,12}*1152
if this polytope has a name.
Group : SmallGroup(1152,155788)
Rank : 4
Schlafli Type : {3,8,12}
Number of vertices, edges, etc : 6, 24, 96, 12
Order of s0s1s2s3 : 12
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 : {3,4,12}*576, {3,8,6}*576
3-fold quotients : {3,8,4}*384
4-fold quotients : {3,4,6}*288
6-fold quotients : {3,4,4}*192b, {3,8,2}*192
8-fold quotients : {3,2,12}*144
12-fold quotients : {3,4,2}*96
16-fold quotients : {3,2,6}*72
24-fold quotients : {3,2,4}*48, {3,4,2}*48
32-fold quotients : {3,2,3}*36
48-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 7)( 6, 8)( 11, 12)( 13, 15)( 14, 16)( 19, 20)( 21, 23)
( 22, 24)( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 55)( 30, 56)( 31, 53)
( 32, 54)( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 37, 63)( 38, 64)( 39, 61)
( 40, 62)( 41, 65)( 42, 66)( 43, 68)( 44, 67)( 45, 71)( 46, 72)( 47, 69)
( 48, 70)( 75, 76)( 77, 79)( 78, 80)( 83, 84)( 85, 87)( 86, 88)( 91, 92)
( 93, 95)( 94, 96)( 97,121)( 98,122)( 99,124)(100,123)(101,127)(102,128)
(103,125)(104,126)(105,129)(106,130)(107,132)(108,131)(109,135)(110,136)
(111,133)(112,134)(113,137)(114,138)(115,140)(116,139)(117,143)(118,144)
(119,141)(120,142)(145,146)(149,152)(150,151)(153,154)(157,160)(158,159)
(161,162)(165,168)(166,167)(169,194)(170,193)(171,195)(172,196)(173,200)
(174,199)(175,198)(176,197)(177,202)(178,201)(179,203)(180,204)(181,208)
(182,207)(183,206)(184,205)(185,210)(186,209)(187,211)(188,212)(189,216)
(190,215)(191,214)(192,213)(217,218)(221,224)(222,223)(225,226)(229,232)
(230,231)(233,234)(237,240)(238,239)(241,266)(242,265)(243,267)(244,268)
(245,272)(246,271)(247,270)(248,269)(249,274)(250,273)(251,275)(252,276)
(253,280)(254,279)(255,278)(256,277)(257,282)(258,281)(259,283)(260,284)
(261,288)(262,287)(263,286)(264,285);;
s1 := ( 1, 25)( 2, 26)( 3, 31)( 4, 32)( 5, 30)( 6, 29)( 7, 27)( 8, 28)
( 9, 33)( 10, 34)( 11, 39)( 12, 40)( 13, 38)( 14, 37)( 15, 35)( 16, 36)
( 17, 41)( 18, 42)( 19, 47)( 20, 48)( 21, 46)( 22, 45)( 23, 43)( 24, 44)
( 51, 55)( 52, 56)( 53, 54)( 59, 63)( 60, 64)( 61, 62)( 67, 71)( 68, 72)
( 69, 70)( 73, 97)( 74, 98)( 75,103)( 76,104)( 77,102)( 78,101)( 79, 99)
( 80,100)( 81,105)( 82,106)( 83,111)( 84,112)( 85,110)( 86,109)( 87,107)
( 88,108)( 89,113)( 90,114)( 91,119)( 92,120)( 93,118)( 94,117)( 95,115)
( 96,116)(123,127)(124,128)(125,126)(131,135)(132,136)(133,134)(139,143)
(140,144)(141,142)(145,170)(146,169)(147,176)(148,175)(149,173)(150,174)
(151,172)(152,171)(153,178)(154,177)(155,184)(156,183)(157,181)(158,182)
(159,180)(160,179)(161,186)(162,185)(163,192)(164,191)(165,189)(166,190)
(167,188)(168,187)(193,194)(195,200)(196,199)(201,202)(203,208)(204,207)
(209,210)(211,216)(212,215)(217,242)(218,241)(219,248)(220,247)(221,245)
(222,246)(223,244)(224,243)(225,250)(226,249)(227,256)(228,255)(229,253)
(230,254)(231,252)(232,251)(233,258)(234,257)(235,264)(236,263)(237,261)
(238,262)(239,260)(240,259)(265,266)(267,272)(268,271)(273,274)(275,280)
(276,279)(281,282)(283,288)(284,287);;
s2 := ( 1,147)( 2,148)( 3,146)( 4,145)( 5,151)( 6,152)( 7,150)( 8,149)
( 9,163)( 10,164)( 11,162)( 12,161)( 13,167)( 14,168)( 15,166)( 16,165)
( 17,155)( 18,156)( 19,154)( 20,153)( 21,159)( 22,160)( 23,158)( 24,157)
( 25,171)( 26,172)( 27,170)( 28,169)( 29,175)( 30,176)( 31,174)( 32,173)
( 33,187)( 34,188)( 35,186)( 36,185)( 37,191)( 38,192)( 39,190)( 40,189)
( 41,179)( 42,180)( 43,178)( 44,177)( 45,183)( 46,184)( 47,182)( 48,181)
( 49,195)( 50,196)( 51,194)( 52,193)( 53,199)( 54,200)( 55,198)( 56,197)
( 57,211)( 58,212)( 59,210)( 60,209)( 61,215)( 62,216)( 63,214)( 64,213)
( 65,203)( 66,204)( 67,202)( 68,201)( 69,207)( 70,208)( 71,206)( 72,205)
( 73,219)( 74,220)( 75,218)( 76,217)( 77,223)( 78,224)( 79,222)( 80,221)
( 81,235)( 82,236)( 83,234)( 84,233)( 85,239)( 86,240)( 87,238)( 88,237)
( 89,227)( 90,228)( 91,226)( 92,225)( 93,231)( 94,232)( 95,230)( 96,229)
( 97,243)( 98,244)( 99,242)(100,241)(101,247)(102,248)(103,246)(104,245)
(105,259)(106,260)(107,258)(108,257)(109,263)(110,264)(111,262)(112,261)
(113,251)(114,252)(115,250)(116,249)(117,255)(118,256)(119,254)(120,253)
(121,267)(122,268)(123,266)(124,265)(125,271)(126,272)(127,270)(128,269)
(129,283)(130,284)(131,282)(132,281)(133,287)(134,288)(135,286)(136,285)
(137,275)(138,276)(139,274)(140,273)(141,279)(142,280)(143,278)(144,277);;
s3 := ( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 13)( 6, 14)( 7, 15)( 8, 16)
( 25, 33)( 26, 34)( 27, 35)( 28, 36)( 29, 37)( 30, 38)( 31, 39)( 32, 40)
( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)( 56, 64)
( 73, 81)( 74, 82)( 75, 83)( 76, 84)( 77, 85)( 78, 86)( 79, 87)( 80, 88)
( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)(104,112)
(121,129)(122,130)(123,131)(124,132)(125,133)(126,134)(127,135)(128,136)
(145,225)(146,226)(147,227)(148,228)(149,229)(150,230)(151,231)(152,232)
(153,217)(154,218)(155,219)(156,220)(157,221)(158,222)(159,223)(160,224)
(161,233)(162,234)(163,235)(164,236)(165,237)(166,238)(167,239)(168,240)
(169,249)(170,250)(171,251)(172,252)(173,253)(174,254)(175,255)(176,256)
(177,241)(178,242)(179,243)(180,244)(181,245)(182,246)(183,247)(184,248)
(185,257)(186,258)(187,259)(188,260)(189,261)(190,262)(191,263)(192,264)
(193,273)(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)(200,280)
(201,265)(202,266)(203,267)(204,268)(205,269)(206,270)(207,271)(208,272)
(209,281)(210,282)(211,283)(212,284)(213,285)(214,286)(215,287)(216,288);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*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(288)!( 3, 4)( 5, 7)( 6, 8)( 11, 12)( 13, 15)( 14, 16)( 19, 20)
( 21, 23)( 22, 24)( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 55)( 30, 56)
( 31, 53)( 32, 54)( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 37, 63)( 38, 64)
( 39, 61)( 40, 62)( 41, 65)( 42, 66)( 43, 68)( 44, 67)( 45, 71)( 46, 72)
( 47, 69)( 48, 70)( 75, 76)( 77, 79)( 78, 80)( 83, 84)( 85, 87)( 86, 88)
( 91, 92)( 93, 95)( 94, 96)( 97,121)( 98,122)( 99,124)(100,123)(101,127)
(102,128)(103,125)(104,126)(105,129)(106,130)(107,132)(108,131)(109,135)
(110,136)(111,133)(112,134)(113,137)(114,138)(115,140)(116,139)(117,143)
(118,144)(119,141)(120,142)(145,146)(149,152)(150,151)(153,154)(157,160)
(158,159)(161,162)(165,168)(166,167)(169,194)(170,193)(171,195)(172,196)
(173,200)(174,199)(175,198)(176,197)(177,202)(178,201)(179,203)(180,204)
(181,208)(182,207)(183,206)(184,205)(185,210)(186,209)(187,211)(188,212)
(189,216)(190,215)(191,214)(192,213)(217,218)(221,224)(222,223)(225,226)
(229,232)(230,231)(233,234)(237,240)(238,239)(241,266)(242,265)(243,267)
(244,268)(245,272)(246,271)(247,270)(248,269)(249,274)(250,273)(251,275)
(252,276)(253,280)(254,279)(255,278)(256,277)(257,282)(258,281)(259,283)
(260,284)(261,288)(262,287)(263,286)(264,285);
s1 := Sym(288)!( 1, 25)( 2, 26)( 3, 31)( 4, 32)( 5, 30)( 6, 29)( 7, 27)
( 8, 28)( 9, 33)( 10, 34)( 11, 39)( 12, 40)( 13, 38)( 14, 37)( 15, 35)
( 16, 36)( 17, 41)( 18, 42)( 19, 47)( 20, 48)( 21, 46)( 22, 45)( 23, 43)
( 24, 44)( 51, 55)( 52, 56)( 53, 54)( 59, 63)( 60, 64)( 61, 62)( 67, 71)
( 68, 72)( 69, 70)( 73, 97)( 74, 98)( 75,103)( 76,104)( 77,102)( 78,101)
( 79, 99)( 80,100)( 81,105)( 82,106)( 83,111)( 84,112)( 85,110)( 86,109)
( 87,107)( 88,108)( 89,113)( 90,114)( 91,119)( 92,120)( 93,118)( 94,117)
( 95,115)( 96,116)(123,127)(124,128)(125,126)(131,135)(132,136)(133,134)
(139,143)(140,144)(141,142)(145,170)(146,169)(147,176)(148,175)(149,173)
(150,174)(151,172)(152,171)(153,178)(154,177)(155,184)(156,183)(157,181)
(158,182)(159,180)(160,179)(161,186)(162,185)(163,192)(164,191)(165,189)
(166,190)(167,188)(168,187)(193,194)(195,200)(196,199)(201,202)(203,208)
(204,207)(209,210)(211,216)(212,215)(217,242)(218,241)(219,248)(220,247)
(221,245)(222,246)(223,244)(224,243)(225,250)(226,249)(227,256)(228,255)
(229,253)(230,254)(231,252)(232,251)(233,258)(234,257)(235,264)(236,263)
(237,261)(238,262)(239,260)(240,259)(265,266)(267,272)(268,271)(273,274)
(275,280)(276,279)(281,282)(283,288)(284,287);
s2 := Sym(288)!( 1,147)( 2,148)( 3,146)( 4,145)( 5,151)( 6,152)( 7,150)
( 8,149)( 9,163)( 10,164)( 11,162)( 12,161)( 13,167)( 14,168)( 15,166)
( 16,165)( 17,155)( 18,156)( 19,154)( 20,153)( 21,159)( 22,160)( 23,158)
( 24,157)( 25,171)( 26,172)( 27,170)( 28,169)( 29,175)( 30,176)( 31,174)
( 32,173)( 33,187)( 34,188)( 35,186)( 36,185)( 37,191)( 38,192)( 39,190)
( 40,189)( 41,179)( 42,180)( 43,178)( 44,177)( 45,183)( 46,184)( 47,182)
( 48,181)( 49,195)( 50,196)( 51,194)( 52,193)( 53,199)( 54,200)( 55,198)
( 56,197)( 57,211)( 58,212)( 59,210)( 60,209)( 61,215)( 62,216)( 63,214)
( 64,213)( 65,203)( 66,204)( 67,202)( 68,201)( 69,207)( 70,208)( 71,206)
( 72,205)( 73,219)( 74,220)( 75,218)( 76,217)( 77,223)( 78,224)( 79,222)
( 80,221)( 81,235)( 82,236)( 83,234)( 84,233)( 85,239)( 86,240)( 87,238)
( 88,237)( 89,227)( 90,228)( 91,226)( 92,225)( 93,231)( 94,232)( 95,230)
( 96,229)( 97,243)( 98,244)( 99,242)(100,241)(101,247)(102,248)(103,246)
(104,245)(105,259)(106,260)(107,258)(108,257)(109,263)(110,264)(111,262)
(112,261)(113,251)(114,252)(115,250)(116,249)(117,255)(118,256)(119,254)
(120,253)(121,267)(122,268)(123,266)(124,265)(125,271)(126,272)(127,270)
(128,269)(129,283)(130,284)(131,282)(132,281)(133,287)(134,288)(135,286)
(136,285)(137,275)(138,276)(139,274)(140,273)(141,279)(142,280)(143,278)
(144,277);
s3 := Sym(288)!( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 13)( 6, 14)( 7, 15)
( 8, 16)( 25, 33)( 26, 34)( 27, 35)( 28, 36)( 29, 37)( 30, 38)( 31, 39)
( 32, 40)( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)
( 56, 64)( 73, 81)( 74, 82)( 75, 83)( 76, 84)( 77, 85)( 78, 86)( 79, 87)
( 80, 88)( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)
(104,112)(121,129)(122,130)(123,131)(124,132)(125,133)(126,134)(127,135)
(128,136)(145,225)(146,226)(147,227)(148,228)(149,229)(150,230)(151,231)
(152,232)(153,217)(154,218)(155,219)(156,220)(157,221)(158,222)(159,223)
(160,224)(161,233)(162,234)(163,235)(164,236)(165,237)(166,238)(167,239)
(168,240)(169,249)(170,250)(171,251)(172,252)(173,253)(174,254)(175,255)
(176,256)(177,241)(178,242)(179,243)(180,244)(181,245)(182,246)(183,247)
(184,248)(185,257)(186,258)(187,259)(188,260)(189,261)(190,262)(191,263)
(192,264)(193,273)(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)
(200,280)(201,265)(202,266)(203,267)(204,268)(205,269)(206,270)(207,271)
(208,272)(209,281)(210,282)(211,283)(212,284)(213,285)(214,286)(215,287)
(216,288);
poly := sub<Sym(288)|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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*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