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Polytope of Type {3,6,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,8}*1152
if this polytope has a name.
Group : SmallGroup(1152,156074)
Rank : 4
Schlafli Type : {3,6,8}
Number of vertices, edges, etc : 12, 36, 96, 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 : {3,6,4}*576a
   3-fold quotients : {3,6,8}*384
   4-fold quotients : {3,6,8}*288, {3,6,2}*288
   6-fold quotients : {3,6,4}*192
   8-fold quotients : {3,6,4}*144
   12-fold quotients : {3,2,8}*96, {3,6,2}*96
   16-fold quotients : {3,6,2}*72
   24-fold quotients : {3,2,4}*48, {3,3,2}*48
   48-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 25)( 14, 26)( 15, 28)
( 16, 27)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 29)( 22, 30)( 23, 32)
( 24, 31)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 61)( 50, 62)
( 51, 64)( 52, 63)( 53, 69)( 54, 70)( 55, 72)( 56, 71)( 57, 65)( 58, 66)
( 59, 68)( 60, 67)( 75, 76)( 77, 81)( 78, 82)( 79, 84)( 80, 83)( 85, 97)
( 86, 98)( 87,100)( 88, 99)( 89,105)( 90,106)( 91,108)( 92,107)( 93,101)
( 94,102)( 95,104)( 96,103)(111,112)(113,117)(114,118)(115,120)(116,119)
(121,133)(122,134)(123,136)(124,135)(125,141)(126,142)(127,144)(128,143)
(129,137)(130,138)(131,140)(132,139)(147,148)(149,153)(150,154)(151,156)
(152,155)(157,169)(158,170)(159,172)(160,171)(161,177)(162,178)(163,180)
(164,179)(165,173)(166,174)(167,176)(168,175)(183,184)(185,189)(186,190)
(187,192)(188,191)(193,205)(194,206)(195,208)(196,207)(197,213)(198,214)
(199,216)(200,215)(201,209)(202,210)(203,212)(204,211)(219,220)(221,225)
(222,226)(223,228)(224,227)(229,241)(230,242)(231,244)(232,243)(233,249)
(234,250)(235,252)(236,251)(237,245)(238,246)(239,248)(240,247)(255,256)
(257,261)(258,262)(259,264)(260,263)(265,277)(266,278)(267,280)(268,279)
(269,285)(270,286)(271,288)(272,287)(273,281)(274,282)(275,284)(276,283);;
s1 := (  1, 17)(  2, 20)(  3, 19)(  4, 18)(  5, 13)(  6, 16)(  7, 15)(  8, 14)
(  9, 21)( 10, 24)( 11, 23)( 12, 22)( 25, 29)( 26, 32)( 27, 31)( 28, 30)
( 34, 36)( 37, 53)( 38, 56)( 39, 55)( 40, 54)( 41, 49)( 42, 52)( 43, 51)
( 44, 50)( 45, 57)( 46, 60)( 47, 59)( 48, 58)( 61, 65)( 62, 68)( 63, 67)
( 64, 66)( 70, 72)( 73, 89)( 74, 92)( 75, 91)( 76, 90)( 77, 85)( 78, 88)
( 79, 87)( 80, 86)( 81, 93)( 82, 96)( 83, 95)( 84, 94)( 97,101)( 98,104)
( 99,103)(100,102)(106,108)(109,125)(110,128)(111,127)(112,126)(113,121)
(114,124)(115,123)(116,122)(117,129)(118,132)(119,131)(120,130)(133,137)
(134,140)(135,139)(136,138)(142,144)(145,161)(146,164)(147,163)(148,162)
(149,157)(150,160)(151,159)(152,158)(153,165)(154,168)(155,167)(156,166)
(169,173)(170,176)(171,175)(172,174)(178,180)(181,197)(182,200)(183,199)
(184,198)(185,193)(186,196)(187,195)(188,194)(189,201)(190,204)(191,203)
(192,202)(205,209)(206,212)(207,211)(208,210)(214,216)(217,233)(218,236)
(219,235)(220,234)(221,229)(222,232)(223,231)(224,230)(225,237)(226,240)
(227,239)(228,238)(241,245)(242,248)(243,247)(244,246)(250,252)(253,269)
(254,272)(255,271)(256,270)(257,265)(258,268)(259,267)(260,266)(261,273)
(262,276)(263,275)(264,274)(277,281)(278,284)(279,283)(280,282)(286,288);;
s2 := (  1,  2)(  5,  6)(  9, 10)( 13, 26)( 14, 25)( 15, 27)( 16, 28)( 17, 30)
( 18, 29)( 19, 31)( 20, 32)( 21, 34)( 22, 33)( 23, 35)( 24, 36)( 37, 38)
( 41, 42)( 45, 46)( 49, 62)( 50, 61)( 51, 63)( 52, 64)( 53, 66)( 54, 65)
( 55, 67)( 56, 68)( 57, 70)( 58, 69)( 59, 71)( 60, 72)( 73,110)( 74,109)
( 75,111)( 76,112)( 77,114)( 78,113)( 79,115)( 80,116)( 81,118)( 82,117)
( 83,119)( 84,120)( 85,134)( 86,133)( 87,135)( 88,136)( 89,138)( 90,137)
( 91,139)( 92,140)( 93,142)( 94,141)( 95,143)( 96,144)( 97,122)( 98,121)
( 99,123)(100,124)(101,126)(102,125)(103,127)(104,128)(105,130)(106,129)
(107,131)(108,132)(145,218)(146,217)(147,219)(148,220)(149,222)(150,221)
(151,223)(152,224)(153,226)(154,225)(155,227)(156,228)(157,242)(158,241)
(159,243)(160,244)(161,246)(162,245)(163,247)(164,248)(165,250)(166,249)
(167,251)(168,252)(169,230)(170,229)(171,231)(172,232)(173,234)(174,233)
(175,235)(176,236)(177,238)(178,237)(179,239)(180,240)(181,254)(182,253)
(183,255)(184,256)(185,258)(186,257)(187,259)(188,260)(189,262)(190,261)
(191,263)(192,264)(193,278)(194,277)(195,279)(196,280)(197,282)(198,281)
(199,283)(200,284)(201,286)(202,285)(203,287)(204,288)(205,266)(206,265)
(207,267)(208,268)(209,270)(210,269)(211,271)(212,272)(213,274)(214,273)
(215,275)(216,276);;
s3 := (  1,145)(  2,146)(  3,147)(  4,148)(  5,149)(  6,150)(  7,151)(  8,152)
(  9,153)( 10,154)( 11,155)( 12,156)( 13,157)( 14,158)( 15,159)( 16,160)
( 17,161)( 18,162)( 19,163)( 20,164)( 21,165)( 22,166)( 23,167)( 24,168)
( 25,169)( 26,170)( 27,171)( 28,172)( 29,173)( 30,174)( 31,175)( 32,176)
( 33,177)( 34,178)( 35,179)( 36,180)( 37,181)( 38,182)( 39,183)( 40,184)
( 41,185)( 42,186)( 43,187)( 44,188)( 45,189)( 46,190)( 47,191)( 48,192)
( 49,193)( 50,194)( 51,195)( 52,196)( 53,197)( 54,198)( 55,199)( 56,200)
( 57,201)( 58,202)( 59,203)( 60,204)( 61,205)( 62,206)( 63,207)( 64,208)
( 65,209)( 66,210)( 67,211)( 68,212)( 69,213)( 70,214)( 71,215)( 72,216)
( 73,253)( 74,254)( 75,255)( 76,256)( 77,257)( 78,258)( 79,259)( 80,260)
( 81,261)( 82,262)( 83,263)( 84,264)( 85,265)( 86,266)( 87,267)( 88,268)
( 89,269)( 90,270)( 91,271)( 92,272)( 93,273)( 94,274)( 95,275)( 96,276)
( 97,277)( 98,278)( 99,279)(100,280)(101,281)(102,282)(103,283)(104,284)
(105,285)(106,286)(107,287)(108,288)(109,217)(110,218)(111,219)(112,220)
(113,221)(114,222)(115,223)(116,224)(117,225)(118,226)(119,227)(120,228)
(121,229)(122,230)(123,231)(124,232)(125,233)(126,234)(127,235)(128,236)
(129,237)(130,238)(131,239)(132,240)(133,241)(134,242)(135,243)(136,244)
(137,245)(138,246)(139,247)(140,248)(141,249)(142,250)(143,251)(144,252);;
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, 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*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(288)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 25)( 14, 26)
( 15, 28)( 16, 27)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 29)( 22, 30)
( 23, 32)( 24, 31)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 61)
( 50, 62)( 51, 64)( 52, 63)( 53, 69)( 54, 70)( 55, 72)( 56, 71)( 57, 65)
( 58, 66)( 59, 68)( 60, 67)( 75, 76)( 77, 81)( 78, 82)( 79, 84)( 80, 83)
( 85, 97)( 86, 98)( 87,100)( 88, 99)( 89,105)( 90,106)( 91,108)( 92,107)
( 93,101)( 94,102)( 95,104)( 96,103)(111,112)(113,117)(114,118)(115,120)
(116,119)(121,133)(122,134)(123,136)(124,135)(125,141)(126,142)(127,144)
(128,143)(129,137)(130,138)(131,140)(132,139)(147,148)(149,153)(150,154)
(151,156)(152,155)(157,169)(158,170)(159,172)(160,171)(161,177)(162,178)
(163,180)(164,179)(165,173)(166,174)(167,176)(168,175)(183,184)(185,189)
(186,190)(187,192)(188,191)(193,205)(194,206)(195,208)(196,207)(197,213)
(198,214)(199,216)(200,215)(201,209)(202,210)(203,212)(204,211)(219,220)
(221,225)(222,226)(223,228)(224,227)(229,241)(230,242)(231,244)(232,243)
(233,249)(234,250)(235,252)(236,251)(237,245)(238,246)(239,248)(240,247)
(255,256)(257,261)(258,262)(259,264)(260,263)(265,277)(266,278)(267,280)
(268,279)(269,285)(270,286)(271,288)(272,287)(273,281)(274,282)(275,284)
(276,283);
s1 := Sym(288)!(  1, 17)(  2, 20)(  3, 19)(  4, 18)(  5, 13)(  6, 16)(  7, 15)
(  8, 14)(  9, 21)( 10, 24)( 11, 23)( 12, 22)( 25, 29)( 26, 32)( 27, 31)
( 28, 30)( 34, 36)( 37, 53)( 38, 56)( 39, 55)( 40, 54)( 41, 49)( 42, 52)
( 43, 51)( 44, 50)( 45, 57)( 46, 60)( 47, 59)( 48, 58)( 61, 65)( 62, 68)
( 63, 67)( 64, 66)( 70, 72)( 73, 89)( 74, 92)( 75, 91)( 76, 90)( 77, 85)
( 78, 88)( 79, 87)( 80, 86)( 81, 93)( 82, 96)( 83, 95)( 84, 94)( 97,101)
( 98,104)( 99,103)(100,102)(106,108)(109,125)(110,128)(111,127)(112,126)
(113,121)(114,124)(115,123)(116,122)(117,129)(118,132)(119,131)(120,130)
(133,137)(134,140)(135,139)(136,138)(142,144)(145,161)(146,164)(147,163)
(148,162)(149,157)(150,160)(151,159)(152,158)(153,165)(154,168)(155,167)
(156,166)(169,173)(170,176)(171,175)(172,174)(178,180)(181,197)(182,200)
(183,199)(184,198)(185,193)(186,196)(187,195)(188,194)(189,201)(190,204)
(191,203)(192,202)(205,209)(206,212)(207,211)(208,210)(214,216)(217,233)
(218,236)(219,235)(220,234)(221,229)(222,232)(223,231)(224,230)(225,237)
(226,240)(227,239)(228,238)(241,245)(242,248)(243,247)(244,246)(250,252)
(253,269)(254,272)(255,271)(256,270)(257,265)(258,268)(259,267)(260,266)
(261,273)(262,276)(263,275)(264,274)(277,281)(278,284)(279,283)(280,282)
(286,288);
s2 := Sym(288)!(  1,  2)(  5,  6)(  9, 10)( 13, 26)( 14, 25)( 15, 27)( 16, 28)
( 17, 30)( 18, 29)( 19, 31)( 20, 32)( 21, 34)( 22, 33)( 23, 35)( 24, 36)
( 37, 38)( 41, 42)( 45, 46)( 49, 62)( 50, 61)( 51, 63)( 52, 64)( 53, 66)
( 54, 65)( 55, 67)( 56, 68)( 57, 70)( 58, 69)( 59, 71)( 60, 72)( 73,110)
( 74,109)( 75,111)( 76,112)( 77,114)( 78,113)( 79,115)( 80,116)( 81,118)
( 82,117)( 83,119)( 84,120)( 85,134)( 86,133)( 87,135)( 88,136)( 89,138)
( 90,137)( 91,139)( 92,140)( 93,142)( 94,141)( 95,143)( 96,144)( 97,122)
( 98,121)( 99,123)(100,124)(101,126)(102,125)(103,127)(104,128)(105,130)
(106,129)(107,131)(108,132)(145,218)(146,217)(147,219)(148,220)(149,222)
(150,221)(151,223)(152,224)(153,226)(154,225)(155,227)(156,228)(157,242)
(158,241)(159,243)(160,244)(161,246)(162,245)(163,247)(164,248)(165,250)
(166,249)(167,251)(168,252)(169,230)(170,229)(171,231)(172,232)(173,234)
(174,233)(175,235)(176,236)(177,238)(178,237)(179,239)(180,240)(181,254)
(182,253)(183,255)(184,256)(185,258)(186,257)(187,259)(188,260)(189,262)
(190,261)(191,263)(192,264)(193,278)(194,277)(195,279)(196,280)(197,282)
(198,281)(199,283)(200,284)(201,286)(202,285)(203,287)(204,288)(205,266)
(206,265)(207,267)(208,268)(209,270)(210,269)(211,271)(212,272)(213,274)
(214,273)(215,275)(216,276);
s3 := Sym(288)!(  1,145)(  2,146)(  3,147)(  4,148)(  5,149)(  6,150)(  7,151)
(  8,152)(  9,153)( 10,154)( 11,155)( 12,156)( 13,157)( 14,158)( 15,159)
( 16,160)( 17,161)( 18,162)( 19,163)( 20,164)( 21,165)( 22,166)( 23,167)
( 24,168)( 25,169)( 26,170)( 27,171)( 28,172)( 29,173)( 30,174)( 31,175)
( 32,176)( 33,177)( 34,178)( 35,179)( 36,180)( 37,181)( 38,182)( 39,183)
( 40,184)( 41,185)( 42,186)( 43,187)( 44,188)( 45,189)( 46,190)( 47,191)
( 48,192)( 49,193)( 50,194)( 51,195)( 52,196)( 53,197)( 54,198)( 55,199)
( 56,200)( 57,201)( 58,202)( 59,203)( 60,204)( 61,205)( 62,206)( 63,207)
( 64,208)( 65,209)( 66,210)( 67,211)( 68,212)( 69,213)( 70,214)( 71,215)
( 72,216)( 73,253)( 74,254)( 75,255)( 76,256)( 77,257)( 78,258)( 79,259)
( 80,260)( 81,261)( 82,262)( 83,263)( 84,264)( 85,265)( 86,266)( 87,267)
( 88,268)( 89,269)( 90,270)( 91,271)( 92,272)( 93,273)( 94,274)( 95,275)
( 96,276)( 97,277)( 98,278)( 99,279)(100,280)(101,281)(102,282)(103,283)
(104,284)(105,285)(106,286)(107,287)(108,288)(109,217)(110,218)(111,219)
(112,220)(113,221)(114,222)(115,223)(116,224)(117,225)(118,226)(119,227)
(120,228)(121,229)(122,230)(123,231)(124,232)(125,233)(126,234)(127,235)
(128,236)(129,237)(130,238)(131,239)(132,240)(133,241)(134,242)(135,243)
(136,244)(137,245)(138,246)(139,247)(140,248)(141,249)(142,250)(143,251)
(144,252);
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, 
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*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >; 
 
References : None.
to this polytope