Polytope of Type {3,4,4,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4,4,3}*1152
Also Known As : {{3,4},{4,4|2},{4,3}}. if this polytope has another name.
Group : SmallGroup(1152,157851)
Rank : 5
Schlafli Type : {3,4,4,3}
Number of vertices, edges, etc : 6, 12, 32, 12, 6
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
4-fold quotients : {3,2,4,3}*288, {3,4,2,3}*288
8-fold quotients : {3,2,4,3}*144, {3,4,2,3}*144
16-fold quotients : {3,2,2,3}*72
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)
( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 49, 97)( 50, 98)( 51,100)( 52, 99)
( 53,101)( 54,102)( 55,104)( 56,103)( 57,105)( 58,106)( 59,108)( 60,107)
( 61,109)( 62,110)( 63,112)( 64,111)( 65,113)( 66,114)( 67,116)( 68,115)
( 69,117)( 70,118)( 71,120)( 72,119)( 73,121)( 74,122)( 75,124)( 76,123)
( 77,125)( 78,126)( 79,128)( 80,127)( 81,129)( 82,130)( 83,132)( 84,131)
( 85,133)( 86,134)( 87,136)( 88,135)( 89,137)( 90,138)( 91,140)( 92,139)
( 93,141)( 94,142)( 95,144)( 96,143)(147,148)(151,152)(155,156)(159,160)
(163,164)(167,168)(171,172)(175,176)(179,180)(183,184)(187,188)(191,192)
(193,241)(194,242)(195,244)(196,243)(197,245)(198,246)(199,248)(200,247)
(201,249)(202,250)(203,252)(204,251)(205,253)(206,254)(207,256)(208,255)
(209,257)(210,258)(211,260)(212,259)(213,261)(214,262)(215,264)(216,263)
(217,265)(218,266)(219,268)(220,267)(221,269)(222,270)(223,272)(224,271)
(225,273)(226,274)(227,276)(228,275)(229,277)(230,278)(231,280)(232,279)
(233,281)(234,282)(235,284)(236,283)(237,285)(238,286)(239,288)(240,287);;
s1 := ( 1, 49)( 2, 52)( 3, 51)( 4, 50)( 5, 53)( 6, 56)( 7, 55)( 8, 54)
( 9, 57)( 10, 60)( 11, 59)( 12, 58)( 13, 61)( 14, 64)( 15, 63)( 16, 62)
( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21, 69)( 22, 72)( 23, 71)( 24, 70)
( 25, 73)( 26, 76)( 27, 75)( 28, 74)( 29, 77)( 30, 80)( 31, 79)( 32, 78)
( 33, 81)( 34, 84)( 35, 83)( 36, 82)( 37, 85)( 38, 88)( 39, 87)( 40, 86)
( 41, 89)( 42, 92)( 43, 91)( 44, 90)( 45, 93)( 46, 96)( 47, 95)( 48, 94)
( 98,100)(102,104)(106,108)(110,112)(114,116)(118,120)(122,124)(126,128)
(130,132)(134,136)(138,140)(142,144)(145,193)(146,196)(147,195)(148,194)
(149,197)(150,200)(151,199)(152,198)(153,201)(154,204)(155,203)(156,202)
(157,205)(158,208)(159,207)(160,206)(161,209)(162,212)(163,211)(164,210)
(165,213)(166,216)(167,215)(168,214)(169,217)(170,220)(171,219)(172,218)
(173,221)(174,224)(175,223)(176,222)(177,225)(178,228)(179,227)(180,226)
(181,229)(182,232)(183,231)(184,230)(185,233)(186,236)(187,235)(188,234)
(189,237)(190,240)(191,239)(192,238)(242,244)(246,248)(250,252)(254,256)
(258,260)(262,264)(266,268)(270,272)(274,276)(278,280)(282,284)(286,288);;
s2 := ( 1,154)( 2,153)( 3,156)( 4,155)( 5,158)( 6,157)( 7,160)( 8,159)
( 9,146)( 10,145)( 11,148)( 12,147)( 13,150)( 14,149)( 15,152)( 16,151)
( 17,170)( 18,169)( 19,172)( 20,171)( 21,174)( 22,173)( 23,176)( 24,175)
( 25,162)( 26,161)( 27,164)( 28,163)( 29,166)( 30,165)( 31,168)( 32,167)
( 33,186)( 34,185)( 35,188)( 36,187)( 37,190)( 38,189)( 39,192)( 40,191)
( 41,178)( 42,177)( 43,180)( 44,179)( 45,182)( 46,181)( 47,184)( 48,183)
( 49,202)( 50,201)( 51,204)( 52,203)( 53,206)( 54,205)( 55,208)( 56,207)
( 57,194)( 58,193)( 59,196)( 60,195)( 61,198)( 62,197)( 63,200)( 64,199)
( 65,218)( 66,217)( 67,220)( 68,219)( 69,222)( 70,221)( 71,224)( 72,223)
( 73,210)( 74,209)( 75,212)( 76,211)( 77,214)( 78,213)( 79,216)( 80,215)
( 81,234)( 82,233)( 83,236)( 84,235)( 85,238)( 86,237)( 87,240)( 88,239)
( 89,226)( 90,225)( 91,228)( 92,227)( 93,230)( 94,229)( 95,232)( 96,231)
( 97,250)( 98,249)( 99,252)(100,251)(101,254)(102,253)(103,256)(104,255)
(105,242)(106,241)(107,244)(108,243)(109,246)(110,245)(111,248)(112,247)
(113,266)(114,265)(115,268)(116,267)(117,270)(118,269)(119,272)(120,271)
(121,258)(122,257)(123,260)(124,259)(125,262)(126,261)(127,264)(128,263)
(129,282)(130,281)(131,284)(132,283)(133,286)(134,285)(135,288)(136,287)
(137,274)(138,273)(139,276)(140,275)(141,278)(142,277)(143,280)(144,279);;
s3 := ( 9, 13)( 10, 14)( 11, 15)( 12, 16)( 17, 33)( 18, 34)( 19, 35)( 20, 36)
( 21, 37)( 22, 38)( 23, 39)( 24, 40)( 25, 45)( 26, 46)( 27, 47)( 28, 48)
( 29, 41)( 30, 42)( 31, 43)( 32, 44)( 57, 61)( 58, 62)( 59, 63)( 60, 64)
( 65, 81)( 66, 82)( 67, 83)( 68, 84)( 69, 85)( 70, 86)( 71, 87)( 72, 88)
( 73, 93)( 74, 94)( 75, 95)( 76, 96)( 77, 89)( 78, 90)( 79, 91)( 80, 92)
(105,109)(106,110)(107,111)(108,112)(113,129)(114,130)(115,131)(116,132)
(117,133)(118,134)(119,135)(120,136)(121,141)(122,142)(123,143)(124,144)
(125,137)(126,138)(127,139)(128,140)(153,157)(154,158)(155,159)(156,160)
(161,177)(162,178)(163,179)(164,180)(165,181)(166,182)(167,183)(168,184)
(169,189)(170,190)(171,191)(172,192)(173,185)(174,186)(175,187)(176,188)
(201,205)(202,206)(203,207)(204,208)(209,225)(210,226)(211,227)(212,228)
(213,229)(214,230)(215,231)(216,232)(217,237)(218,238)(219,239)(220,240)
(221,233)(222,234)(223,235)(224,236)(249,253)(250,254)(251,255)(252,256)
(257,273)(258,274)(259,275)(260,276)(261,277)(262,278)(263,279)(264,280)
(265,285)(266,286)(267,287)(268,288)(269,281)(270,282)(271,283)(272,284);;
s4 := ( 1, 17)( 2, 18)( 3, 19)( 4, 20)( 5, 29)( 6, 30)( 7, 31)( 8, 32)
( 9, 25)( 10, 26)( 11, 27)( 12, 28)( 13, 21)( 14, 22)( 15, 23)( 16, 24)
( 37, 45)( 38, 46)( 39, 47)( 40, 48)( 49, 65)( 50, 66)( 51, 67)( 52, 68)
( 53, 77)( 54, 78)( 55, 79)( 56, 80)( 57, 73)( 58, 74)( 59, 75)( 60, 76)
( 61, 69)( 62, 70)( 63, 71)( 64, 72)( 85, 93)( 86, 94)( 87, 95)( 88, 96)
( 97,113)( 98,114)( 99,115)(100,116)(101,125)(102,126)(103,127)(104,128)
(105,121)(106,122)(107,123)(108,124)(109,117)(110,118)(111,119)(112,120)
(133,141)(134,142)(135,143)(136,144)(145,161)(146,162)(147,163)(148,164)
(149,173)(150,174)(151,175)(152,176)(153,169)(154,170)(155,171)(156,172)
(157,165)(158,166)(159,167)(160,168)(181,189)(182,190)(183,191)(184,192)
(193,209)(194,210)(195,211)(196,212)(197,221)(198,222)(199,223)(200,224)
(201,217)(202,218)(203,219)(204,220)(205,213)(206,214)(207,215)(208,216)
(229,237)(230,238)(231,239)(232,240)(241,257)(242,258)(243,259)(244,260)
(245,269)(246,270)(247,271)(248,272)(249,265)(250,266)(251,267)(252,268)
(253,261)(254,262)(255,263)(256,264)(277,285)(278,286)(279,287)(280,288);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1,
s3*s4*s3*s4*s3*s4, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(288)!( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)
( 31, 32)( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 49, 97)( 50, 98)( 51,100)
( 52, 99)( 53,101)( 54,102)( 55,104)( 56,103)( 57,105)( 58,106)( 59,108)
( 60,107)( 61,109)( 62,110)( 63,112)( 64,111)( 65,113)( 66,114)( 67,116)
( 68,115)( 69,117)( 70,118)( 71,120)( 72,119)( 73,121)( 74,122)( 75,124)
( 76,123)( 77,125)( 78,126)( 79,128)( 80,127)( 81,129)( 82,130)( 83,132)
( 84,131)( 85,133)( 86,134)( 87,136)( 88,135)( 89,137)( 90,138)( 91,140)
( 92,139)( 93,141)( 94,142)( 95,144)( 96,143)(147,148)(151,152)(155,156)
(159,160)(163,164)(167,168)(171,172)(175,176)(179,180)(183,184)(187,188)
(191,192)(193,241)(194,242)(195,244)(196,243)(197,245)(198,246)(199,248)
(200,247)(201,249)(202,250)(203,252)(204,251)(205,253)(206,254)(207,256)
(208,255)(209,257)(210,258)(211,260)(212,259)(213,261)(214,262)(215,264)
(216,263)(217,265)(218,266)(219,268)(220,267)(221,269)(222,270)(223,272)
(224,271)(225,273)(226,274)(227,276)(228,275)(229,277)(230,278)(231,280)
(232,279)(233,281)(234,282)(235,284)(236,283)(237,285)(238,286)(239,288)
(240,287);
s1 := Sym(288)!( 1, 49)( 2, 52)( 3, 51)( 4, 50)( 5, 53)( 6, 56)( 7, 55)
( 8, 54)( 9, 57)( 10, 60)( 11, 59)( 12, 58)( 13, 61)( 14, 64)( 15, 63)
( 16, 62)( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21, 69)( 22, 72)( 23, 71)
( 24, 70)( 25, 73)( 26, 76)( 27, 75)( 28, 74)( 29, 77)( 30, 80)( 31, 79)
( 32, 78)( 33, 81)( 34, 84)( 35, 83)( 36, 82)( 37, 85)( 38, 88)( 39, 87)
( 40, 86)( 41, 89)( 42, 92)( 43, 91)( 44, 90)( 45, 93)( 46, 96)( 47, 95)
( 48, 94)( 98,100)(102,104)(106,108)(110,112)(114,116)(118,120)(122,124)
(126,128)(130,132)(134,136)(138,140)(142,144)(145,193)(146,196)(147,195)
(148,194)(149,197)(150,200)(151,199)(152,198)(153,201)(154,204)(155,203)
(156,202)(157,205)(158,208)(159,207)(160,206)(161,209)(162,212)(163,211)
(164,210)(165,213)(166,216)(167,215)(168,214)(169,217)(170,220)(171,219)
(172,218)(173,221)(174,224)(175,223)(176,222)(177,225)(178,228)(179,227)
(180,226)(181,229)(182,232)(183,231)(184,230)(185,233)(186,236)(187,235)
(188,234)(189,237)(190,240)(191,239)(192,238)(242,244)(246,248)(250,252)
(254,256)(258,260)(262,264)(266,268)(270,272)(274,276)(278,280)(282,284)
(286,288);
s2 := Sym(288)!( 1,154)( 2,153)( 3,156)( 4,155)( 5,158)( 6,157)( 7,160)
( 8,159)( 9,146)( 10,145)( 11,148)( 12,147)( 13,150)( 14,149)( 15,152)
( 16,151)( 17,170)( 18,169)( 19,172)( 20,171)( 21,174)( 22,173)( 23,176)
( 24,175)( 25,162)( 26,161)( 27,164)( 28,163)( 29,166)( 30,165)( 31,168)
( 32,167)( 33,186)( 34,185)( 35,188)( 36,187)( 37,190)( 38,189)( 39,192)
( 40,191)( 41,178)( 42,177)( 43,180)( 44,179)( 45,182)( 46,181)( 47,184)
( 48,183)( 49,202)( 50,201)( 51,204)( 52,203)( 53,206)( 54,205)( 55,208)
( 56,207)( 57,194)( 58,193)( 59,196)( 60,195)( 61,198)( 62,197)( 63,200)
( 64,199)( 65,218)( 66,217)( 67,220)( 68,219)( 69,222)( 70,221)( 71,224)
( 72,223)( 73,210)( 74,209)( 75,212)( 76,211)( 77,214)( 78,213)( 79,216)
( 80,215)( 81,234)( 82,233)( 83,236)( 84,235)( 85,238)( 86,237)( 87,240)
( 88,239)( 89,226)( 90,225)( 91,228)( 92,227)( 93,230)( 94,229)( 95,232)
( 96,231)( 97,250)( 98,249)( 99,252)(100,251)(101,254)(102,253)(103,256)
(104,255)(105,242)(106,241)(107,244)(108,243)(109,246)(110,245)(111,248)
(112,247)(113,266)(114,265)(115,268)(116,267)(117,270)(118,269)(119,272)
(120,271)(121,258)(122,257)(123,260)(124,259)(125,262)(126,261)(127,264)
(128,263)(129,282)(130,281)(131,284)(132,283)(133,286)(134,285)(135,288)
(136,287)(137,274)(138,273)(139,276)(140,275)(141,278)(142,277)(143,280)
(144,279);
s3 := Sym(288)!( 9, 13)( 10, 14)( 11, 15)( 12, 16)( 17, 33)( 18, 34)( 19, 35)
( 20, 36)( 21, 37)( 22, 38)( 23, 39)( 24, 40)( 25, 45)( 26, 46)( 27, 47)
( 28, 48)( 29, 41)( 30, 42)( 31, 43)( 32, 44)( 57, 61)( 58, 62)( 59, 63)
( 60, 64)( 65, 81)( 66, 82)( 67, 83)( 68, 84)( 69, 85)( 70, 86)( 71, 87)
( 72, 88)( 73, 93)( 74, 94)( 75, 95)( 76, 96)( 77, 89)( 78, 90)( 79, 91)
( 80, 92)(105,109)(106,110)(107,111)(108,112)(113,129)(114,130)(115,131)
(116,132)(117,133)(118,134)(119,135)(120,136)(121,141)(122,142)(123,143)
(124,144)(125,137)(126,138)(127,139)(128,140)(153,157)(154,158)(155,159)
(156,160)(161,177)(162,178)(163,179)(164,180)(165,181)(166,182)(167,183)
(168,184)(169,189)(170,190)(171,191)(172,192)(173,185)(174,186)(175,187)
(176,188)(201,205)(202,206)(203,207)(204,208)(209,225)(210,226)(211,227)
(212,228)(213,229)(214,230)(215,231)(216,232)(217,237)(218,238)(219,239)
(220,240)(221,233)(222,234)(223,235)(224,236)(249,253)(250,254)(251,255)
(252,256)(257,273)(258,274)(259,275)(260,276)(261,277)(262,278)(263,279)
(264,280)(265,285)(266,286)(267,287)(268,288)(269,281)(270,282)(271,283)
(272,284);
s4 := Sym(288)!( 1, 17)( 2, 18)( 3, 19)( 4, 20)( 5, 29)( 6, 30)( 7, 31)
( 8, 32)( 9, 25)( 10, 26)( 11, 27)( 12, 28)( 13, 21)( 14, 22)( 15, 23)
( 16, 24)( 37, 45)( 38, 46)( 39, 47)( 40, 48)( 49, 65)( 50, 66)( 51, 67)
( 52, 68)( 53, 77)( 54, 78)( 55, 79)( 56, 80)( 57, 73)( 58, 74)( 59, 75)
( 60, 76)( 61, 69)( 62, 70)( 63, 71)( 64, 72)( 85, 93)( 86, 94)( 87, 95)
( 88, 96)( 97,113)( 98,114)( 99,115)(100,116)(101,125)(102,126)(103,127)
(104,128)(105,121)(106,122)(107,123)(108,124)(109,117)(110,118)(111,119)
(112,120)(133,141)(134,142)(135,143)(136,144)(145,161)(146,162)(147,163)
(148,164)(149,173)(150,174)(151,175)(152,176)(153,169)(154,170)(155,171)
(156,172)(157,165)(158,166)(159,167)(160,168)(181,189)(182,190)(183,191)
(184,192)(193,209)(194,210)(195,211)(196,212)(197,221)(198,222)(199,223)
(200,224)(201,217)(202,218)(203,219)(204,220)(205,213)(206,214)(207,215)
(208,216)(229,237)(230,238)(231,239)(232,240)(241,257)(242,258)(243,259)
(244,260)(245,269)(246,270)(247,271)(248,272)(249,265)(250,266)(251,267)
(252,268)(253,261)(254,262)(255,263)(256,264)(277,285)(278,286)(279,287)
(280,288);
poly := sub<Sym(288)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, s3*s4*s3*s4*s3*s4,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope