Polytope of Type {6,3,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3,4}*1152b
if this polytope has a name.
Group : SmallGroup(1152,157851)
Rank : 4
Schlafli Type : {6,3,4}
Number of vertices, edges, etc : 24, 72, 48, 8
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Locally Toroidal
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 : {6,3,4}*576
4-fold quotients : {6,3,4}*288, {6,3,2}*288
8-fold quotients : {6,3,4}*144
12-fold quotients : {2,3,4}*96, {6,3,2}*96
16-fold quotients : {6,3,2}*72
24-fold quotients : {2,3,4}*48, {3,3,2}*48
48-fold quotients : {2,3,2}*24
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, 61)( 10, 64)( 11, 63)( 12, 62)( 13, 57)( 14, 60)( 15, 59)( 16, 58)
( 17, 81)( 18, 84)( 19, 83)( 20, 82)( 21, 85)( 22, 88)( 23, 87)( 24, 86)
( 25, 93)( 26, 96)( 27, 95)( 28, 94)( 29, 89)( 30, 92)( 31, 91)( 32, 90)
( 33, 65)( 34, 68)( 35, 67)( 36, 66)( 37, 69)( 38, 72)( 39, 71)( 40, 70)
( 41, 77)( 42, 80)( 43, 79)( 44, 78)( 45, 73)( 46, 76)( 47, 75)( 48, 74)
( 98,100)(102,104)(105,109)(106,112)(107,111)(108,110)(113,129)(114,132)
(115,131)(116,130)(117,133)(118,136)(119,135)(120,134)(121,141)(122,144)
(123,143)(124,142)(125,137)(126,140)(127,139)(128,138)(145,193)(146,196)
(147,195)(148,194)(149,197)(150,200)(151,199)(152,198)(153,205)(154,208)
(155,207)(156,206)(157,201)(158,204)(159,203)(160,202)(161,225)(162,228)
(163,227)(164,226)(165,229)(166,232)(167,231)(168,230)(169,237)(170,240)
(171,239)(172,238)(173,233)(174,236)(175,235)(176,234)(177,209)(178,212)
(179,211)(180,210)(181,213)(182,216)(183,215)(184,214)(185,221)(186,224)
(187,223)(188,222)(189,217)(190,220)(191,219)(192,218)(242,244)(246,248)
(249,253)(250,256)(251,255)(252,254)(257,273)(258,276)(259,275)(260,274)
(261,277)(262,280)(263,279)(264,278)(265,285)(266,288)(267,287)(268,286)
(269,281)(270,284)(271,283)(272,282);;
s2 := ( 1, 18)( 2, 17)( 3, 19)( 4, 20)( 5, 30)( 6, 29)( 7, 31)( 8, 32)
( 9, 26)( 10, 25)( 11, 27)( 12, 28)( 13, 22)( 14, 21)( 15, 23)( 16, 24)
( 33, 34)( 37, 46)( 38, 45)( 39, 47)( 40, 48)( 41, 42)( 49,114)( 50,113)
( 51,115)( 52,116)( 53,126)( 54,125)( 55,127)( 56,128)( 57,122)( 58,121)
( 59,123)( 60,124)( 61,118)( 62,117)( 63,119)( 64,120)( 65, 98)( 66, 97)
( 67, 99)( 68,100)( 69,110)( 70,109)( 71,111)( 72,112)( 73,106)( 74,105)
( 75,107)( 76,108)( 77,102)( 78,101)( 79,103)( 80,104)( 81,130)( 82,129)
( 83,131)( 84,132)( 85,142)( 86,141)( 87,143)( 88,144)( 89,138)( 90,137)
( 91,139)( 92,140)( 93,134)( 94,133)( 95,135)( 96,136)(145,162)(146,161)
(147,163)(148,164)(149,174)(150,173)(151,175)(152,176)(153,170)(154,169)
(155,171)(156,172)(157,166)(158,165)(159,167)(160,168)(177,178)(181,190)
(182,189)(183,191)(184,192)(185,186)(193,258)(194,257)(195,259)(196,260)
(197,270)(198,269)(199,271)(200,272)(201,266)(202,265)(203,267)(204,268)
(205,262)(206,261)(207,263)(208,264)(209,242)(210,241)(211,243)(212,244)
(213,254)(214,253)(215,255)(216,256)(217,250)(218,249)(219,251)(220,252)
(221,246)(222,245)(223,247)(224,248)(225,274)(226,273)(227,275)(228,276)
(229,286)(230,285)(231,287)(232,288)(233,282)(234,281)(235,283)(236,284)
(237,278)(238,277)(239,279)(240,280);;
s3 := ( 1,149)( 2,150)( 3,151)( 4,152)( 5,145)( 6,146)( 7,147)( 8,148)
( 9,157)( 10,158)( 11,159)( 12,160)( 13,153)( 14,154)( 15,155)( 16,156)
( 17,165)( 18,166)( 19,167)( 20,168)( 21,161)( 22,162)( 23,163)( 24,164)
( 25,173)( 26,174)( 27,175)( 28,176)( 29,169)( 30,170)( 31,171)( 32,172)
( 33,181)( 34,182)( 35,183)( 36,184)( 37,177)( 38,178)( 39,179)( 40,180)
( 41,189)( 42,190)( 43,191)( 44,192)( 45,185)( 46,186)( 47,187)( 48,188)
( 49,197)( 50,198)( 51,199)( 52,200)( 53,193)( 54,194)( 55,195)( 56,196)
( 57,205)( 58,206)( 59,207)( 60,208)( 61,201)( 62,202)( 63,203)( 64,204)
( 65,213)( 66,214)( 67,215)( 68,216)( 69,209)( 70,210)( 71,211)( 72,212)
( 73,221)( 74,222)( 75,223)( 76,224)( 77,217)( 78,218)( 79,219)( 80,220)
( 81,229)( 82,230)( 83,231)( 84,232)( 85,225)( 86,226)( 87,227)( 88,228)
( 89,237)( 90,238)( 91,239)( 92,240)( 93,233)( 94,234)( 95,235)( 96,236)
( 97,245)( 98,246)( 99,247)(100,248)(101,241)(102,242)(103,243)(104,244)
(105,253)(106,254)(107,255)(108,256)(109,249)(110,250)(111,251)(112,252)
(113,261)(114,262)(115,263)(116,264)(117,257)(118,258)(119,259)(120,260)
(121,269)(122,270)(123,271)(124,272)(125,265)(126,266)(127,267)(128,268)
(129,277)(130,278)(131,279)(132,280)(133,273)(134,274)(135,275)(136,276)
(137,285)(138,286)(139,287)(140,288)(141,281)(142,282)(143,283)(144,284);;
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, s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
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, 61)( 10, 64)( 11, 63)( 12, 62)( 13, 57)( 14, 60)( 15, 59)
( 16, 58)( 17, 81)( 18, 84)( 19, 83)( 20, 82)( 21, 85)( 22, 88)( 23, 87)
( 24, 86)( 25, 93)( 26, 96)( 27, 95)( 28, 94)( 29, 89)( 30, 92)( 31, 91)
( 32, 90)( 33, 65)( 34, 68)( 35, 67)( 36, 66)( 37, 69)( 38, 72)( 39, 71)
( 40, 70)( 41, 77)( 42, 80)( 43, 79)( 44, 78)( 45, 73)( 46, 76)( 47, 75)
( 48, 74)( 98,100)(102,104)(105,109)(106,112)(107,111)(108,110)(113,129)
(114,132)(115,131)(116,130)(117,133)(118,136)(119,135)(120,134)(121,141)
(122,144)(123,143)(124,142)(125,137)(126,140)(127,139)(128,138)(145,193)
(146,196)(147,195)(148,194)(149,197)(150,200)(151,199)(152,198)(153,205)
(154,208)(155,207)(156,206)(157,201)(158,204)(159,203)(160,202)(161,225)
(162,228)(163,227)(164,226)(165,229)(166,232)(167,231)(168,230)(169,237)
(170,240)(171,239)(172,238)(173,233)(174,236)(175,235)(176,234)(177,209)
(178,212)(179,211)(180,210)(181,213)(182,216)(183,215)(184,214)(185,221)
(186,224)(187,223)(188,222)(189,217)(190,220)(191,219)(192,218)(242,244)
(246,248)(249,253)(250,256)(251,255)(252,254)(257,273)(258,276)(259,275)
(260,274)(261,277)(262,280)(263,279)(264,278)(265,285)(266,288)(267,287)
(268,286)(269,281)(270,284)(271,283)(272,282);
s2 := Sym(288)!( 1, 18)( 2, 17)( 3, 19)( 4, 20)( 5, 30)( 6, 29)( 7, 31)
( 8, 32)( 9, 26)( 10, 25)( 11, 27)( 12, 28)( 13, 22)( 14, 21)( 15, 23)
( 16, 24)( 33, 34)( 37, 46)( 38, 45)( 39, 47)( 40, 48)( 41, 42)( 49,114)
( 50,113)( 51,115)( 52,116)( 53,126)( 54,125)( 55,127)( 56,128)( 57,122)
( 58,121)( 59,123)( 60,124)( 61,118)( 62,117)( 63,119)( 64,120)( 65, 98)
( 66, 97)( 67, 99)( 68,100)( 69,110)( 70,109)( 71,111)( 72,112)( 73,106)
( 74,105)( 75,107)( 76,108)( 77,102)( 78,101)( 79,103)( 80,104)( 81,130)
( 82,129)( 83,131)( 84,132)( 85,142)( 86,141)( 87,143)( 88,144)( 89,138)
( 90,137)( 91,139)( 92,140)( 93,134)( 94,133)( 95,135)( 96,136)(145,162)
(146,161)(147,163)(148,164)(149,174)(150,173)(151,175)(152,176)(153,170)
(154,169)(155,171)(156,172)(157,166)(158,165)(159,167)(160,168)(177,178)
(181,190)(182,189)(183,191)(184,192)(185,186)(193,258)(194,257)(195,259)
(196,260)(197,270)(198,269)(199,271)(200,272)(201,266)(202,265)(203,267)
(204,268)(205,262)(206,261)(207,263)(208,264)(209,242)(210,241)(211,243)
(212,244)(213,254)(214,253)(215,255)(216,256)(217,250)(218,249)(219,251)
(220,252)(221,246)(222,245)(223,247)(224,248)(225,274)(226,273)(227,275)
(228,276)(229,286)(230,285)(231,287)(232,288)(233,282)(234,281)(235,283)
(236,284)(237,278)(238,277)(239,279)(240,280);
s3 := Sym(288)!( 1,149)( 2,150)( 3,151)( 4,152)( 5,145)( 6,146)( 7,147)
( 8,148)( 9,157)( 10,158)( 11,159)( 12,160)( 13,153)( 14,154)( 15,155)
( 16,156)( 17,165)( 18,166)( 19,167)( 20,168)( 21,161)( 22,162)( 23,163)
( 24,164)( 25,173)( 26,174)( 27,175)( 28,176)( 29,169)( 30,170)( 31,171)
( 32,172)( 33,181)( 34,182)( 35,183)( 36,184)( 37,177)( 38,178)( 39,179)
( 40,180)( 41,189)( 42,190)( 43,191)( 44,192)( 45,185)( 46,186)( 47,187)
( 48,188)( 49,197)( 50,198)( 51,199)( 52,200)( 53,193)( 54,194)( 55,195)
( 56,196)( 57,205)( 58,206)( 59,207)( 60,208)( 61,201)( 62,202)( 63,203)
( 64,204)( 65,213)( 66,214)( 67,215)( 68,216)( 69,209)( 70,210)( 71,211)
( 72,212)( 73,221)( 74,222)( 75,223)( 76,224)( 77,217)( 78,218)( 79,219)
( 80,220)( 81,229)( 82,230)( 83,231)( 84,232)( 85,225)( 86,226)( 87,227)
( 88,228)( 89,237)( 90,238)( 91,239)( 92,240)( 93,233)( 94,234)( 95,235)
( 96,236)( 97,245)( 98,246)( 99,247)(100,248)(101,241)(102,242)(103,243)
(104,244)(105,253)(106,254)(107,255)(108,256)(109,249)(110,250)(111,251)
(112,252)(113,261)(114,262)(115,263)(116,264)(117,257)(118,258)(119,259)
(120,260)(121,269)(122,270)(123,271)(124,272)(125,265)(126,266)(127,267)
(128,268)(129,277)(130,278)(131,279)(132,280)(133,273)(134,274)(135,275)
(136,276)(137,285)(138,286)(139,287)(140,288)(141,281)(142,282)(143,283)
(144,284);
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,
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 >;
References : None.
to this polytope