Polytope of Type {4,6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,12}*1152a
if this polytope has a name.
Group : SmallGroup(1152,157550)
Rank : 4
Schlafli Type : {4,6,12}
Number of vertices, edges, etc : 8, 24, 72, 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 : {4,6,12}*576d, {4,6,6}*576a
3-fold quotients : {4,6,4}*384b
4-fold quotients : {2,6,12}*288a, {4,6,6}*288d
6-fold quotients : {4,6,4}*192c, {4,6,2}*192
8-fold quotients : {2,6,6}*144a
12-fold quotients : {2,2,12}*96, {2,6,4}*96a, {4,3,2}*96, {4,6,2}*96b, {4,6,2}*96c
24-fold quotients : {4,3,2}*48, {2,2,6}*48, {2,6,2}*48
36-fold quotients : {2,2,4}*32
48-fold quotients : {2,2,3}*24, {2,3,2}*24
72-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,147)( 2,148)( 3,145)( 4,146)( 5,151)( 6,152)( 7,149)( 8,150)
( 9,155)( 10,156)( 11,153)( 12,154)( 13,159)( 14,160)( 15,157)( 16,158)
( 17,163)( 18,164)( 19,161)( 20,162)( 21,167)( 22,168)( 23,165)( 24,166)
( 25,171)( 26,172)( 27,169)( 28,170)( 29,175)( 30,176)( 31,173)( 32,174)
( 33,179)( 34,180)( 35,177)( 36,178)( 37,183)( 38,184)( 39,181)( 40,182)
( 41,187)( 42,188)( 43,185)( 44,186)( 45,191)( 46,192)( 47,189)( 48,190)
( 49,195)( 50,196)( 51,193)( 52,194)( 53,199)( 54,200)( 55,197)( 56,198)
( 57,203)( 58,204)( 59,201)( 60,202)( 61,207)( 62,208)( 63,205)( 64,206)
( 65,211)( 66,212)( 67,209)( 68,210)( 69,215)( 70,216)( 71,213)( 72,214)
( 73,219)( 74,220)( 75,217)( 76,218)( 77,223)( 78,224)( 79,221)( 80,222)
( 81,227)( 82,228)( 83,225)( 84,226)( 85,231)( 86,232)( 87,229)( 88,230)
( 89,235)( 90,236)( 91,233)( 92,234)( 93,239)( 94,240)( 95,237)( 96,238)
( 97,243)( 98,244)( 99,241)(100,242)(101,247)(102,248)(103,245)(104,246)
(105,251)(106,252)(107,249)(108,250)(109,255)(110,256)(111,253)(112,254)
(113,259)(114,260)(115,257)(116,258)(117,263)(118,264)(119,261)(120,262)
(121,267)(122,268)(123,265)(124,266)(125,271)(126,272)(127,269)(128,270)
(129,275)(130,276)(131,273)(132,274)(133,279)(134,280)(135,277)(136,278)
(137,283)(138,284)(139,281)(140,282)(141,287)(142,288)(143,285)(144,286);;
s1 := ( 3, 4)( 7, 8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)
( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 39, 40)
( 43, 44)( 47, 48)( 49, 61)( 50, 62)( 51, 64)( 52, 63)( 53, 65)( 54, 66)
( 55, 68)( 56, 67)( 57, 69)( 58, 70)( 59, 72)( 60, 71)( 75, 76)( 79, 80)
( 83, 84)( 85, 97)( 86, 98)( 87,100)( 88, 99)( 89,101)( 90,102)( 91,104)
( 92,103)( 93,105)( 94,106)( 95,108)( 96,107)(111,112)(115,116)(119,120)
(121,133)(122,134)(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)
(129,141)(130,142)(131,144)(132,143)(147,148)(151,152)(155,156)(157,169)
(158,170)(159,172)(160,171)(161,173)(162,174)(163,176)(164,175)(165,177)
(166,178)(167,180)(168,179)(183,184)(187,188)(191,192)(193,205)(194,206)
(195,208)(196,207)(197,209)(198,210)(199,212)(200,211)(201,213)(202,214)
(203,216)(204,215)(219,220)(223,224)(227,228)(229,241)(230,242)(231,244)
(232,243)(233,245)(234,246)(235,248)(236,247)(237,249)(238,250)(239,252)
(240,251)(255,256)(259,260)(263,264)(265,277)(266,278)(267,280)(268,279)
(269,281)(270,282)(271,284)(272,283)(273,285)(274,286)(275,288)(276,287);;
s2 := ( 1, 13)( 2, 16)( 3, 15)( 4, 14)( 5, 21)( 6, 24)( 7, 23)( 8, 22)
( 9, 17)( 10, 20)( 11, 19)( 12, 18)( 26, 28)( 29, 33)( 30, 36)( 31, 35)
( 32, 34)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 57)( 42, 60)( 43, 59)
( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)( 62, 64)( 65, 69)( 66, 72)
( 67, 71)( 68, 70)( 73,121)( 74,124)( 75,123)( 76,122)( 77,129)( 78,132)
( 79,131)( 80,130)( 81,125)( 82,128)( 83,127)( 84,126)( 85,109)( 86,112)
( 87,111)( 88,110)( 89,117)( 90,120)( 91,119)( 92,118)( 93,113)( 94,116)
( 95,115)( 96,114)( 97,133)( 98,136)( 99,135)(100,134)(101,141)(102,144)
(103,143)(104,142)(105,137)(106,140)(107,139)(108,138)(145,157)(146,160)
(147,159)(148,158)(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)
(155,163)(156,162)(170,172)(173,177)(174,180)(175,179)(176,178)(181,193)
(182,196)(183,195)(184,194)(185,201)(186,204)(187,203)(188,202)(189,197)
(190,200)(191,199)(192,198)(206,208)(209,213)(210,216)(211,215)(212,214)
(217,265)(218,268)(219,267)(220,266)(221,273)(222,276)(223,275)(224,274)
(225,269)(226,272)(227,271)(228,270)(229,253)(230,256)(231,255)(232,254)
(233,261)(234,264)(235,263)(236,262)(237,257)(238,260)(239,259)(240,258)
(241,277)(242,280)(243,279)(244,278)(245,285)(246,288)(247,287)(248,286)
(249,281)(250,284)(251,283)(252,282);;
s3 := ( 1, 77)( 2, 78)( 3, 79)( 4, 80)( 5, 73)( 6, 74)( 7, 75)( 8, 76)
( 9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 89)( 14, 90)( 15, 91)( 16, 92)
( 17, 85)( 18, 86)( 19, 87)( 20, 88)( 21, 93)( 22, 94)( 23, 95)( 24, 96)
( 25,101)( 26,102)( 27,103)( 28,104)( 29, 97)( 30, 98)( 31, 99)( 32,100)
( 33,105)( 34,106)( 35,107)( 36,108)( 37,113)( 38,114)( 39,115)( 40,116)
( 41,109)( 42,110)( 43,111)( 44,112)( 45,117)( 46,118)( 47,119)( 48,120)
( 49,125)( 50,126)( 51,127)( 52,128)( 53,121)( 54,122)( 55,123)( 56,124)
( 57,129)( 58,130)( 59,131)( 60,132)( 61,137)( 62,138)( 63,139)( 64,140)
( 65,133)( 66,134)( 67,135)( 68,136)( 69,141)( 70,142)( 71,143)( 72,144)
(145,221)(146,222)(147,223)(148,224)(149,217)(150,218)(151,219)(152,220)
(153,225)(154,226)(155,227)(156,228)(157,233)(158,234)(159,235)(160,236)
(161,229)(162,230)(163,231)(164,232)(165,237)(166,238)(167,239)(168,240)
(169,245)(170,246)(171,247)(172,248)(173,241)(174,242)(175,243)(176,244)
(177,249)(178,250)(179,251)(180,252)(181,257)(182,258)(183,259)(184,260)
(185,253)(186,254)(187,255)(188,256)(189,261)(190,262)(191,263)(192,264)
(193,269)(194,270)(195,271)(196,272)(197,265)(198,266)(199,267)(200,268)
(201,273)(202,274)(203,275)(204,276)(205,281)(206,282)(207,283)(208,284)
(209,277)(210,278)(211,279)(212,280)(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*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
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(288)!( 1,147)( 2,148)( 3,145)( 4,146)( 5,151)( 6,152)( 7,149)
( 8,150)( 9,155)( 10,156)( 11,153)( 12,154)( 13,159)( 14,160)( 15,157)
( 16,158)( 17,163)( 18,164)( 19,161)( 20,162)( 21,167)( 22,168)( 23,165)
( 24,166)( 25,171)( 26,172)( 27,169)( 28,170)( 29,175)( 30,176)( 31,173)
( 32,174)( 33,179)( 34,180)( 35,177)( 36,178)( 37,183)( 38,184)( 39,181)
( 40,182)( 41,187)( 42,188)( 43,185)( 44,186)( 45,191)( 46,192)( 47,189)
( 48,190)( 49,195)( 50,196)( 51,193)( 52,194)( 53,199)( 54,200)( 55,197)
( 56,198)( 57,203)( 58,204)( 59,201)( 60,202)( 61,207)( 62,208)( 63,205)
( 64,206)( 65,211)( 66,212)( 67,209)( 68,210)( 69,215)( 70,216)( 71,213)
( 72,214)( 73,219)( 74,220)( 75,217)( 76,218)( 77,223)( 78,224)( 79,221)
( 80,222)( 81,227)( 82,228)( 83,225)( 84,226)( 85,231)( 86,232)( 87,229)
( 88,230)( 89,235)( 90,236)( 91,233)( 92,234)( 93,239)( 94,240)( 95,237)
( 96,238)( 97,243)( 98,244)( 99,241)(100,242)(101,247)(102,248)(103,245)
(104,246)(105,251)(106,252)(107,249)(108,250)(109,255)(110,256)(111,253)
(112,254)(113,259)(114,260)(115,257)(116,258)(117,263)(118,264)(119,261)
(120,262)(121,267)(122,268)(123,265)(124,266)(125,271)(126,272)(127,269)
(128,270)(129,275)(130,276)(131,273)(132,274)(133,279)(134,280)(135,277)
(136,278)(137,283)(138,284)(139,281)(140,282)(141,287)(142,288)(143,285)
(144,286);
s1 := Sym(288)!( 3, 4)( 7, 8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)
( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)
( 39, 40)( 43, 44)( 47, 48)( 49, 61)( 50, 62)( 51, 64)( 52, 63)( 53, 65)
( 54, 66)( 55, 68)( 56, 67)( 57, 69)( 58, 70)( 59, 72)( 60, 71)( 75, 76)
( 79, 80)( 83, 84)( 85, 97)( 86, 98)( 87,100)( 88, 99)( 89,101)( 90,102)
( 91,104)( 92,103)( 93,105)( 94,106)( 95,108)( 96,107)(111,112)(115,116)
(119,120)(121,133)(122,134)(123,136)(124,135)(125,137)(126,138)(127,140)
(128,139)(129,141)(130,142)(131,144)(132,143)(147,148)(151,152)(155,156)
(157,169)(158,170)(159,172)(160,171)(161,173)(162,174)(163,176)(164,175)
(165,177)(166,178)(167,180)(168,179)(183,184)(187,188)(191,192)(193,205)
(194,206)(195,208)(196,207)(197,209)(198,210)(199,212)(200,211)(201,213)
(202,214)(203,216)(204,215)(219,220)(223,224)(227,228)(229,241)(230,242)
(231,244)(232,243)(233,245)(234,246)(235,248)(236,247)(237,249)(238,250)
(239,252)(240,251)(255,256)(259,260)(263,264)(265,277)(266,278)(267,280)
(268,279)(269,281)(270,282)(271,284)(272,283)(273,285)(274,286)(275,288)
(276,287);
s2 := Sym(288)!( 1, 13)( 2, 16)( 3, 15)( 4, 14)( 5, 21)( 6, 24)( 7, 23)
( 8, 22)( 9, 17)( 10, 20)( 11, 19)( 12, 18)( 26, 28)( 29, 33)( 30, 36)
( 31, 35)( 32, 34)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 57)( 42, 60)
( 43, 59)( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)( 62, 64)( 65, 69)
( 66, 72)( 67, 71)( 68, 70)( 73,121)( 74,124)( 75,123)( 76,122)( 77,129)
( 78,132)( 79,131)( 80,130)( 81,125)( 82,128)( 83,127)( 84,126)( 85,109)
( 86,112)( 87,111)( 88,110)( 89,117)( 90,120)( 91,119)( 92,118)( 93,113)
( 94,116)( 95,115)( 96,114)( 97,133)( 98,136)( 99,135)(100,134)(101,141)
(102,144)(103,143)(104,142)(105,137)(106,140)(107,139)(108,138)(145,157)
(146,160)(147,159)(148,158)(149,165)(150,168)(151,167)(152,166)(153,161)
(154,164)(155,163)(156,162)(170,172)(173,177)(174,180)(175,179)(176,178)
(181,193)(182,196)(183,195)(184,194)(185,201)(186,204)(187,203)(188,202)
(189,197)(190,200)(191,199)(192,198)(206,208)(209,213)(210,216)(211,215)
(212,214)(217,265)(218,268)(219,267)(220,266)(221,273)(222,276)(223,275)
(224,274)(225,269)(226,272)(227,271)(228,270)(229,253)(230,256)(231,255)
(232,254)(233,261)(234,264)(235,263)(236,262)(237,257)(238,260)(239,259)
(240,258)(241,277)(242,280)(243,279)(244,278)(245,285)(246,288)(247,287)
(248,286)(249,281)(250,284)(251,283)(252,282);
s3 := Sym(288)!( 1, 77)( 2, 78)( 3, 79)( 4, 80)( 5, 73)( 6, 74)( 7, 75)
( 8, 76)( 9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 89)( 14, 90)( 15, 91)
( 16, 92)( 17, 85)( 18, 86)( 19, 87)( 20, 88)( 21, 93)( 22, 94)( 23, 95)
( 24, 96)( 25,101)( 26,102)( 27,103)( 28,104)( 29, 97)( 30, 98)( 31, 99)
( 32,100)( 33,105)( 34,106)( 35,107)( 36,108)( 37,113)( 38,114)( 39,115)
( 40,116)( 41,109)( 42,110)( 43,111)( 44,112)( 45,117)( 46,118)( 47,119)
( 48,120)( 49,125)( 50,126)( 51,127)( 52,128)( 53,121)( 54,122)( 55,123)
( 56,124)( 57,129)( 58,130)( 59,131)( 60,132)( 61,137)( 62,138)( 63,139)
( 64,140)( 65,133)( 66,134)( 67,135)( 68,136)( 69,141)( 70,142)( 71,143)
( 72,144)(145,221)(146,222)(147,223)(148,224)(149,217)(150,218)(151,219)
(152,220)(153,225)(154,226)(155,227)(156,228)(157,233)(158,234)(159,235)
(160,236)(161,229)(162,230)(163,231)(164,232)(165,237)(166,238)(167,239)
(168,240)(169,245)(170,246)(171,247)(172,248)(173,241)(174,242)(175,243)
(176,244)(177,249)(178,250)(179,251)(180,252)(181,257)(182,258)(183,259)
(184,260)(185,253)(186,254)(187,255)(188,256)(189,261)(190,262)(191,263)
(192,264)(193,269)(194,270)(195,271)(196,272)(197,265)(198,266)(199,267)
(200,268)(201,273)(202,274)(203,275)(204,276)(205,281)(206,282)(207,283)
(208,284)(209,277)(210,278)(211,279)(212,280)(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*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
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