Polytope of Type {12,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,4}*1152e
if this polytope has a name.
Group : SmallGroup(1152,157864)
Rank : 4
Schlafli Type : {12,6,4}
Number of vertices, edges, etc : 12, 72, 24, 8
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Non-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 : {12,6,4}*576f
3-fold quotients : {4,6,4}*384f
4-fold quotients : {12,6,2}*288d
6-fold quotients : {4,3,4}*192a, {4,6,4}*192f, {4,6,4}*192g
12-fold quotients : {4,6,2}*96b, {4,3,4}*96
24-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 13)( 6, 14)( 7, 15)( 8, 16)
( 17, 41)( 18, 42)( 19, 43)( 20, 44)( 21, 45)( 22, 46)( 23, 47)( 24, 48)
( 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)
( 65, 89)( 66, 90)( 67, 91)( 68, 92)( 69, 93)( 70, 94)( 71, 95)( 72, 96)
( 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)
(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)(120,144)
(121,129)(122,130)(123,131)(124,132)(125,133)(126,134)(127,135)(128,136)
(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)
(161,185)(162,186)(163,187)(164,188)(165,189)(166,190)(167,191)(168,192)
(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,183)(176,184)
(193,201)(194,202)(195,203)(196,204)(197,205)(198,206)(199,207)(200,208)
(209,233)(210,234)(211,235)(212,236)(213,237)(214,238)(215,239)(216,240)
(217,225)(218,226)(219,227)(220,228)(221,229)(222,230)(223,231)(224,232)
(241,249)(242,250)(243,251)(244,252)(245,253)(246,254)(247,255)(248,256)
(257,281)(258,282)(259,283)(260,284)(261,285)(262,286)(263,287)(264,288)
(265,273)(266,274)(267,275)(268,276)(269,277)(270,278)(271,279)(272,280);;
s1 := ( 1, 17)( 2, 18)( 3, 20)( 4, 19)( 5, 21)( 6, 22)( 7, 24)( 8, 23)
( 9, 29)( 10, 30)( 11, 32)( 12, 31)( 13, 25)( 14, 26)( 15, 28)( 16, 27)
( 35, 36)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49,113)( 50,114)
( 51,116)( 52,115)( 53,117)( 54,118)( 55,120)( 56,119)( 57,125)( 58,126)
( 59,128)( 60,127)( 61,121)( 62,122)( 63,124)( 64,123)( 65, 97)( 66, 98)
( 67,100)( 68, 99)( 69,101)( 70,102)( 71,104)( 72,103)( 73,109)( 74,110)
( 75,112)( 76,111)( 77,105)( 78,106)( 79,108)( 80,107)( 81,129)( 82,130)
( 83,132)( 84,131)( 85,133)( 86,134)( 87,136)( 88,135)( 89,141)( 90,142)
( 91,144)( 92,143)( 93,137)( 94,138)( 95,140)( 96,139)(145,161)(146,162)
(147,164)(148,163)(149,165)(150,166)(151,168)(152,167)(153,173)(154,174)
(155,176)(156,175)(157,169)(158,170)(159,172)(160,171)(179,180)(183,184)
(185,189)(186,190)(187,192)(188,191)(193,257)(194,258)(195,260)(196,259)
(197,261)(198,262)(199,264)(200,263)(201,269)(202,270)(203,272)(204,271)
(205,265)(206,266)(207,268)(208,267)(209,241)(210,242)(211,244)(212,243)
(213,245)(214,246)(215,248)(216,247)(217,253)(218,254)(219,256)(220,255)
(221,249)(222,250)(223,252)(224,251)(225,273)(226,274)(227,276)(228,275)
(229,277)(230,278)(231,280)(232,279)(233,285)(234,286)(235,288)(236,287)
(237,281)(238,282)(239,284)(240,283);;
s2 := ( 1, 49)( 2, 52)( 3, 51)( 4, 50)( 5, 61)( 6, 64)( 7, 63)( 8, 62)
( 9, 57)( 10, 60)( 11, 59)( 12, 58)( 13, 53)( 14, 56)( 15, 55)( 16, 54)
( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21, 77)( 22, 80)( 23, 79)( 24, 78)
( 25, 73)( 26, 76)( 27, 75)( 28, 74)( 29, 69)( 30, 72)( 31, 71)( 32, 70)
( 33, 81)( 34, 84)( 35, 83)( 36, 82)( 37, 93)( 38, 96)( 39, 95)( 40, 94)
( 41, 89)( 42, 92)( 43, 91)( 44, 90)( 45, 85)( 46, 88)( 47, 87)( 48, 86)
( 98,100)(101,109)(102,112)(103,111)(104,110)(106,108)(114,116)(117,125)
(118,128)(119,127)(120,126)(122,124)(130,132)(133,141)(134,144)(135,143)
(136,142)(138,140)(145,193)(146,196)(147,195)(148,194)(149,205)(150,208)
(151,207)(152,206)(153,201)(154,204)(155,203)(156,202)(157,197)(158,200)
(159,199)(160,198)(161,209)(162,212)(163,211)(164,210)(165,221)(166,224)
(167,223)(168,222)(169,217)(170,220)(171,219)(172,218)(173,213)(174,216)
(175,215)(176,214)(177,225)(178,228)(179,227)(180,226)(181,237)(182,240)
(183,239)(184,238)(185,233)(186,236)(187,235)(188,234)(189,229)(190,232)
(191,231)(192,230)(242,244)(245,253)(246,256)(247,255)(248,254)(250,252)
(258,260)(261,269)(262,272)(263,271)(264,270)(266,268)(274,276)(277,285)
(278,288)(279,287)(280,286)(282,284);;
s3 := ( 1,146)( 2,145)( 3,148)( 4,147)( 5,150)( 6,149)( 7,152)( 8,151)
( 9,154)( 10,153)( 11,156)( 12,155)( 13,158)( 14,157)( 15,160)( 16,159)
( 17,162)( 18,161)( 19,164)( 20,163)( 21,166)( 22,165)( 23,168)( 24,167)
( 25,170)( 26,169)( 27,172)( 28,171)( 29,174)( 30,173)( 31,176)( 32,175)
( 33,178)( 34,177)( 35,180)( 36,179)( 37,182)( 38,181)( 39,184)( 40,183)
( 41,186)( 42,185)( 43,188)( 44,187)( 45,190)( 46,189)( 47,192)( 48,191)
( 49,194)( 50,193)( 51,196)( 52,195)( 53,198)( 54,197)( 55,200)( 56,199)
( 57,202)( 58,201)( 59,204)( 60,203)( 61,206)( 62,205)( 63,208)( 64,207)
( 65,210)( 66,209)( 67,212)( 68,211)( 69,214)( 70,213)( 71,216)( 72,215)
( 73,218)( 74,217)( 75,220)( 76,219)( 77,222)( 78,221)( 79,224)( 80,223)
( 81,226)( 82,225)( 83,228)( 84,227)( 85,230)( 86,229)( 87,232)( 88,231)
( 89,234)( 90,233)( 91,236)( 92,235)( 93,238)( 94,237)( 95,240)( 96,239)
( 97,242)( 98,241)( 99,244)(100,243)(101,246)(102,245)(103,248)(104,247)
(105,250)(106,249)(107,252)(108,251)(109,254)(110,253)(111,256)(112,255)
(113,258)(114,257)(115,260)(116,259)(117,262)(118,261)(119,264)(120,263)
(121,266)(122,265)(123,268)(124,267)(125,270)(126,269)(127,272)(128,271)
(129,274)(130,273)(131,276)(132,275)(133,278)(134,277)(135,280)(136,279)
(137,282)(138,281)(139,284)(140,283)(141,286)(142,285)(143,288)(144,287);;
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, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s2*s0*s1*s2*s0*s1*s2, s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(288)!( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 13)( 6, 14)( 7, 15)
( 8, 16)( 17, 41)( 18, 42)( 19, 43)( 20, 44)( 21, 45)( 22, 46)( 23, 47)
( 24, 48)( 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)( 65, 89)( 66, 90)( 67, 91)( 68, 92)( 69, 93)( 70, 94)( 71, 95)
( 72, 96)( 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)(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)
(120,144)(121,129)(122,130)(123,131)(124,132)(125,133)(126,134)(127,135)
(128,136)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)
(152,160)(161,185)(162,186)(163,187)(164,188)(165,189)(166,190)(167,191)
(168,192)(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,183)
(176,184)(193,201)(194,202)(195,203)(196,204)(197,205)(198,206)(199,207)
(200,208)(209,233)(210,234)(211,235)(212,236)(213,237)(214,238)(215,239)
(216,240)(217,225)(218,226)(219,227)(220,228)(221,229)(222,230)(223,231)
(224,232)(241,249)(242,250)(243,251)(244,252)(245,253)(246,254)(247,255)
(248,256)(257,281)(258,282)(259,283)(260,284)(261,285)(262,286)(263,287)
(264,288)(265,273)(266,274)(267,275)(268,276)(269,277)(270,278)(271,279)
(272,280);
s1 := Sym(288)!( 1, 17)( 2, 18)( 3, 20)( 4, 19)( 5, 21)( 6, 22)( 7, 24)
( 8, 23)( 9, 29)( 10, 30)( 11, 32)( 12, 31)( 13, 25)( 14, 26)( 15, 28)
( 16, 27)( 35, 36)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49,113)
( 50,114)( 51,116)( 52,115)( 53,117)( 54,118)( 55,120)( 56,119)( 57,125)
( 58,126)( 59,128)( 60,127)( 61,121)( 62,122)( 63,124)( 64,123)( 65, 97)
( 66, 98)( 67,100)( 68, 99)( 69,101)( 70,102)( 71,104)( 72,103)( 73,109)
( 74,110)( 75,112)( 76,111)( 77,105)( 78,106)( 79,108)( 80,107)( 81,129)
( 82,130)( 83,132)( 84,131)( 85,133)( 86,134)( 87,136)( 88,135)( 89,141)
( 90,142)( 91,144)( 92,143)( 93,137)( 94,138)( 95,140)( 96,139)(145,161)
(146,162)(147,164)(148,163)(149,165)(150,166)(151,168)(152,167)(153,173)
(154,174)(155,176)(156,175)(157,169)(158,170)(159,172)(160,171)(179,180)
(183,184)(185,189)(186,190)(187,192)(188,191)(193,257)(194,258)(195,260)
(196,259)(197,261)(198,262)(199,264)(200,263)(201,269)(202,270)(203,272)
(204,271)(205,265)(206,266)(207,268)(208,267)(209,241)(210,242)(211,244)
(212,243)(213,245)(214,246)(215,248)(216,247)(217,253)(218,254)(219,256)
(220,255)(221,249)(222,250)(223,252)(224,251)(225,273)(226,274)(227,276)
(228,275)(229,277)(230,278)(231,280)(232,279)(233,285)(234,286)(235,288)
(236,287)(237,281)(238,282)(239,284)(240,283);
s2 := Sym(288)!( 1, 49)( 2, 52)( 3, 51)( 4, 50)( 5, 61)( 6, 64)( 7, 63)
( 8, 62)( 9, 57)( 10, 60)( 11, 59)( 12, 58)( 13, 53)( 14, 56)( 15, 55)
( 16, 54)( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21, 77)( 22, 80)( 23, 79)
( 24, 78)( 25, 73)( 26, 76)( 27, 75)( 28, 74)( 29, 69)( 30, 72)( 31, 71)
( 32, 70)( 33, 81)( 34, 84)( 35, 83)( 36, 82)( 37, 93)( 38, 96)( 39, 95)
( 40, 94)( 41, 89)( 42, 92)( 43, 91)( 44, 90)( 45, 85)( 46, 88)( 47, 87)
( 48, 86)( 98,100)(101,109)(102,112)(103,111)(104,110)(106,108)(114,116)
(117,125)(118,128)(119,127)(120,126)(122,124)(130,132)(133,141)(134,144)
(135,143)(136,142)(138,140)(145,193)(146,196)(147,195)(148,194)(149,205)
(150,208)(151,207)(152,206)(153,201)(154,204)(155,203)(156,202)(157,197)
(158,200)(159,199)(160,198)(161,209)(162,212)(163,211)(164,210)(165,221)
(166,224)(167,223)(168,222)(169,217)(170,220)(171,219)(172,218)(173,213)
(174,216)(175,215)(176,214)(177,225)(178,228)(179,227)(180,226)(181,237)
(182,240)(183,239)(184,238)(185,233)(186,236)(187,235)(188,234)(189,229)
(190,232)(191,231)(192,230)(242,244)(245,253)(246,256)(247,255)(248,254)
(250,252)(258,260)(261,269)(262,272)(263,271)(264,270)(266,268)(274,276)
(277,285)(278,288)(279,287)(280,286)(282,284);
s3 := Sym(288)!( 1,146)( 2,145)( 3,148)( 4,147)( 5,150)( 6,149)( 7,152)
( 8,151)( 9,154)( 10,153)( 11,156)( 12,155)( 13,158)( 14,157)( 15,160)
( 16,159)( 17,162)( 18,161)( 19,164)( 20,163)( 21,166)( 22,165)( 23,168)
( 24,167)( 25,170)( 26,169)( 27,172)( 28,171)( 29,174)( 30,173)( 31,176)
( 32,175)( 33,178)( 34,177)( 35,180)( 36,179)( 37,182)( 38,181)( 39,184)
( 40,183)( 41,186)( 42,185)( 43,188)( 44,187)( 45,190)( 46,189)( 47,192)
( 48,191)( 49,194)( 50,193)( 51,196)( 52,195)( 53,198)( 54,197)( 55,200)
( 56,199)( 57,202)( 58,201)( 59,204)( 60,203)( 61,206)( 62,205)( 63,208)
( 64,207)( 65,210)( 66,209)( 67,212)( 68,211)( 69,214)( 70,213)( 71,216)
( 72,215)( 73,218)( 74,217)( 75,220)( 76,219)( 77,222)( 78,221)( 79,224)
( 80,223)( 81,226)( 82,225)( 83,228)( 84,227)( 85,230)( 86,229)( 87,232)
( 88,231)( 89,234)( 90,233)( 91,236)( 92,235)( 93,238)( 94,237)( 95,240)
( 96,239)( 97,242)( 98,241)( 99,244)(100,243)(101,246)(102,245)(103,248)
(104,247)(105,250)(106,249)(107,252)(108,251)(109,254)(110,253)(111,256)
(112,255)(113,258)(114,257)(115,260)(116,259)(117,262)(118,261)(119,264)
(120,263)(121,266)(122,265)(123,268)(124,267)(125,270)(126,269)(127,272)
(128,271)(129,274)(130,273)(131,276)(132,275)(133,278)(134,277)(135,280)
(136,279)(137,282)(138,281)(139,284)(140,283)(141,286)(142,285)(143,288)
(144,287);
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,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s0*s1*s2*s0*s1*s2,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1 >;
References : None.
to this polytope