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