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