Polytope of Type {12,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,3}*1296b
if this polytope has a name.
Group : SmallGroup(1296,840)
Rank : 4
Schlafli Type : {12,6,3}
Number of vertices, edges, etc : 12, 108, 27, 9
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 6
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 : {6,6,3}*648b
3-fold quotients : {12,6,3}*432a
4-fold quotients : {3,6,3}*324a
6-fold quotients : {6,6,3}*216a
9-fold quotients : {12,2,3}*144
12-fold quotients : {3,6,3}*108
18-fold quotients : {6,2,3}*72
27-fold quotients : {4,2,3}*48
36-fold quotients : {3,2,3}*36
54-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 10, 19)( 11, 21)( 12, 20)( 13, 25)
( 14, 27)( 15, 26)( 16, 22)( 17, 24)( 18, 23)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)( 43, 49)
( 44, 51)( 45, 50)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 73)( 65, 75)
( 66, 74)( 67, 79)( 68, 81)( 69, 80)( 70, 76)( 71, 78)( 72, 77)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 91,100)( 92,102)( 93,101)( 94,106)( 95,108)
( 96,107)( 97,103)( 98,105)( 99,104)(110,111)(112,115)(113,117)(114,116)
(118,127)(119,129)(120,128)(121,133)(122,135)(123,134)(124,130)(125,132)
(126,131)(137,138)(139,142)(140,144)(141,143)(145,154)(146,156)(147,155)
(148,160)(149,162)(150,161)(151,157)(152,159)(153,158)(163,244)(164,246)
(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)(171,248)(172,262)
(173,264)(174,263)(175,268)(176,270)(177,269)(178,265)(179,267)(180,266)
(181,253)(182,255)(183,254)(184,259)(185,261)(186,260)(187,256)(188,258)
(189,257)(190,271)(191,273)(192,272)(193,277)(194,279)(195,278)(196,274)
(197,276)(198,275)(199,289)(200,291)(201,290)(202,295)(203,297)(204,296)
(205,292)(206,294)(207,293)(208,280)(209,282)(210,281)(211,286)(212,288)
(213,287)(214,283)(215,285)(216,284)(217,298)(218,300)(219,299)(220,304)
(221,306)(222,305)(223,301)(224,303)(225,302)(226,316)(227,318)(228,317)
(229,322)(230,324)(231,323)(232,319)(233,321)(234,320)(235,307)(236,309)
(237,308)(238,313)(239,315)(240,314)(241,310)(242,312)(243,311);;
s1 := ( 1,172)( 2,174)( 3,173)( 4,178)( 5,180)( 6,179)( 7,175)( 8,177)
( 9,176)( 10,163)( 11,165)( 12,164)( 13,169)( 14,171)( 15,170)( 16,166)
( 17,168)( 18,167)( 19,181)( 20,183)( 21,182)( 22,187)( 23,189)( 24,188)
( 25,184)( 26,186)( 27,185)( 28,199)( 29,201)( 30,200)( 31,205)( 32,207)
( 33,206)( 34,202)( 35,204)( 36,203)( 37,190)( 38,192)( 39,191)( 40,196)
( 41,198)( 42,197)( 43,193)( 44,195)( 45,194)( 46,208)( 47,210)( 48,209)
( 49,214)( 50,216)( 51,215)( 52,211)( 53,213)( 54,212)( 55,226)( 56,228)
( 57,227)( 58,232)( 59,234)( 60,233)( 61,229)( 62,231)( 63,230)( 64,217)
( 65,219)( 66,218)( 67,223)( 68,225)( 69,224)( 70,220)( 71,222)( 72,221)
( 73,235)( 74,237)( 75,236)( 76,241)( 77,243)( 78,242)( 79,238)( 80,240)
( 81,239)( 82,253)( 83,255)( 84,254)( 85,259)( 86,261)( 87,260)( 88,256)
( 89,258)( 90,257)( 91,244)( 92,246)( 93,245)( 94,250)( 95,252)( 96,251)
( 97,247)( 98,249)( 99,248)(100,262)(101,264)(102,263)(103,268)(104,270)
(105,269)(106,265)(107,267)(108,266)(109,280)(110,282)(111,281)(112,286)
(113,288)(114,287)(115,283)(116,285)(117,284)(118,271)(119,273)(120,272)
(121,277)(122,279)(123,278)(124,274)(125,276)(126,275)(127,289)(128,291)
(129,290)(130,295)(131,297)(132,296)(133,292)(134,294)(135,293)(136,307)
(137,309)(138,308)(139,313)(140,315)(141,314)(142,310)(143,312)(144,311)
(145,298)(146,300)(147,299)(148,304)(149,306)(150,305)(151,301)(152,303)
(153,302)(154,316)(155,318)(156,317)(157,322)(158,324)(159,323)(160,319)
(161,321)(162,320);;
s2 := ( 1, 28)( 2, 29)( 3, 30)( 4, 34)( 5, 35)( 6, 36)( 7, 31)( 8, 32)
( 9, 33)( 10, 40)( 11, 41)( 12, 42)( 13, 37)( 14, 38)( 15, 39)( 16, 43)
( 17, 44)( 18, 45)( 19, 52)( 20, 53)( 21, 54)( 22, 49)( 23, 50)( 24, 51)
( 25, 46)( 26, 47)( 27, 48)( 58, 61)( 59, 62)( 60, 63)( 64, 67)( 65, 68)
( 66, 69)( 73, 79)( 74, 80)( 75, 81)( 82,109)( 83,110)( 84,111)( 85,115)
( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)( 91,121)( 92,122)( 93,123)
( 94,118)( 95,119)( 96,120)( 97,124)( 98,125)( 99,126)(100,133)(101,134)
(102,135)(103,130)(104,131)(105,132)(106,127)(107,128)(108,129)(139,142)
(140,143)(141,144)(145,148)(146,149)(147,150)(154,160)(155,161)(156,162)
(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)
(171,195)(172,202)(173,203)(174,204)(175,199)(176,200)(177,201)(178,205)
(179,206)(180,207)(181,214)(182,215)(183,216)(184,211)(185,212)(186,213)
(187,208)(188,209)(189,210)(220,223)(221,224)(222,225)(226,229)(227,230)
(228,231)(235,241)(236,242)(237,243)(244,271)(245,272)(246,273)(247,277)
(248,278)(249,279)(250,274)(251,275)(252,276)(253,283)(254,284)(255,285)
(256,280)(257,281)(258,282)(259,286)(260,287)(261,288)(262,295)(263,296)
(264,297)(265,292)(266,293)(267,294)(268,289)(269,290)(270,291)(301,304)
(302,305)(303,306)(307,310)(308,311)(309,312)(316,322)(317,323)(318,324);;
s3 := ( 4, 8)( 5, 9)( 6, 7)( 13, 17)( 14, 18)( 15, 16)( 22, 26)( 23, 27)
( 24, 25)( 28, 55)( 29, 56)( 30, 57)( 31, 62)( 32, 63)( 33, 61)( 34, 60)
( 35, 58)( 36, 59)( 37, 64)( 38, 65)( 39, 66)( 40, 71)( 41, 72)( 42, 70)
( 43, 69)( 44, 67)( 45, 68)( 46, 73)( 47, 74)( 48, 75)( 49, 80)( 50, 81)
( 51, 79)( 52, 78)( 53, 76)( 54, 77)( 85, 89)( 86, 90)( 87, 88)( 94, 98)
( 95, 99)( 96, 97)(103,107)(104,108)(105,106)(109,136)(110,137)(111,138)
(112,143)(113,144)(114,142)(115,141)(116,139)(117,140)(118,145)(119,146)
(120,147)(121,152)(122,153)(123,151)(124,150)(125,148)(126,149)(127,154)
(128,155)(129,156)(130,161)(131,162)(132,160)(133,159)(134,157)(135,158)
(166,170)(167,171)(168,169)(175,179)(176,180)(177,178)(184,188)(185,189)
(186,187)(190,217)(191,218)(192,219)(193,224)(194,225)(195,223)(196,222)
(197,220)(198,221)(199,226)(200,227)(201,228)(202,233)(203,234)(204,232)
(205,231)(206,229)(207,230)(208,235)(209,236)(210,237)(211,242)(212,243)
(213,241)(214,240)(215,238)(216,239)(247,251)(248,252)(249,250)(256,260)
(257,261)(258,259)(265,269)(266,270)(267,268)(271,298)(272,299)(273,300)
(274,305)(275,306)(276,304)(277,303)(278,301)(279,302)(280,307)(281,308)
(282,309)(283,314)(284,315)(285,313)(286,312)(287,310)(288,311)(289,316)
(290,317)(291,318)(292,323)(293,324)(294,322)(295,321)(296,319)(297,320);;
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*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(324)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 10, 19)( 11, 21)( 12, 20)
( 13, 25)( 14, 27)( 15, 26)( 16, 22)( 17, 24)( 18, 23)( 29, 30)( 31, 34)
( 32, 36)( 33, 35)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)
( 43, 49)( 44, 51)( 45, 50)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 73)
( 65, 75)( 66, 74)( 67, 79)( 68, 81)( 69, 80)( 70, 76)( 71, 78)( 72, 77)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 91,100)( 92,102)( 93,101)( 94,106)
( 95,108)( 96,107)( 97,103)( 98,105)( 99,104)(110,111)(112,115)(113,117)
(114,116)(118,127)(119,129)(120,128)(121,133)(122,135)(123,134)(124,130)
(125,132)(126,131)(137,138)(139,142)(140,144)(141,143)(145,154)(146,156)
(147,155)(148,160)(149,162)(150,161)(151,157)(152,159)(153,158)(163,244)
(164,246)(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)(171,248)
(172,262)(173,264)(174,263)(175,268)(176,270)(177,269)(178,265)(179,267)
(180,266)(181,253)(182,255)(183,254)(184,259)(185,261)(186,260)(187,256)
(188,258)(189,257)(190,271)(191,273)(192,272)(193,277)(194,279)(195,278)
(196,274)(197,276)(198,275)(199,289)(200,291)(201,290)(202,295)(203,297)
(204,296)(205,292)(206,294)(207,293)(208,280)(209,282)(210,281)(211,286)
(212,288)(213,287)(214,283)(215,285)(216,284)(217,298)(218,300)(219,299)
(220,304)(221,306)(222,305)(223,301)(224,303)(225,302)(226,316)(227,318)
(228,317)(229,322)(230,324)(231,323)(232,319)(233,321)(234,320)(235,307)
(236,309)(237,308)(238,313)(239,315)(240,314)(241,310)(242,312)(243,311);
s1 := Sym(324)!( 1,172)( 2,174)( 3,173)( 4,178)( 5,180)( 6,179)( 7,175)
( 8,177)( 9,176)( 10,163)( 11,165)( 12,164)( 13,169)( 14,171)( 15,170)
( 16,166)( 17,168)( 18,167)( 19,181)( 20,183)( 21,182)( 22,187)( 23,189)
( 24,188)( 25,184)( 26,186)( 27,185)( 28,199)( 29,201)( 30,200)( 31,205)
( 32,207)( 33,206)( 34,202)( 35,204)( 36,203)( 37,190)( 38,192)( 39,191)
( 40,196)( 41,198)( 42,197)( 43,193)( 44,195)( 45,194)( 46,208)( 47,210)
( 48,209)( 49,214)( 50,216)( 51,215)( 52,211)( 53,213)( 54,212)( 55,226)
( 56,228)( 57,227)( 58,232)( 59,234)( 60,233)( 61,229)( 62,231)( 63,230)
( 64,217)( 65,219)( 66,218)( 67,223)( 68,225)( 69,224)( 70,220)( 71,222)
( 72,221)( 73,235)( 74,237)( 75,236)( 76,241)( 77,243)( 78,242)( 79,238)
( 80,240)( 81,239)( 82,253)( 83,255)( 84,254)( 85,259)( 86,261)( 87,260)
( 88,256)( 89,258)( 90,257)( 91,244)( 92,246)( 93,245)( 94,250)( 95,252)
( 96,251)( 97,247)( 98,249)( 99,248)(100,262)(101,264)(102,263)(103,268)
(104,270)(105,269)(106,265)(107,267)(108,266)(109,280)(110,282)(111,281)
(112,286)(113,288)(114,287)(115,283)(116,285)(117,284)(118,271)(119,273)
(120,272)(121,277)(122,279)(123,278)(124,274)(125,276)(126,275)(127,289)
(128,291)(129,290)(130,295)(131,297)(132,296)(133,292)(134,294)(135,293)
(136,307)(137,309)(138,308)(139,313)(140,315)(141,314)(142,310)(143,312)
(144,311)(145,298)(146,300)(147,299)(148,304)(149,306)(150,305)(151,301)
(152,303)(153,302)(154,316)(155,318)(156,317)(157,322)(158,324)(159,323)
(160,319)(161,321)(162,320);
s2 := Sym(324)!( 1, 28)( 2, 29)( 3, 30)( 4, 34)( 5, 35)( 6, 36)( 7, 31)
( 8, 32)( 9, 33)( 10, 40)( 11, 41)( 12, 42)( 13, 37)( 14, 38)( 15, 39)
( 16, 43)( 17, 44)( 18, 45)( 19, 52)( 20, 53)( 21, 54)( 22, 49)( 23, 50)
( 24, 51)( 25, 46)( 26, 47)( 27, 48)( 58, 61)( 59, 62)( 60, 63)( 64, 67)
( 65, 68)( 66, 69)( 73, 79)( 74, 80)( 75, 81)( 82,109)( 83,110)( 84,111)
( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)( 91,121)( 92,122)
( 93,123)( 94,118)( 95,119)( 96,120)( 97,124)( 98,125)( 99,126)(100,133)
(101,134)(102,135)(103,130)(104,131)(105,132)(106,127)(107,128)(108,129)
(139,142)(140,143)(141,144)(145,148)(146,149)(147,150)(154,160)(155,161)
(156,162)(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)
(170,194)(171,195)(172,202)(173,203)(174,204)(175,199)(176,200)(177,201)
(178,205)(179,206)(180,207)(181,214)(182,215)(183,216)(184,211)(185,212)
(186,213)(187,208)(188,209)(189,210)(220,223)(221,224)(222,225)(226,229)
(227,230)(228,231)(235,241)(236,242)(237,243)(244,271)(245,272)(246,273)
(247,277)(248,278)(249,279)(250,274)(251,275)(252,276)(253,283)(254,284)
(255,285)(256,280)(257,281)(258,282)(259,286)(260,287)(261,288)(262,295)
(263,296)(264,297)(265,292)(266,293)(267,294)(268,289)(269,290)(270,291)
(301,304)(302,305)(303,306)(307,310)(308,311)(309,312)(316,322)(317,323)
(318,324);
s3 := Sym(324)!( 4, 8)( 5, 9)( 6, 7)( 13, 17)( 14, 18)( 15, 16)( 22, 26)
( 23, 27)( 24, 25)( 28, 55)( 29, 56)( 30, 57)( 31, 62)( 32, 63)( 33, 61)
( 34, 60)( 35, 58)( 36, 59)( 37, 64)( 38, 65)( 39, 66)( 40, 71)( 41, 72)
( 42, 70)( 43, 69)( 44, 67)( 45, 68)( 46, 73)( 47, 74)( 48, 75)( 49, 80)
( 50, 81)( 51, 79)( 52, 78)( 53, 76)( 54, 77)( 85, 89)( 86, 90)( 87, 88)
( 94, 98)( 95, 99)( 96, 97)(103,107)(104,108)(105,106)(109,136)(110,137)
(111,138)(112,143)(113,144)(114,142)(115,141)(116,139)(117,140)(118,145)
(119,146)(120,147)(121,152)(122,153)(123,151)(124,150)(125,148)(126,149)
(127,154)(128,155)(129,156)(130,161)(131,162)(132,160)(133,159)(134,157)
(135,158)(166,170)(167,171)(168,169)(175,179)(176,180)(177,178)(184,188)
(185,189)(186,187)(190,217)(191,218)(192,219)(193,224)(194,225)(195,223)
(196,222)(197,220)(198,221)(199,226)(200,227)(201,228)(202,233)(203,234)
(204,232)(205,231)(206,229)(207,230)(208,235)(209,236)(210,237)(211,242)
(212,243)(213,241)(214,240)(215,238)(216,239)(247,251)(248,252)(249,250)
(256,260)(257,261)(258,259)(265,269)(266,270)(267,268)(271,298)(272,299)
(273,300)(274,305)(275,306)(276,304)(277,303)(278,301)(279,302)(280,307)
(281,308)(282,309)(283,314)(284,315)(285,313)(286,312)(287,310)(288,311)
(289,316)(290,317)(291,318)(292,323)(293,324)(294,322)(295,321)(296,319)
(297,320);
poly := sub<Sym(324)|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*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope