Polytope of Type {12,6,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,9}*1296a
if this polytope has a name.
Group : SmallGroup(1296,811)
Rank : 4
Schlafli Type : {12,6,9}
Number of vertices, edges, etc : 12, 36, 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 : {6,6,9}*648a
3-fold quotients : {12,2,9}*432, {12,6,3}*432a
4-fold quotients : {3,6,9}*324
6-fold quotients : {6,2,9}*216, {6,6,3}*216a
9-fold quotients : {4,2,9}*144, {12,2,3}*144
12-fold quotients : {3,2,9}*108, {3,6,3}*108
18-fold quotients : {2,2,9}*72, {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)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)
( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)
( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)
(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)
(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)
(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161)
(163,244)(164,246)(165,245)(166,247)(167,249)(168,248)(169,250)(170,252)
(171,251)(172,262)(173,264)(174,263)(175,265)(176,267)(177,266)(178,268)
(179,270)(180,269)(181,253)(182,255)(183,254)(184,256)(185,258)(186,257)
(187,259)(188,261)(189,260)(190,271)(191,273)(192,272)(193,274)(194,276)
(195,275)(196,277)(197,279)(198,278)(199,289)(200,291)(201,290)(202,292)
(203,294)(204,293)(205,295)(206,297)(207,296)(208,280)(209,282)(210,281)
(211,283)(212,285)(213,284)(214,286)(215,288)(216,287)(217,298)(218,300)
(219,299)(220,301)(221,303)(222,302)(223,304)(224,306)(225,305)(226,316)
(227,318)(228,317)(229,319)(230,321)(231,320)(232,322)(233,324)(234,323)
(235,307)(236,309)(237,308)(238,310)(239,312)(240,311)(241,313)(242,315)
(243,314);;
s1 := ( 1,172)( 2,174)( 3,173)( 4,175)( 5,177)( 6,176)( 7,178)( 8,180)
( 9,179)( 10,163)( 11,165)( 12,164)( 13,166)( 14,168)( 15,167)( 16,169)
( 17,171)( 18,170)( 19,181)( 20,183)( 21,182)( 22,184)( 23,186)( 24,185)
( 25,187)( 26,189)( 27,188)( 28,199)( 29,201)( 30,200)( 31,202)( 32,204)
( 33,203)( 34,205)( 35,207)( 36,206)( 37,190)( 38,192)( 39,191)( 40,193)
( 41,195)( 42,194)( 43,196)( 44,198)( 45,197)( 46,208)( 47,210)( 48,209)
( 49,211)( 50,213)( 51,212)( 52,214)( 53,216)( 54,215)( 55,226)( 56,228)
( 57,227)( 58,229)( 59,231)( 60,230)( 61,232)( 62,234)( 63,233)( 64,217)
( 65,219)( 66,218)( 67,220)( 68,222)( 69,221)( 70,223)( 71,225)( 72,224)
( 73,235)( 74,237)( 75,236)( 76,238)( 77,240)( 78,239)( 79,241)( 80,243)
( 81,242)( 82,253)( 83,255)( 84,254)( 85,256)( 86,258)( 87,257)( 88,259)
( 89,261)( 90,260)( 91,244)( 92,246)( 93,245)( 94,247)( 95,249)( 96,248)
( 97,250)( 98,252)( 99,251)(100,262)(101,264)(102,263)(103,265)(104,267)
(105,266)(106,268)(107,270)(108,269)(109,280)(110,282)(111,281)(112,283)
(113,285)(114,284)(115,286)(116,288)(117,287)(118,271)(119,273)(120,272)
(121,274)(122,276)(123,275)(124,277)(125,279)(126,278)(127,289)(128,291)
(129,290)(130,292)(131,294)(132,293)(133,295)(134,297)(135,296)(136,307)
(137,309)(138,308)(139,310)(140,312)(141,311)(142,313)(143,315)(144,314)
(145,298)(146,300)(147,299)(148,301)(149,303)(150,302)(151,304)(152,306)
(153,305)(154,316)(155,318)(156,317)(157,319)(158,321)(159,320)(160,322)
(161,324)(162,323);;
s2 := ( 1, 28)( 2, 30)( 3, 29)( 4, 34)( 5, 36)( 6, 35)( 7, 31)( 8, 33)
( 9, 32)( 10, 38)( 11, 37)( 12, 39)( 13, 44)( 14, 43)( 15, 45)( 16, 41)
( 17, 40)( 18, 42)( 19, 48)( 20, 47)( 21, 46)( 22, 54)( 23, 53)( 24, 52)
( 25, 51)( 26, 50)( 27, 49)( 55, 61)( 56, 63)( 57, 62)( 59, 60)( 64, 71)
( 65, 70)( 66, 72)( 67, 68)( 73, 81)( 74, 80)( 75, 79)( 76, 78)( 82,109)
( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)( 90,113)
( 91,119)( 92,118)( 93,120)( 94,125)( 95,124)( 96,126)( 97,122)( 98,121)
( 99,123)(100,129)(101,128)(102,127)(103,135)(104,134)(105,133)(106,132)
(107,131)(108,130)(136,142)(137,144)(138,143)(140,141)(145,152)(146,151)
(147,153)(148,149)(154,162)(155,161)(156,160)(157,159)(163,190)(164,192)
(165,191)(166,196)(167,198)(168,197)(169,193)(170,195)(171,194)(172,200)
(173,199)(174,201)(175,206)(176,205)(177,207)(178,203)(179,202)(180,204)
(181,210)(182,209)(183,208)(184,216)(185,215)(186,214)(187,213)(188,212)
(189,211)(217,223)(218,225)(219,224)(221,222)(226,233)(227,232)(228,234)
(229,230)(235,243)(236,242)(237,241)(238,240)(244,271)(245,273)(246,272)
(247,277)(248,279)(249,278)(250,274)(251,276)(252,275)(253,281)(254,280)
(255,282)(256,287)(257,286)(258,288)(259,284)(260,283)(261,285)(262,291)
(263,290)(264,289)(265,297)(266,296)(267,295)(268,294)(269,293)(270,292)
(298,304)(299,306)(300,305)(302,303)(307,314)(308,313)(309,315)(310,311)
(316,324)(317,323)(318,322)(319,321);;
s3 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 61)( 29, 63)( 30, 62)( 31, 58)
( 32, 60)( 33, 59)( 34, 55)( 35, 57)( 36, 56)( 37, 70)( 38, 72)( 39, 71)
( 40, 67)( 41, 69)( 42, 68)( 43, 64)( 44, 66)( 45, 65)( 46, 79)( 47, 81)
( 48, 80)( 49, 76)( 50, 78)( 51, 77)( 52, 73)( 53, 75)( 54, 74)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,142)(110,144)(111,143)(112,139)(113,141)
(114,140)(115,136)(116,138)(117,137)(118,151)(119,153)(120,152)(121,148)
(122,150)(123,149)(124,145)(125,147)(126,146)(127,160)(128,162)(129,161)
(130,157)(131,159)(132,158)(133,154)(134,156)(135,155)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(190,223)(191,225)(192,224)(193,220)(194,222)(195,221)
(196,217)(197,219)(198,218)(199,232)(200,234)(201,233)(202,229)(203,231)
(204,230)(205,226)(206,228)(207,227)(208,241)(209,243)(210,242)(211,238)
(212,240)(213,239)(214,235)(215,237)(216,236)(245,246)(247,250)(248,252)
(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)(266,270)
(267,269)(271,304)(272,306)(273,305)(274,301)(275,303)(276,302)(277,298)
(278,300)(279,299)(280,313)(281,315)(282,314)(283,310)(284,312)(285,311)
(286,307)(287,309)(288,308)(289,322)(290,324)(291,323)(292,319)(293,321)
(294,320)(295,316)(296,318)(297,317);;
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*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)
( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)
( 91,100)( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)
( 99,107)(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)
(122,132)(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)
(145,154)(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)
(153,161)(163,244)(164,246)(165,245)(166,247)(167,249)(168,248)(169,250)
(170,252)(171,251)(172,262)(173,264)(174,263)(175,265)(176,267)(177,266)
(178,268)(179,270)(180,269)(181,253)(182,255)(183,254)(184,256)(185,258)
(186,257)(187,259)(188,261)(189,260)(190,271)(191,273)(192,272)(193,274)
(194,276)(195,275)(196,277)(197,279)(198,278)(199,289)(200,291)(201,290)
(202,292)(203,294)(204,293)(205,295)(206,297)(207,296)(208,280)(209,282)
(210,281)(211,283)(212,285)(213,284)(214,286)(215,288)(216,287)(217,298)
(218,300)(219,299)(220,301)(221,303)(222,302)(223,304)(224,306)(225,305)
(226,316)(227,318)(228,317)(229,319)(230,321)(231,320)(232,322)(233,324)
(234,323)(235,307)(236,309)(237,308)(238,310)(239,312)(240,311)(241,313)
(242,315)(243,314);
s1 := Sym(324)!( 1,172)( 2,174)( 3,173)( 4,175)( 5,177)( 6,176)( 7,178)
( 8,180)( 9,179)( 10,163)( 11,165)( 12,164)( 13,166)( 14,168)( 15,167)
( 16,169)( 17,171)( 18,170)( 19,181)( 20,183)( 21,182)( 22,184)( 23,186)
( 24,185)( 25,187)( 26,189)( 27,188)( 28,199)( 29,201)( 30,200)( 31,202)
( 32,204)( 33,203)( 34,205)( 35,207)( 36,206)( 37,190)( 38,192)( 39,191)
( 40,193)( 41,195)( 42,194)( 43,196)( 44,198)( 45,197)( 46,208)( 47,210)
( 48,209)( 49,211)( 50,213)( 51,212)( 52,214)( 53,216)( 54,215)( 55,226)
( 56,228)( 57,227)( 58,229)( 59,231)( 60,230)( 61,232)( 62,234)( 63,233)
( 64,217)( 65,219)( 66,218)( 67,220)( 68,222)( 69,221)( 70,223)( 71,225)
( 72,224)( 73,235)( 74,237)( 75,236)( 76,238)( 77,240)( 78,239)( 79,241)
( 80,243)( 81,242)( 82,253)( 83,255)( 84,254)( 85,256)( 86,258)( 87,257)
( 88,259)( 89,261)( 90,260)( 91,244)( 92,246)( 93,245)( 94,247)( 95,249)
( 96,248)( 97,250)( 98,252)( 99,251)(100,262)(101,264)(102,263)(103,265)
(104,267)(105,266)(106,268)(107,270)(108,269)(109,280)(110,282)(111,281)
(112,283)(113,285)(114,284)(115,286)(116,288)(117,287)(118,271)(119,273)
(120,272)(121,274)(122,276)(123,275)(124,277)(125,279)(126,278)(127,289)
(128,291)(129,290)(130,292)(131,294)(132,293)(133,295)(134,297)(135,296)
(136,307)(137,309)(138,308)(139,310)(140,312)(141,311)(142,313)(143,315)
(144,314)(145,298)(146,300)(147,299)(148,301)(149,303)(150,302)(151,304)
(152,306)(153,305)(154,316)(155,318)(156,317)(157,319)(158,321)(159,320)
(160,322)(161,324)(162,323);
s2 := Sym(324)!( 1, 28)( 2, 30)( 3, 29)( 4, 34)( 5, 36)( 6, 35)( 7, 31)
( 8, 33)( 9, 32)( 10, 38)( 11, 37)( 12, 39)( 13, 44)( 14, 43)( 15, 45)
( 16, 41)( 17, 40)( 18, 42)( 19, 48)( 20, 47)( 21, 46)( 22, 54)( 23, 53)
( 24, 52)( 25, 51)( 26, 50)( 27, 49)( 55, 61)( 56, 63)( 57, 62)( 59, 60)
( 64, 71)( 65, 70)( 66, 72)( 67, 68)( 73, 81)( 74, 80)( 75, 79)( 76, 78)
( 82,109)( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)
( 90,113)( 91,119)( 92,118)( 93,120)( 94,125)( 95,124)( 96,126)( 97,122)
( 98,121)( 99,123)(100,129)(101,128)(102,127)(103,135)(104,134)(105,133)
(106,132)(107,131)(108,130)(136,142)(137,144)(138,143)(140,141)(145,152)
(146,151)(147,153)(148,149)(154,162)(155,161)(156,160)(157,159)(163,190)
(164,192)(165,191)(166,196)(167,198)(168,197)(169,193)(170,195)(171,194)
(172,200)(173,199)(174,201)(175,206)(176,205)(177,207)(178,203)(179,202)
(180,204)(181,210)(182,209)(183,208)(184,216)(185,215)(186,214)(187,213)
(188,212)(189,211)(217,223)(218,225)(219,224)(221,222)(226,233)(227,232)
(228,234)(229,230)(235,243)(236,242)(237,241)(238,240)(244,271)(245,273)
(246,272)(247,277)(248,279)(249,278)(250,274)(251,276)(252,275)(253,281)
(254,280)(255,282)(256,287)(257,286)(258,288)(259,284)(260,283)(261,285)
(262,291)(263,290)(264,289)(265,297)(266,296)(267,295)(268,294)(269,293)
(270,292)(298,304)(299,306)(300,305)(302,303)(307,314)(308,313)(309,315)
(310,311)(316,324)(317,323)(318,322)(319,321);
s3 := Sym(324)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 61)( 29, 63)( 30, 62)
( 31, 58)( 32, 60)( 33, 59)( 34, 55)( 35, 57)( 36, 56)( 37, 70)( 38, 72)
( 39, 71)( 40, 67)( 41, 69)( 42, 68)( 43, 64)( 44, 66)( 45, 65)( 46, 79)
( 47, 81)( 48, 80)( 49, 76)( 50, 78)( 51, 77)( 52, 73)( 53, 75)( 54, 74)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,142)(110,144)(111,143)(112,139)
(113,141)(114,140)(115,136)(116,138)(117,137)(118,151)(119,153)(120,152)
(121,148)(122,150)(123,149)(124,145)(125,147)(126,146)(127,160)(128,162)
(129,161)(130,157)(131,159)(132,158)(133,154)(134,156)(135,155)(164,165)
(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)
(184,187)(185,189)(186,188)(190,223)(191,225)(192,224)(193,220)(194,222)
(195,221)(196,217)(197,219)(198,218)(199,232)(200,234)(201,233)(202,229)
(203,231)(204,230)(205,226)(206,228)(207,227)(208,241)(209,243)(210,242)
(211,238)(212,240)(213,239)(214,235)(215,237)(216,236)(245,246)(247,250)
(248,252)(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)
(266,270)(267,269)(271,304)(272,306)(273,305)(274,301)(275,303)(276,302)
(277,298)(278,300)(279,299)(280,313)(281,315)(282,314)(283,310)(284,312)
(285,311)(286,307)(287,309)(288,308)(289,322)(290,324)(291,323)(292,319)
(293,321)(294,320)(295,316)(296,318)(297,317);
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*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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