Polytope of Type {9,6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6,12}*1296a
if this polytope has a name.
Group : SmallGroup(1296,811)
Rank : 4
Schlafli Type : {9,6,12}
Number of vertices, edges, etc : 9, 27, 36, 12
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 : {9,6,6}*648a
3-fold quotients : {9,2,12}*432, {3,6,12}*432a
4-fold quotients : {9,6,3}*324
6-fold quotients : {9,2,6}*216, {3,6,6}*216a
9-fold quotients : {9,2,4}*144, {3,2,12}*144
12-fold quotients : {9,2,3}*108, {3,6,3}*108
18-fold quotients : {9,2,2}*72, {3,2,6}*72
27-fold quotients : {3,2,4}*48
36-fold quotients : {3,2,3}*36
54-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 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);;
s1 := ( 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);;
s2 := ( 1, 10)( 2, 12)( 3, 11)( 4, 13)( 5, 15)( 6, 14)( 7, 16)( 8, 18)
( 9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 40)
( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 47, 48)( 50, 51)( 53, 54)
( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)( 62, 72)
( 63, 71)( 74, 75)( 77, 78)( 80, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 94)
( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(101,102)(104,105)(107,108)
(109,118)(110,120)(111,119)(112,121)(113,123)(114,122)(115,124)(116,126)
(117,125)(128,129)(131,132)(134,135)(136,145)(137,147)(138,146)(139,148)
(140,150)(141,149)(142,151)(143,153)(144,152)(155,156)(158,159)(161,162)
(163,253)(164,255)(165,254)(166,256)(167,258)(168,257)(169,259)(170,261)
(171,260)(172,244)(173,246)(174,245)(175,247)(176,249)(177,248)(178,250)
(179,252)(180,251)(181,262)(182,264)(183,263)(184,265)(185,267)(186,266)
(187,268)(188,270)(189,269)(190,280)(191,282)(192,281)(193,283)(194,285)
(195,284)(196,286)(197,288)(198,287)(199,271)(200,273)(201,272)(202,274)
(203,276)(204,275)(205,277)(206,279)(207,278)(208,289)(209,291)(210,290)
(211,292)(212,294)(213,293)(214,295)(215,297)(216,296)(217,307)(218,309)
(219,308)(220,310)(221,312)(222,311)(223,313)(224,315)(225,314)(226,298)
(227,300)(228,299)(229,301)(230,303)(231,302)(232,304)(233,306)(234,305)
(235,316)(236,318)(237,317)(238,319)(239,321)(240,320)(241,322)(242,324)
(243,323);;
s3 := ( 1,163)( 2,165)( 3,164)( 4,166)( 5,168)( 6,167)( 7,169)( 8,171)
( 9,170)( 10,181)( 11,183)( 12,182)( 13,184)( 14,186)( 15,185)( 16,187)
( 17,189)( 18,188)( 19,172)( 20,174)( 21,173)( 22,175)( 23,177)( 24,176)
( 25,178)( 26,180)( 27,179)( 28,190)( 29,192)( 30,191)( 31,193)( 32,195)
( 33,194)( 34,196)( 35,198)( 36,197)( 37,208)( 38,210)( 39,209)( 40,211)
( 41,213)( 42,212)( 43,214)( 44,216)( 45,215)( 46,199)( 47,201)( 48,200)
( 49,202)( 50,204)( 51,203)( 52,205)( 53,207)( 54,206)( 55,217)( 56,219)
( 57,218)( 58,220)( 59,222)( 60,221)( 61,223)( 62,225)( 63,224)( 64,235)
( 65,237)( 66,236)( 67,238)( 68,240)( 69,239)( 70,241)( 71,243)( 72,242)
( 73,226)( 74,228)( 75,227)( 76,229)( 77,231)( 78,230)( 79,232)( 80,234)
( 81,233)( 82,244)( 83,246)( 84,245)( 85,247)( 86,249)( 87,248)( 88,250)
( 89,252)( 90,251)( 91,262)( 92,264)( 93,263)( 94,265)( 95,267)( 96,266)
( 97,268)( 98,270)( 99,269)(100,253)(101,255)(102,254)(103,256)(104,258)
(105,257)(106,259)(107,261)(108,260)(109,271)(110,273)(111,272)(112,274)
(113,276)(114,275)(115,277)(116,279)(117,278)(118,289)(119,291)(120,290)
(121,292)(122,294)(123,293)(124,295)(125,297)(126,296)(127,280)(128,282)
(129,281)(130,283)(131,285)(132,284)(133,286)(134,288)(135,287)(136,298)
(137,300)(138,299)(139,301)(140,303)(141,302)(142,304)(143,306)(144,305)
(145,316)(146,318)(147,317)(148,319)(149,321)(150,320)(151,322)(152,324)
(153,323)(154,307)(155,309)(156,308)(157,310)(158,312)(159,311)(160,313)
(161,315)(162,314);;
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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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);
s1 := 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);
s2 := Sym(324)!( 1, 10)( 2, 12)( 3, 11)( 4, 13)( 5, 15)( 6, 14)( 7, 16)
( 8, 18)( 9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 40)( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 47, 48)( 50, 51)
( 53, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)
( 62, 72)( 63, 71)( 74, 75)( 77, 78)( 80, 81)( 82, 91)( 83, 93)( 84, 92)
( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(101,102)(104,105)
(107,108)(109,118)(110,120)(111,119)(112,121)(113,123)(114,122)(115,124)
(116,126)(117,125)(128,129)(131,132)(134,135)(136,145)(137,147)(138,146)
(139,148)(140,150)(141,149)(142,151)(143,153)(144,152)(155,156)(158,159)
(161,162)(163,253)(164,255)(165,254)(166,256)(167,258)(168,257)(169,259)
(170,261)(171,260)(172,244)(173,246)(174,245)(175,247)(176,249)(177,248)
(178,250)(179,252)(180,251)(181,262)(182,264)(183,263)(184,265)(185,267)
(186,266)(187,268)(188,270)(189,269)(190,280)(191,282)(192,281)(193,283)
(194,285)(195,284)(196,286)(197,288)(198,287)(199,271)(200,273)(201,272)
(202,274)(203,276)(204,275)(205,277)(206,279)(207,278)(208,289)(209,291)
(210,290)(211,292)(212,294)(213,293)(214,295)(215,297)(216,296)(217,307)
(218,309)(219,308)(220,310)(221,312)(222,311)(223,313)(224,315)(225,314)
(226,298)(227,300)(228,299)(229,301)(230,303)(231,302)(232,304)(233,306)
(234,305)(235,316)(236,318)(237,317)(238,319)(239,321)(240,320)(241,322)
(242,324)(243,323);
s3 := Sym(324)!( 1,163)( 2,165)( 3,164)( 4,166)( 5,168)( 6,167)( 7,169)
( 8,171)( 9,170)( 10,181)( 11,183)( 12,182)( 13,184)( 14,186)( 15,185)
( 16,187)( 17,189)( 18,188)( 19,172)( 20,174)( 21,173)( 22,175)( 23,177)
( 24,176)( 25,178)( 26,180)( 27,179)( 28,190)( 29,192)( 30,191)( 31,193)
( 32,195)( 33,194)( 34,196)( 35,198)( 36,197)( 37,208)( 38,210)( 39,209)
( 40,211)( 41,213)( 42,212)( 43,214)( 44,216)( 45,215)( 46,199)( 47,201)
( 48,200)( 49,202)( 50,204)( 51,203)( 52,205)( 53,207)( 54,206)( 55,217)
( 56,219)( 57,218)( 58,220)( 59,222)( 60,221)( 61,223)( 62,225)( 63,224)
( 64,235)( 65,237)( 66,236)( 67,238)( 68,240)( 69,239)( 70,241)( 71,243)
( 72,242)( 73,226)( 74,228)( 75,227)( 76,229)( 77,231)( 78,230)( 79,232)
( 80,234)( 81,233)( 82,244)( 83,246)( 84,245)( 85,247)( 86,249)( 87,248)
( 88,250)( 89,252)( 90,251)( 91,262)( 92,264)( 93,263)( 94,265)( 95,267)
( 96,266)( 97,268)( 98,270)( 99,269)(100,253)(101,255)(102,254)(103,256)
(104,258)(105,257)(106,259)(107,261)(108,260)(109,271)(110,273)(111,272)
(112,274)(113,276)(114,275)(115,277)(116,279)(117,278)(118,289)(119,291)
(120,290)(121,292)(122,294)(123,293)(124,295)(125,297)(126,296)(127,280)
(128,282)(129,281)(130,283)(131,285)(132,284)(133,286)(134,288)(135,287)
(136,298)(137,300)(138,299)(139,301)(140,303)(141,302)(142,304)(143,306)
(144,305)(145,316)(146,318)(147,317)(148,319)(149,321)(150,320)(151,322)
(152,324)(153,323)(154,307)(155,309)(156,308)(157,310)(158,312)(159,311)
(160,313)(161,315)(162,314);
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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope