Polytope of Type {3,6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,12}*1296a
if this polytope has a name.
Group : SmallGroup(1296,839)
Rank : 4
Schlafli Type : {3,6,12}
Number of vertices, edges, etc : 3, 27, 108, 36
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 : {3,6,6}*648b
3-fold quotients : {3,6,12}*432a
4-fold quotients : {3,6,3}*324a
6-fold quotients : {3,6,6}*216a
9-fold quotients : {3,2,12}*144
12-fold quotients : {3,6,3}*108
18-fold quotients : {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, 55)( 29, 57)( 30, 56)( 31, 61)
( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)( 39, 65)
( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)( 47, 75)
( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)
(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)
(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)
(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(190,217)(191,219)(192,218)(193,223)(194,225)(195,224)
(196,220)(197,222)(198,221)(199,226)(200,228)(201,227)(202,232)(203,234)
(204,233)(205,229)(206,231)(207,230)(208,235)(209,237)(210,236)(211,241)
(212,243)(213,242)(214,238)(215,240)(216,239)(245,246)(247,250)(248,252)
(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)(266,270)
(267,269)(271,298)(272,300)(273,299)(274,304)(275,306)(276,305)(277,301)
(278,303)(279,302)(280,307)(281,309)(282,308)(283,313)(284,315)(285,314)
(286,310)(287,312)(288,311)(289,316)(290,318)(291,317)(292,322)(293,324)
(294,323)(295,319)(296,321)(297,320);;
s1 := ( 1, 28)( 2, 30)( 3, 29)( 4, 34)( 5, 36)( 6, 35)( 7, 31)( 8, 33)
( 9, 32)( 10, 40)( 11, 42)( 12, 41)( 13, 37)( 14, 39)( 15, 38)( 16, 43)
( 17, 45)( 18, 44)( 19, 53)( 20, 52)( 21, 54)( 22, 50)( 23, 49)( 24, 51)
( 25, 47)( 26, 46)( 27, 48)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 67)
( 65, 69)( 66, 68)( 71, 72)( 73, 80)( 74, 79)( 75, 81)( 76, 77)( 82,109)
( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)( 90,113)
( 91,121)( 92,123)( 93,122)( 94,118)( 95,120)( 96,119)( 97,124)( 98,126)
( 99,125)(100,134)(101,133)(102,135)(103,131)(104,130)(105,132)(106,128)
(107,127)(108,129)(137,138)(139,142)(140,144)(141,143)(145,148)(146,150)
(147,149)(152,153)(154,161)(155,160)(156,162)(157,158)(163,190)(164,192)
(165,191)(166,196)(167,198)(168,197)(169,193)(170,195)(171,194)(172,202)
(173,204)(174,203)(175,199)(176,201)(177,200)(178,205)(179,207)(180,206)
(181,215)(182,214)(183,216)(184,212)(185,211)(186,213)(187,209)(188,208)
(189,210)(218,219)(220,223)(221,225)(222,224)(226,229)(227,231)(228,230)
(233,234)(235,242)(236,241)(237,243)(238,239)(244,271)(245,273)(246,272)
(247,277)(248,279)(249,278)(250,274)(251,276)(252,275)(253,283)(254,285)
(255,284)(256,280)(257,282)(258,281)(259,286)(260,288)(261,287)(262,296)
(263,295)(264,297)(265,293)(266,292)(267,294)(268,290)(269,289)(270,291)
(299,300)(301,304)(302,306)(303,305)(307,310)(308,312)(309,311)(314,315)
(316,323)(317,322)(318,324)(319,320);;
s2 := ( 1, 10)( 2, 11)( 3, 12)( 4, 16)( 5, 17)( 6, 18)( 7, 13)( 8, 14)
( 9, 15)( 22, 25)( 23, 26)( 24, 27)( 28, 37)( 29, 38)( 30, 39)( 31, 43)
( 32, 44)( 33, 45)( 34, 40)( 35, 41)( 36, 42)( 49, 52)( 50, 53)( 51, 54)
( 55, 64)( 56, 65)( 57, 66)( 58, 70)( 59, 71)( 60, 72)( 61, 67)( 62, 68)
( 63, 69)( 76, 79)( 77, 80)( 78, 81)( 82, 91)( 83, 92)( 84, 93)( 85, 97)
( 86, 98)( 87, 99)( 88, 94)( 89, 95)( 90, 96)(103,106)(104,107)(105,108)
(109,118)(110,119)(111,120)(112,124)(113,125)(114,126)(115,121)(116,122)
(117,123)(130,133)(131,134)(132,135)(136,145)(137,146)(138,147)(139,151)
(140,152)(141,153)(142,148)(143,149)(144,150)(157,160)(158,161)(159,162)
(163,253)(164,254)(165,255)(166,259)(167,260)(168,261)(169,256)(170,257)
(171,258)(172,244)(173,245)(174,246)(175,250)(176,251)(177,252)(178,247)
(179,248)(180,249)(181,262)(182,263)(183,264)(184,268)(185,269)(186,270)
(187,265)(188,266)(189,267)(190,280)(191,281)(192,282)(193,286)(194,287)
(195,288)(196,283)(197,284)(198,285)(199,271)(200,272)(201,273)(202,277)
(203,278)(204,279)(205,274)(206,275)(207,276)(208,289)(209,290)(210,291)
(211,295)(212,296)(213,297)(214,292)(215,293)(216,294)(217,307)(218,308)
(219,309)(220,313)(221,314)(222,315)(223,310)(224,311)(225,312)(226,298)
(227,299)(228,300)(229,304)(230,305)(231,306)(232,301)(233,302)(234,303)
(235,316)(236,317)(237,318)(238,322)(239,323)(240,324)(241,319)(242,320)
(243,321);;
s3 := ( 1,163)( 2,164)( 3,165)( 4,170)( 5,171)( 6,169)( 7,168)( 8,166)
( 9,167)( 10,181)( 11,182)( 12,183)( 13,188)( 14,189)( 15,187)( 16,186)
( 17,184)( 18,185)( 19,172)( 20,173)( 21,174)( 22,179)( 23,180)( 24,178)
( 25,177)( 26,175)( 27,176)( 28,190)( 29,191)( 30,192)( 31,197)( 32,198)
( 33,196)( 34,195)( 35,193)( 36,194)( 37,208)( 38,209)( 39,210)( 40,215)
( 41,216)( 42,214)( 43,213)( 44,211)( 45,212)( 46,199)( 47,200)( 48,201)
( 49,206)( 50,207)( 51,205)( 52,204)( 53,202)( 54,203)( 55,217)( 56,218)
( 57,219)( 58,224)( 59,225)( 60,223)( 61,222)( 62,220)( 63,221)( 64,235)
( 65,236)( 66,237)( 67,242)( 68,243)( 69,241)( 70,240)( 71,238)( 72,239)
( 73,226)( 74,227)( 75,228)( 76,233)( 77,234)( 78,232)( 79,231)( 80,229)
( 81,230)( 82,244)( 83,245)( 84,246)( 85,251)( 86,252)( 87,250)( 88,249)
( 89,247)( 90,248)( 91,262)( 92,263)( 93,264)( 94,269)( 95,270)( 96,268)
( 97,267)( 98,265)( 99,266)(100,253)(101,254)(102,255)(103,260)(104,261)
(105,259)(106,258)(107,256)(108,257)(109,271)(110,272)(111,273)(112,278)
(113,279)(114,277)(115,276)(116,274)(117,275)(118,289)(119,290)(120,291)
(121,296)(122,297)(123,295)(124,294)(125,292)(126,293)(127,280)(128,281)
(129,282)(130,287)(131,288)(132,286)(133,285)(134,283)(135,284)(136,298)
(137,299)(138,300)(139,305)(140,306)(141,304)(142,303)(143,301)(144,302)
(145,316)(146,317)(147,318)(148,323)(149,324)(150,322)(151,321)(152,319)
(153,320)(154,307)(155,308)(156,309)(157,314)(158,315)(159,313)(160,312)
(161,310)(162,311);;
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,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
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, 55)( 29, 57)( 30, 56)
( 31, 61)( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)
( 39, 65)( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)
( 47, 75)( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)
(113,144)(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)
(121,151)(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)
(129,155)(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(164,165)
(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)
(184,187)(185,189)(186,188)(190,217)(191,219)(192,218)(193,223)(194,225)
(195,224)(196,220)(197,222)(198,221)(199,226)(200,228)(201,227)(202,232)
(203,234)(204,233)(205,229)(206,231)(207,230)(208,235)(209,237)(210,236)
(211,241)(212,243)(213,242)(214,238)(215,240)(216,239)(245,246)(247,250)
(248,252)(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)
(266,270)(267,269)(271,298)(272,300)(273,299)(274,304)(275,306)(276,305)
(277,301)(278,303)(279,302)(280,307)(281,309)(282,308)(283,313)(284,315)
(285,314)(286,310)(287,312)(288,311)(289,316)(290,318)(291,317)(292,322)
(293,324)(294,323)(295,319)(296,321)(297,320);
s1 := Sym(324)!( 1, 28)( 2, 30)( 3, 29)( 4, 34)( 5, 36)( 6, 35)( 7, 31)
( 8, 33)( 9, 32)( 10, 40)( 11, 42)( 12, 41)( 13, 37)( 14, 39)( 15, 38)
( 16, 43)( 17, 45)( 18, 44)( 19, 53)( 20, 52)( 21, 54)( 22, 50)( 23, 49)
( 24, 51)( 25, 47)( 26, 46)( 27, 48)( 56, 57)( 58, 61)( 59, 63)( 60, 62)
( 64, 67)( 65, 69)( 66, 68)( 71, 72)( 73, 80)( 74, 79)( 75, 81)( 76, 77)
( 82,109)( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)
( 90,113)( 91,121)( 92,123)( 93,122)( 94,118)( 95,120)( 96,119)( 97,124)
( 98,126)( 99,125)(100,134)(101,133)(102,135)(103,131)(104,130)(105,132)
(106,128)(107,127)(108,129)(137,138)(139,142)(140,144)(141,143)(145,148)
(146,150)(147,149)(152,153)(154,161)(155,160)(156,162)(157,158)(163,190)
(164,192)(165,191)(166,196)(167,198)(168,197)(169,193)(170,195)(171,194)
(172,202)(173,204)(174,203)(175,199)(176,201)(177,200)(178,205)(179,207)
(180,206)(181,215)(182,214)(183,216)(184,212)(185,211)(186,213)(187,209)
(188,208)(189,210)(218,219)(220,223)(221,225)(222,224)(226,229)(227,231)
(228,230)(233,234)(235,242)(236,241)(237,243)(238,239)(244,271)(245,273)
(246,272)(247,277)(248,279)(249,278)(250,274)(251,276)(252,275)(253,283)
(254,285)(255,284)(256,280)(257,282)(258,281)(259,286)(260,288)(261,287)
(262,296)(263,295)(264,297)(265,293)(266,292)(267,294)(268,290)(269,289)
(270,291)(299,300)(301,304)(302,306)(303,305)(307,310)(308,312)(309,311)
(314,315)(316,323)(317,322)(318,324)(319,320);
s2 := Sym(324)!( 1, 10)( 2, 11)( 3, 12)( 4, 16)( 5, 17)( 6, 18)( 7, 13)
( 8, 14)( 9, 15)( 22, 25)( 23, 26)( 24, 27)( 28, 37)( 29, 38)( 30, 39)
( 31, 43)( 32, 44)( 33, 45)( 34, 40)( 35, 41)( 36, 42)( 49, 52)( 50, 53)
( 51, 54)( 55, 64)( 56, 65)( 57, 66)( 58, 70)( 59, 71)( 60, 72)( 61, 67)
( 62, 68)( 63, 69)( 76, 79)( 77, 80)( 78, 81)( 82, 91)( 83, 92)( 84, 93)
( 85, 97)( 86, 98)( 87, 99)( 88, 94)( 89, 95)( 90, 96)(103,106)(104,107)
(105,108)(109,118)(110,119)(111,120)(112,124)(113,125)(114,126)(115,121)
(116,122)(117,123)(130,133)(131,134)(132,135)(136,145)(137,146)(138,147)
(139,151)(140,152)(141,153)(142,148)(143,149)(144,150)(157,160)(158,161)
(159,162)(163,253)(164,254)(165,255)(166,259)(167,260)(168,261)(169,256)
(170,257)(171,258)(172,244)(173,245)(174,246)(175,250)(176,251)(177,252)
(178,247)(179,248)(180,249)(181,262)(182,263)(183,264)(184,268)(185,269)
(186,270)(187,265)(188,266)(189,267)(190,280)(191,281)(192,282)(193,286)
(194,287)(195,288)(196,283)(197,284)(198,285)(199,271)(200,272)(201,273)
(202,277)(203,278)(204,279)(205,274)(206,275)(207,276)(208,289)(209,290)
(210,291)(211,295)(212,296)(213,297)(214,292)(215,293)(216,294)(217,307)
(218,308)(219,309)(220,313)(221,314)(222,315)(223,310)(224,311)(225,312)
(226,298)(227,299)(228,300)(229,304)(230,305)(231,306)(232,301)(233,302)
(234,303)(235,316)(236,317)(237,318)(238,322)(239,323)(240,324)(241,319)
(242,320)(243,321);
s3 := Sym(324)!( 1,163)( 2,164)( 3,165)( 4,170)( 5,171)( 6,169)( 7,168)
( 8,166)( 9,167)( 10,181)( 11,182)( 12,183)( 13,188)( 14,189)( 15,187)
( 16,186)( 17,184)( 18,185)( 19,172)( 20,173)( 21,174)( 22,179)( 23,180)
( 24,178)( 25,177)( 26,175)( 27,176)( 28,190)( 29,191)( 30,192)( 31,197)
( 32,198)( 33,196)( 34,195)( 35,193)( 36,194)( 37,208)( 38,209)( 39,210)
( 40,215)( 41,216)( 42,214)( 43,213)( 44,211)( 45,212)( 46,199)( 47,200)
( 48,201)( 49,206)( 50,207)( 51,205)( 52,204)( 53,202)( 54,203)( 55,217)
( 56,218)( 57,219)( 58,224)( 59,225)( 60,223)( 61,222)( 62,220)( 63,221)
( 64,235)( 65,236)( 66,237)( 67,242)( 68,243)( 69,241)( 70,240)( 71,238)
( 72,239)( 73,226)( 74,227)( 75,228)( 76,233)( 77,234)( 78,232)( 79,231)
( 80,229)( 81,230)( 82,244)( 83,245)( 84,246)( 85,251)( 86,252)( 87,250)
( 88,249)( 89,247)( 90,248)( 91,262)( 92,263)( 93,264)( 94,269)( 95,270)
( 96,268)( 97,267)( 98,265)( 99,266)(100,253)(101,254)(102,255)(103,260)
(104,261)(105,259)(106,258)(107,256)(108,257)(109,271)(110,272)(111,273)
(112,278)(113,279)(114,277)(115,276)(116,274)(117,275)(118,289)(119,290)
(120,291)(121,296)(122,297)(123,295)(124,294)(125,292)(126,293)(127,280)
(128,281)(129,282)(130,287)(131,288)(132,286)(133,285)(134,283)(135,284)
(136,298)(137,299)(138,300)(139,305)(140,306)(141,304)(142,303)(143,301)
(144,302)(145,316)(146,317)(147,318)(148,323)(149,324)(150,322)(151,321)
(152,319)(153,320)(154,307)(155,308)(156,309)(157,314)(158,315)(159,313)
(160,312)(161,310)(162,311);
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,
s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
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