Polytope of Type {12,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18}*1296i
if this polytope has a name.
Group : SmallGroup(1296,1786)
Rank : 3
Schlafli Type : {12,18}
Number of vertices, edges, etc : 36, 324, 54
Order of s0s1s2 : 3
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {12,6}*432d
4-fold quotients : {6,18}*324c
9-fold quotients : {12,6}*144d
12-fold quotients : {6,6}*108
27-fold quotients : {4,6}*48b
54-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 31)( 14, 32)
( 15, 29)( 16, 30)( 17, 35)( 18, 36)( 19, 33)( 20, 34)( 21, 27)( 22, 28)
( 23, 25)( 24, 26)( 37, 75)( 38, 76)( 39, 73)( 40, 74)( 41, 79)( 42, 80)
( 43, 77)( 44, 78)( 45, 83)( 46, 84)( 47, 81)( 48, 82)( 49,103)( 50,104)
( 51,101)( 52,102)( 53,107)( 54,108)( 55,105)( 56,106)( 57, 99)( 58,100)
( 59, 97)( 60, 98)( 61, 95)( 62, 96)( 63, 93)( 64, 94)( 65, 87)( 66, 88)
( 67, 85)( 68, 86)( 69, 91)( 70, 92)( 71, 89)( 72, 90)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,139)(122,140)(123,137)(124,138)
(125,143)(126,144)(127,141)(128,142)(129,135)(130,136)(131,133)(132,134)
(145,183)(146,184)(147,181)(148,182)(149,187)(150,188)(151,185)(152,186)
(153,191)(154,192)(155,189)(156,190)(157,211)(158,212)(159,209)(160,210)
(161,215)(162,216)(163,213)(164,214)(165,207)(166,208)(167,205)(168,206)
(169,203)(170,204)(171,201)(172,202)(173,195)(174,196)(175,193)(176,194)
(177,199)(178,200)(179,197)(180,198)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,247)(230,248)(231,245)(232,246)(233,251)(234,252)
(235,249)(236,250)(237,243)(238,244)(239,241)(240,242)(253,291)(254,292)
(255,289)(256,290)(257,295)(258,296)(259,293)(260,294)(261,299)(262,300)
(263,297)(264,298)(265,319)(266,320)(267,317)(268,318)(269,323)(270,324)
(271,321)(272,322)(273,315)(274,316)(275,313)(276,314)(277,311)(278,312)
(279,309)(280,310)(281,303)(282,304)(283,301)(284,302)(285,307)(286,308)
(287,305)(288,306);;
s1 := ( 1, 37)( 2, 38)( 3, 40)( 4, 39)( 5, 45)( 6, 46)( 7, 48)( 8, 47)
( 9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 49)( 14, 50)( 15, 52)( 16, 51)
( 17, 57)( 18, 58)( 19, 60)( 20, 59)( 21, 53)( 22, 54)( 23, 56)( 24, 55)
( 25, 61)( 26, 62)( 27, 64)( 28, 63)( 29, 69)( 30, 70)( 31, 72)( 32, 71)
( 33, 65)( 34, 66)( 35, 68)( 36, 67)( 75, 76)( 77, 81)( 78, 82)( 79, 84)
( 80, 83)( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)( 99,100)(101,105)
(102,106)(103,108)(104,107)(109,261)(110,262)(111,264)(112,263)(113,257)
(114,258)(115,260)(116,259)(117,253)(118,254)(119,256)(120,255)(121,273)
(122,274)(123,276)(124,275)(125,269)(126,270)(127,272)(128,271)(129,265)
(130,266)(131,268)(132,267)(133,285)(134,286)(135,288)(136,287)(137,281)
(138,282)(139,284)(140,283)(141,277)(142,278)(143,280)(144,279)(145,225)
(146,226)(147,228)(148,227)(149,221)(150,222)(151,224)(152,223)(153,217)
(154,218)(155,220)(156,219)(157,237)(158,238)(159,240)(160,239)(161,233)
(162,234)(163,236)(164,235)(165,229)(166,230)(167,232)(168,231)(169,249)
(170,250)(171,252)(172,251)(173,245)(174,246)(175,248)(176,247)(177,241)
(178,242)(179,244)(180,243)(181,297)(182,298)(183,300)(184,299)(185,293)
(186,294)(187,296)(188,295)(189,289)(190,290)(191,292)(192,291)(193,309)
(194,310)(195,312)(196,311)(197,305)(198,306)(199,308)(200,307)(201,301)
(202,302)(203,304)(204,303)(205,321)(206,322)(207,324)(208,323)(209,317)
(210,318)(211,320)(212,319)(213,313)(214,314)(215,316)(216,315);;
s2 := ( 1,109)( 2,112)( 3,111)( 4,110)( 5,117)( 6,120)( 7,119)( 8,118)
( 9,113)( 10,116)( 11,115)( 12,114)( 13,133)( 14,136)( 15,135)( 16,134)
( 17,141)( 18,144)( 19,143)( 20,142)( 21,137)( 22,140)( 23,139)( 24,138)
( 25,121)( 26,124)( 27,123)( 28,122)( 29,129)( 30,132)( 31,131)( 32,130)
( 33,125)( 34,128)( 35,127)( 36,126)( 37,157)( 38,160)( 39,159)( 40,158)
( 41,165)( 42,168)( 43,167)( 44,166)( 45,161)( 46,164)( 47,163)( 48,162)
( 49,145)( 50,148)( 51,147)( 52,146)( 53,153)( 54,156)( 55,155)( 56,154)
( 57,149)( 58,152)( 59,151)( 60,150)( 61,169)( 62,172)( 63,171)( 64,170)
( 65,177)( 66,180)( 67,179)( 68,178)( 69,173)( 70,176)( 71,175)( 72,174)
( 73,209)( 74,212)( 75,211)( 76,210)( 77,205)( 78,208)( 79,207)( 80,206)
( 81,213)( 82,216)( 83,215)( 84,214)( 85,197)( 86,200)( 87,199)( 88,198)
( 89,193)( 90,196)( 91,195)( 92,194)( 93,201)( 94,204)( 95,203)( 96,202)
( 97,185)( 98,188)( 99,187)(100,186)(101,181)(102,184)(103,183)(104,182)
(105,189)(106,192)(107,191)(108,190)(217,225)(218,228)(219,227)(220,226)
(222,224)(229,249)(230,252)(231,251)(232,250)(233,245)(234,248)(235,247)
(236,246)(237,241)(238,244)(239,243)(240,242)(253,273)(254,276)(255,275)
(256,274)(257,269)(258,272)(259,271)(260,270)(261,265)(262,268)(263,267)
(264,266)(277,285)(278,288)(279,287)(280,286)(282,284)(289,313)(290,316)
(291,315)(292,314)(293,321)(294,324)(295,323)(296,322)(297,317)(298,320)
(299,319)(300,318)(302,304)(305,309)(306,312)(307,311)(308,310);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s0*s1*s2*s0*s1*s2,
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*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(324)!( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 31)
( 14, 32)( 15, 29)( 16, 30)( 17, 35)( 18, 36)( 19, 33)( 20, 34)( 21, 27)
( 22, 28)( 23, 25)( 24, 26)( 37, 75)( 38, 76)( 39, 73)( 40, 74)( 41, 79)
( 42, 80)( 43, 77)( 44, 78)( 45, 83)( 46, 84)( 47, 81)( 48, 82)( 49,103)
( 50,104)( 51,101)( 52,102)( 53,107)( 54,108)( 55,105)( 56,106)( 57, 99)
( 58,100)( 59, 97)( 60, 98)( 61, 95)( 62, 96)( 63, 93)( 64, 94)( 65, 87)
( 66, 88)( 67, 85)( 68, 86)( 69, 91)( 70, 92)( 71, 89)( 72, 90)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,139)(122,140)(123,137)
(124,138)(125,143)(126,144)(127,141)(128,142)(129,135)(130,136)(131,133)
(132,134)(145,183)(146,184)(147,181)(148,182)(149,187)(150,188)(151,185)
(152,186)(153,191)(154,192)(155,189)(156,190)(157,211)(158,212)(159,209)
(160,210)(161,215)(162,216)(163,213)(164,214)(165,207)(166,208)(167,205)
(168,206)(169,203)(170,204)(171,201)(172,202)(173,195)(174,196)(175,193)
(176,194)(177,199)(178,200)(179,197)(180,198)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,247)(230,248)(231,245)(232,246)(233,251)
(234,252)(235,249)(236,250)(237,243)(238,244)(239,241)(240,242)(253,291)
(254,292)(255,289)(256,290)(257,295)(258,296)(259,293)(260,294)(261,299)
(262,300)(263,297)(264,298)(265,319)(266,320)(267,317)(268,318)(269,323)
(270,324)(271,321)(272,322)(273,315)(274,316)(275,313)(276,314)(277,311)
(278,312)(279,309)(280,310)(281,303)(282,304)(283,301)(284,302)(285,307)
(286,308)(287,305)(288,306);
s1 := Sym(324)!( 1, 37)( 2, 38)( 3, 40)( 4, 39)( 5, 45)( 6, 46)( 7, 48)
( 8, 47)( 9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 49)( 14, 50)( 15, 52)
( 16, 51)( 17, 57)( 18, 58)( 19, 60)( 20, 59)( 21, 53)( 22, 54)( 23, 56)
( 24, 55)( 25, 61)( 26, 62)( 27, 64)( 28, 63)( 29, 69)( 30, 70)( 31, 72)
( 32, 71)( 33, 65)( 34, 66)( 35, 68)( 36, 67)( 75, 76)( 77, 81)( 78, 82)
( 79, 84)( 80, 83)( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)( 99,100)
(101,105)(102,106)(103,108)(104,107)(109,261)(110,262)(111,264)(112,263)
(113,257)(114,258)(115,260)(116,259)(117,253)(118,254)(119,256)(120,255)
(121,273)(122,274)(123,276)(124,275)(125,269)(126,270)(127,272)(128,271)
(129,265)(130,266)(131,268)(132,267)(133,285)(134,286)(135,288)(136,287)
(137,281)(138,282)(139,284)(140,283)(141,277)(142,278)(143,280)(144,279)
(145,225)(146,226)(147,228)(148,227)(149,221)(150,222)(151,224)(152,223)
(153,217)(154,218)(155,220)(156,219)(157,237)(158,238)(159,240)(160,239)
(161,233)(162,234)(163,236)(164,235)(165,229)(166,230)(167,232)(168,231)
(169,249)(170,250)(171,252)(172,251)(173,245)(174,246)(175,248)(176,247)
(177,241)(178,242)(179,244)(180,243)(181,297)(182,298)(183,300)(184,299)
(185,293)(186,294)(187,296)(188,295)(189,289)(190,290)(191,292)(192,291)
(193,309)(194,310)(195,312)(196,311)(197,305)(198,306)(199,308)(200,307)
(201,301)(202,302)(203,304)(204,303)(205,321)(206,322)(207,324)(208,323)
(209,317)(210,318)(211,320)(212,319)(213,313)(214,314)(215,316)(216,315);
s2 := Sym(324)!( 1,109)( 2,112)( 3,111)( 4,110)( 5,117)( 6,120)( 7,119)
( 8,118)( 9,113)( 10,116)( 11,115)( 12,114)( 13,133)( 14,136)( 15,135)
( 16,134)( 17,141)( 18,144)( 19,143)( 20,142)( 21,137)( 22,140)( 23,139)
( 24,138)( 25,121)( 26,124)( 27,123)( 28,122)( 29,129)( 30,132)( 31,131)
( 32,130)( 33,125)( 34,128)( 35,127)( 36,126)( 37,157)( 38,160)( 39,159)
( 40,158)( 41,165)( 42,168)( 43,167)( 44,166)( 45,161)( 46,164)( 47,163)
( 48,162)( 49,145)( 50,148)( 51,147)( 52,146)( 53,153)( 54,156)( 55,155)
( 56,154)( 57,149)( 58,152)( 59,151)( 60,150)( 61,169)( 62,172)( 63,171)
( 64,170)( 65,177)( 66,180)( 67,179)( 68,178)( 69,173)( 70,176)( 71,175)
( 72,174)( 73,209)( 74,212)( 75,211)( 76,210)( 77,205)( 78,208)( 79,207)
( 80,206)( 81,213)( 82,216)( 83,215)( 84,214)( 85,197)( 86,200)( 87,199)
( 88,198)( 89,193)( 90,196)( 91,195)( 92,194)( 93,201)( 94,204)( 95,203)
( 96,202)( 97,185)( 98,188)( 99,187)(100,186)(101,181)(102,184)(103,183)
(104,182)(105,189)(106,192)(107,191)(108,190)(217,225)(218,228)(219,227)
(220,226)(222,224)(229,249)(230,252)(231,251)(232,250)(233,245)(234,248)
(235,247)(236,246)(237,241)(238,244)(239,243)(240,242)(253,273)(254,276)
(255,275)(256,274)(257,269)(258,272)(259,271)(260,270)(261,265)(262,268)
(263,267)(264,266)(277,285)(278,288)(279,287)(280,286)(282,284)(289,313)
(290,316)(291,315)(292,314)(293,321)(294,324)(295,323)(296,322)(297,317)
(298,320)(299,319)(300,318)(302,304)(305,309)(306,312)(307,311)(308,310);
poly := sub<Sym(324)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s2*s0*s1*s2*s0*s1*s2,
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*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 >;
References : None.
to this polytope