Polytope of Type {12,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18}*1296c
if this polytope has a name.
Group : SmallGroup(1296,867)
Rank : 3
Schlafli Type : {12,18}
Number of vertices, edges, etc : 36, 324, 54
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
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) :
2-fold quotients : {6,18}*648g
3-fold quotients : {12,6}*432b
4-fold quotients : {6,18}*324c
6-fold quotients : {6,6}*216b
9-fold quotients : {12,6}*144a
12-fold quotients : {6,6}*108
18-fold quotients : {6,6}*72a
27-fold quotients : {12,2}*48, {4,6}*48a
54-fold quotients : {2,6}*24, {6,2}*24
81-fold quotients : {4,2}*16
108-fold quotients : {2,3}*12, {3,2}*12
162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 4, 8)( 5, 9)( 6, 7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)( 14, 27)
( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)( 37, 46)
( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)( 45, 50)
( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)( 68, 81)
( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 85, 89)( 86, 90)( 87, 88)( 91,100)
( 92,101)( 93,102)( 94,107)( 95,108)( 96,106)( 97,105)( 98,103)( 99,104)
(112,116)(113,117)(114,115)(118,127)(119,128)(120,129)(121,134)(122,135)
(123,133)(124,132)(125,130)(126,131)(139,143)(140,144)(141,142)(145,154)
(146,155)(147,156)(148,161)(149,162)(150,160)(151,159)(152,157)(153,158)
(163,244)(164,245)(165,246)(166,251)(167,252)(168,250)(169,249)(170,247)
(171,248)(172,262)(173,263)(174,264)(175,269)(176,270)(177,268)(178,267)
(179,265)(180,266)(181,253)(182,254)(183,255)(184,260)(185,261)(186,259)
(187,258)(188,256)(189,257)(190,271)(191,272)(192,273)(193,278)(194,279)
(195,277)(196,276)(197,274)(198,275)(199,289)(200,290)(201,291)(202,296)
(203,297)(204,295)(205,294)(206,292)(207,293)(208,280)(209,281)(210,282)
(211,287)(212,288)(213,286)(214,285)(215,283)(216,284)(217,298)(218,299)
(219,300)(220,305)(221,306)(222,304)(223,303)(224,301)(225,302)(226,316)
(227,317)(228,318)(229,323)(230,324)(231,322)(232,321)(233,319)(234,320)
(235,307)(236,308)(237,309)(238,314)(239,315)(240,313)(241,312)(242,310)
(243,311);;
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)( 28,228)( 29,227)( 30,226)( 31,231)( 32,230)
( 33,229)( 34,234)( 35,233)( 36,232)( 37,219)( 38,218)( 39,217)( 40,222)
( 41,221)( 42,220)( 43,225)( 44,224)( 45,223)( 46,237)( 47,236)( 48,235)
( 49,240)( 50,239)( 51,238)( 52,243)( 53,242)( 54,241)( 55,201)( 56,200)
( 57,199)( 58,204)( 59,203)( 60,202)( 61,207)( 62,206)( 63,205)( 64,192)
( 65,191)( 66,190)( 67,195)( 68,194)( 69,193)( 70,198)( 71,197)( 72,196)
( 73,210)( 74,209)( 75,208)( 76,213)( 77,212)( 78,211)( 79,216)( 80,215)
( 81,214)( 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)(109,309)(110,308)(111,307)(112,312)
(113,311)(114,310)(115,315)(116,314)(117,313)(118,300)(119,299)(120,298)
(121,303)(122,302)(123,301)(124,306)(125,305)(126,304)(127,318)(128,317)
(129,316)(130,321)(131,320)(132,319)(133,324)(134,323)(135,322)(136,282)
(137,281)(138,280)(139,285)(140,284)(141,283)(142,288)(143,287)(144,286)
(145,273)(146,272)(147,271)(148,276)(149,275)(150,274)(151,279)(152,278)
(153,277)(154,291)(155,290)(156,289)(157,294)(158,293)(159,292)(160,297)
(161,296)(162,295);;
s2 := ( 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)( 55, 57)( 58, 63)( 59, 62)( 60, 61)( 64, 69)
( 65, 68)( 66, 67)( 70, 72)( 73, 79)( 74, 81)( 75, 80)( 77, 78)( 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)(136,138)(139,144)(140,143)(141,142)(145,150)(146,149)
(147,148)(151,153)(154,160)(155,162)(156,161)(158,159)(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)(217,219)(220,225)(221,224)(222,223)(226,231)(227,230)(228,229)
(232,234)(235,241)(236,243)(237,242)(239,240)(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)
(298,300)(301,306)(302,305)(303,304)(307,312)(308,311)(309,310)(313,315)
(316,322)(317,324)(318,323)(320,321);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*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*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(324)!( 4, 8)( 5, 9)( 6, 7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)
( 14, 27)( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)
( 37, 46)( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)
( 45, 50)( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)
( 68, 81)( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 85, 89)( 86, 90)( 87, 88)
( 91,100)( 92,101)( 93,102)( 94,107)( 95,108)( 96,106)( 97,105)( 98,103)
( 99,104)(112,116)(113,117)(114,115)(118,127)(119,128)(120,129)(121,134)
(122,135)(123,133)(124,132)(125,130)(126,131)(139,143)(140,144)(141,142)
(145,154)(146,155)(147,156)(148,161)(149,162)(150,160)(151,159)(152,157)
(153,158)(163,244)(164,245)(165,246)(166,251)(167,252)(168,250)(169,249)
(170,247)(171,248)(172,262)(173,263)(174,264)(175,269)(176,270)(177,268)
(178,267)(179,265)(180,266)(181,253)(182,254)(183,255)(184,260)(185,261)
(186,259)(187,258)(188,256)(189,257)(190,271)(191,272)(192,273)(193,278)
(194,279)(195,277)(196,276)(197,274)(198,275)(199,289)(200,290)(201,291)
(202,296)(203,297)(204,295)(205,294)(206,292)(207,293)(208,280)(209,281)
(210,282)(211,287)(212,288)(213,286)(214,285)(215,283)(216,284)(217,298)
(218,299)(219,300)(220,305)(221,306)(222,304)(223,303)(224,301)(225,302)
(226,316)(227,317)(228,318)(229,323)(230,324)(231,322)(232,321)(233,319)
(234,320)(235,307)(236,308)(237,309)(238,314)(239,315)(240,313)(241,312)
(242,310)(243,311);
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)( 28,228)( 29,227)( 30,226)( 31,231)
( 32,230)( 33,229)( 34,234)( 35,233)( 36,232)( 37,219)( 38,218)( 39,217)
( 40,222)( 41,221)( 42,220)( 43,225)( 44,224)( 45,223)( 46,237)( 47,236)
( 48,235)( 49,240)( 50,239)( 51,238)( 52,243)( 53,242)( 54,241)( 55,201)
( 56,200)( 57,199)( 58,204)( 59,203)( 60,202)( 61,207)( 62,206)( 63,205)
( 64,192)( 65,191)( 66,190)( 67,195)( 68,194)( 69,193)( 70,198)( 71,197)
( 72,196)( 73,210)( 74,209)( 75,208)( 76,213)( 77,212)( 78,211)( 79,216)
( 80,215)( 81,214)( 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)(109,309)(110,308)(111,307)
(112,312)(113,311)(114,310)(115,315)(116,314)(117,313)(118,300)(119,299)
(120,298)(121,303)(122,302)(123,301)(124,306)(125,305)(126,304)(127,318)
(128,317)(129,316)(130,321)(131,320)(132,319)(133,324)(134,323)(135,322)
(136,282)(137,281)(138,280)(139,285)(140,284)(141,283)(142,288)(143,287)
(144,286)(145,273)(146,272)(147,271)(148,276)(149,275)(150,274)(151,279)
(152,278)(153,277)(154,291)(155,290)(156,289)(157,294)(158,293)(159,292)
(160,297)(161,296)(162,295);
s2 := 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)( 55, 57)( 58, 63)( 59, 62)( 60, 61)
( 64, 69)( 65, 68)( 66, 67)( 70, 72)( 73, 79)( 74, 81)( 75, 80)( 77, 78)
( 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)(136,138)(139,144)(140,143)(141,142)(145,150)
(146,149)(147,148)(151,153)(154,160)(155,162)(156,161)(158,159)(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)(217,219)(220,225)(221,224)(222,223)(226,231)(227,230)
(228,229)(232,234)(235,241)(236,243)(237,242)(239,240)(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)(298,300)(301,306)(302,305)(303,304)(307,312)(308,311)(309,310)
(313,315)(316,322)(317,324)(318,323)(320,321);
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*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*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;
References : None.
to this polytope