Polytope of Type {12,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18}*1296b
if this polytope has a name.
Group : SmallGroup(1296,853)
Rank : 3
Schlafli Type : {12,18}
Number of vertices, edges, etc : 36, 324, 54
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {6,18}*648f
3-fold quotients : {12,6}*432b
4-fold quotients : {6,18}*324b
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)( 28,227)( 29,226)( 30,228)( 31,230)( 32,229)
( 33,231)( 34,233)( 35,232)( 36,234)( 37,218)( 38,217)( 39,219)( 40,221)
( 41,220)( 42,222)( 43,224)( 44,223)( 45,225)( 46,236)( 47,235)( 48,237)
( 49,239)( 50,238)( 51,240)( 52,242)( 53,241)( 54,243)( 55,200)( 56,199)
( 57,201)( 58,203)( 59,202)( 60,204)( 61,206)( 62,205)( 63,207)( 64,191)
( 65,190)( 66,192)( 67,194)( 68,193)( 69,195)( 70,197)( 71,196)( 72,198)
( 73,209)( 74,208)( 75,210)( 76,212)( 77,211)( 78,213)( 79,215)( 80,214)
( 81,216)( 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)(109,308)(110,307)(111,309)(112,311)
(113,310)(114,312)(115,314)(116,313)(117,315)(118,299)(119,298)(120,300)
(121,302)(122,301)(123,303)(124,305)(125,304)(126,306)(127,317)(128,316)
(129,318)(130,320)(131,319)(132,321)(133,323)(134,322)(135,324)(136,281)
(137,280)(138,282)(139,284)(140,283)(141,285)(142,287)(143,286)(144,288)
(145,272)(146,271)(147,273)(148,275)(149,274)(150,276)(151,278)(152,277)
(153,279)(154,290)(155,289)(156,291)(157,293)(158,292)(159,294)(160,296)
(161,295)(162,297);;
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, 56)( 58, 62)( 59, 61)( 60, 63)( 64, 68)
( 65, 67)( 66, 69)( 70, 71)( 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)
( 91,121)( 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)(136,137)(139,143)(140,142)(141,144)(145,149)(146,148)
(147,150)(151,152)(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)(172,202)
(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)(217,218)(220,224)(221,223)(222,225)(226,230)(227,229)(228,231)
(232,233)(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)(253,283)(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)
(298,299)(301,305)(302,304)(303,306)(307,311)(308,310)(309,312)(313,314)
(316,324)(317,323)(318,322)(319,321);;
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, s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*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 ];;
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)( 28,227)( 29,226)( 30,228)( 31,230)
( 32,229)( 33,231)( 34,233)( 35,232)( 36,234)( 37,218)( 38,217)( 39,219)
( 40,221)( 41,220)( 42,222)( 43,224)( 44,223)( 45,225)( 46,236)( 47,235)
( 48,237)( 49,239)( 50,238)( 51,240)( 52,242)( 53,241)( 54,243)( 55,200)
( 56,199)( 57,201)( 58,203)( 59,202)( 60,204)( 61,206)( 62,205)( 63,207)
( 64,191)( 65,190)( 66,192)( 67,194)( 68,193)( 69,195)( 70,197)( 71,196)
( 72,198)( 73,209)( 74,208)( 75,210)( 76,212)( 77,211)( 78,213)( 79,215)
( 80,214)( 81,216)( 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)(109,308)(110,307)(111,309)
(112,311)(113,310)(114,312)(115,314)(116,313)(117,315)(118,299)(119,298)
(120,300)(121,302)(122,301)(123,303)(124,305)(125,304)(126,306)(127,317)
(128,316)(129,318)(130,320)(131,319)(132,321)(133,323)(134,322)(135,324)
(136,281)(137,280)(138,282)(139,284)(140,283)(141,285)(142,287)(143,286)
(144,288)(145,272)(146,271)(147,273)(148,275)(149,274)(150,276)(151,278)
(152,277)(153,279)(154,290)(155,289)(156,291)(157,293)(158,292)(159,294)
(160,296)(161,295)(162,297);
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, 56)( 58, 62)( 59, 61)( 60, 63)
( 64, 68)( 65, 67)( 66, 69)( 70, 71)( 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)( 91,121)( 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)(136,137)(139,143)(140,142)(141,144)(145,149)
(146,148)(147,150)(151,152)(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)
(172,202)(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)(217,218)(220,224)(221,223)(222,225)(226,230)(227,229)
(228,231)(232,233)(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)(253,283)
(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)(298,299)(301,305)(302,304)(303,306)(307,311)(308,310)(309,312)
(313,314)(316,324)(317,323)(318,322)(319,321);
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, s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*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 >;
References : None.
to this polytope