Polytope of Type {12,54}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,54}*1296a
Also Known As : {12,54|2}. if this polytope has another name.
Group : SmallGroup(1296,825)
Rank : 3
Schlafli Type : {12,54}
Number of vertices, edges, etc : 12, 324, 54
Order of s0s1s2 : 108
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
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 : {6,54}*648a
3-fold quotients : {4,54}*432a, {12,18}*432a
6-fold quotients : {2,54}*216, {6,18}*216a
9-fold quotients : {4,18}*144a, {12,6}*144a
12-fold quotients : {2,27}*108
18-fold quotients : {2,18}*72, {6,6}*72a
27-fold quotients : {12,2}*48, {4,6}*48a
36-fold quotients : {2,9}*36
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 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 69, 78)
( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)
( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)(121,130)
(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)(147,156)
(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(163,244)(164,245)
(165,246)(166,247)(167,248)(168,249)(169,250)(170,251)(171,252)(172,262)
(173,263)(174,264)(175,265)(176,266)(177,267)(178,268)(179,269)(180,270)
(181,253)(182,254)(183,255)(184,256)(185,257)(186,258)(187,259)(188,260)
(189,261)(190,271)(191,272)(192,273)(193,274)(194,275)(195,276)(196,277)
(197,278)(198,279)(199,289)(200,290)(201,291)(202,292)(203,293)(204,294)
(205,295)(206,296)(207,297)(208,280)(209,281)(210,282)(211,283)(212,284)
(213,285)(214,286)(215,287)(216,288)(217,298)(218,299)(219,300)(220,301)
(221,302)(222,303)(223,304)(224,305)(225,306)(226,316)(227,317)(228,318)
(229,319)(230,320)(231,321)(232,322)(233,323)(234,324)(235,307)(236,308)
(237,309)(238,310)(239,311)(240,312)(241,313)(242,314)(243,315);;
s1 := ( 1,172)( 2,174)( 3,173)( 4,180)( 5,179)( 6,178)( 7,177)( 8,176)
( 9,175)( 10,163)( 11,165)( 12,164)( 13,171)( 14,170)( 15,169)( 16,168)
( 17,167)( 18,166)( 19,181)( 20,183)( 21,182)( 22,189)( 23,188)( 24,187)
( 25,186)( 26,185)( 27,184)( 28,234)( 29,233)( 30,232)( 31,231)( 32,230)
( 33,229)( 34,228)( 35,227)( 36,226)( 37,225)( 38,224)( 39,223)( 40,222)
( 41,221)( 42,220)( 43,219)( 44,218)( 45,217)( 46,243)( 47,242)( 48,241)
( 49,240)( 50,239)( 51,238)( 52,237)( 53,236)( 54,235)( 55,207)( 56,206)
( 57,205)( 58,204)( 59,203)( 60,202)( 61,201)( 62,200)( 63,199)( 64,198)
( 65,197)( 66,196)( 67,195)( 68,194)( 69,193)( 70,192)( 71,191)( 72,190)
( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,211)( 79,210)( 80,209)
( 81,208)( 82,253)( 83,255)( 84,254)( 85,261)( 86,260)( 87,259)( 88,258)
( 89,257)( 90,256)( 91,244)( 92,246)( 93,245)( 94,252)( 95,251)( 96,250)
( 97,249)( 98,248)( 99,247)(100,262)(101,264)(102,263)(103,270)(104,269)
(105,268)(106,267)(107,266)(108,265)(109,315)(110,314)(111,313)(112,312)
(113,311)(114,310)(115,309)(116,308)(117,307)(118,306)(119,305)(120,304)
(121,303)(122,302)(123,301)(124,300)(125,299)(126,298)(127,324)(128,323)
(129,322)(130,321)(131,320)(132,319)(133,318)(134,317)(135,316)(136,288)
(137,287)(138,286)(139,285)(140,284)(141,283)(142,282)(143,281)(144,280)
(145,279)(146,278)(147,277)(148,276)(149,275)(150,274)(151,273)(152,272)
(153,271)(154,297)(155,296)(156,295)(157,294)(158,293)(159,292)(160,291)
(161,290)(162,289);;
s2 := ( 1, 28)( 2, 30)( 3, 29)( 4, 36)( 5, 35)( 6, 34)( 7, 33)( 8, 32)
( 9, 31)( 10, 37)( 11, 39)( 12, 38)( 13, 45)( 14, 44)( 15, 43)( 16, 42)
( 17, 41)( 18, 40)( 19, 46)( 20, 48)( 21, 47)( 22, 54)( 23, 53)( 24, 52)
( 25, 51)( 26, 50)( 27, 49)( 55, 63)( 56, 62)( 57, 61)( 58, 60)( 64, 72)
( 65, 71)( 66, 70)( 67, 69)( 73, 81)( 74, 80)( 75, 79)( 76, 78)( 82,109)
( 83,111)( 84,110)( 85,117)( 86,116)( 87,115)( 88,114)( 89,113)( 90,112)
( 91,118)( 92,120)( 93,119)( 94,126)( 95,125)( 96,124)( 97,123)( 98,122)
( 99,121)(100,127)(101,129)(102,128)(103,135)(104,134)(105,133)(106,132)
(107,131)(108,130)(136,144)(137,143)(138,142)(139,141)(145,153)(146,152)
(147,151)(148,150)(154,162)(155,161)(156,160)(157,159)(163,190)(164,192)
(165,191)(166,198)(167,197)(168,196)(169,195)(170,194)(171,193)(172,199)
(173,201)(174,200)(175,207)(176,206)(177,205)(178,204)(179,203)(180,202)
(181,208)(182,210)(183,209)(184,216)(185,215)(186,214)(187,213)(188,212)
(189,211)(217,225)(218,224)(219,223)(220,222)(226,234)(227,233)(228,232)
(229,231)(235,243)(236,242)(237,241)(238,240)(244,271)(245,273)(246,272)
(247,279)(248,278)(249,277)(250,276)(251,275)(252,274)(253,280)(254,282)
(255,281)(256,288)(257,287)(258,286)(259,285)(260,284)(261,283)(262,289)
(263,291)(264,290)(265,297)(266,296)(267,295)(268,294)(269,293)(270,292)
(298,306)(299,305)(300,304)(301,303)(307,315)(308,314)(309,313)(310,312)
(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*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(324)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)
(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)
(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(163,244)
(164,245)(165,246)(166,247)(167,248)(168,249)(169,250)(170,251)(171,252)
(172,262)(173,263)(174,264)(175,265)(176,266)(177,267)(178,268)(179,269)
(180,270)(181,253)(182,254)(183,255)(184,256)(185,257)(186,258)(187,259)
(188,260)(189,261)(190,271)(191,272)(192,273)(193,274)(194,275)(195,276)
(196,277)(197,278)(198,279)(199,289)(200,290)(201,291)(202,292)(203,293)
(204,294)(205,295)(206,296)(207,297)(208,280)(209,281)(210,282)(211,283)
(212,284)(213,285)(214,286)(215,287)(216,288)(217,298)(218,299)(219,300)
(220,301)(221,302)(222,303)(223,304)(224,305)(225,306)(226,316)(227,317)
(228,318)(229,319)(230,320)(231,321)(232,322)(233,323)(234,324)(235,307)
(236,308)(237,309)(238,310)(239,311)(240,312)(241,313)(242,314)(243,315);
s1 := Sym(324)!( 1,172)( 2,174)( 3,173)( 4,180)( 5,179)( 6,178)( 7,177)
( 8,176)( 9,175)( 10,163)( 11,165)( 12,164)( 13,171)( 14,170)( 15,169)
( 16,168)( 17,167)( 18,166)( 19,181)( 20,183)( 21,182)( 22,189)( 23,188)
( 24,187)( 25,186)( 26,185)( 27,184)( 28,234)( 29,233)( 30,232)( 31,231)
( 32,230)( 33,229)( 34,228)( 35,227)( 36,226)( 37,225)( 38,224)( 39,223)
( 40,222)( 41,221)( 42,220)( 43,219)( 44,218)( 45,217)( 46,243)( 47,242)
( 48,241)( 49,240)( 50,239)( 51,238)( 52,237)( 53,236)( 54,235)( 55,207)
( 56,206)( 57,205)( 58,204)( 59,203)( 60,202)( 61,201)( 62,200)( 63,199)
( 64,198)( 65,197)( 66,196)( 67,195)( 68,194)( 69,193)( 70,192)( 71,191)
( 72,190)( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,211)( 79,210)
( 80,209)( 81,208)( 82,253)( 83,255)( 84,254)( 85,261)( 86,260)( 87,259)
( 88,258)( 89,257)( 90,256)( 91,244)( 92,246)( 93,245)( 94,252)( 95,251)
( 96,250)( 97,249)( 98,248)( 99,247)(100,262)(101,264)(102,263)(103,270)
(104,269)(105,268)(106,267)(107,266)(108,265)(109,315)(110,314)(111,313)
(112,312)(113,311)(114,310)(115,309)(116,308)(117,307)(118,306)(119,305)
(120,304)(121,303)(122,302)(123,301)(124,300)(125,299)(126,298)(127,324)
(128,323)(129,322)(130,321)(131,320)(132,319)(133,318)(134,317)(135,316)
(136,288)(137,287)(138,286)(139,285)(140,284)(141,283)(142,282)(143,281)
(144,280)(145,279)(146,278)(147,277)(148,276)(149,275)(150,274)(151,273)
(152,272)(153,271)(154,297)(155,296)(156,295)(157,294)(158,293)(159,292)
(160,291)(161,290)(162,289);
s2 := Sym(324)!( 1, 28)( 2, 30)( 3, 29)( 4, 36)( 5, 35)( 6, 34)( 7, 33)
( 8, 32)( 9, 31)( 10, 37)( 11, 39)( 12, 38)( 13, 45)( 14, 44)( 15, 43)
( 16, 42)( 17, 41)( 18, 40)( 19, 46)( 20, 48)( 21, 47)( 22, 54)( 23, 53)
( 24, 52)( 25, 51)( 26, 50)( 27, 49)( 55, 63)( 56, 62)( 57, 61)( 58, 60)
( 64, 72)( 65, 71)( 66, 70)( 67, 69)( 73, 81)( 74, 80)( 75, 79)( 76, 78)
( 82,109)( 83,111)( 84,110)( 85,117)( 86,116)( 87,115)( 88,114)( 89,113)
( 90,112)( 91,118)( 92,120)( 93,119)( 94,126)( 95,125)( 96,124)( 97,123)
( 98,122)( 99,121)(100,127)(101,129)(102,128)(103,135)(104,134)(105,133)
(106,132)(107,131)(108,130)(136,144)(137,143)(138,142)(139,141)(145,153)
(146,152)(147,151)(148,150)(154,162)(155,161)(156,160)(157,159)(163,190)
(164,192)(165,191)(166,198)(167,197)(168,196)(169,195)(170,194)(171,193)
(172,199)(173,201)(174,200)(175,207)(176,206)(177,205)(178,204)(179,203)
(180,202)(181,208)(182,210)(183,209)(184,216)(185,215)(186,214)(187,213)
(188,212)(189,211)(217,225)(218,224)(219,223)(220,222)(226,234)(227,233)
(228,232)(229,231)(235,243)(236,242)(237,241)(238,240)(244,271)(245,273)
(246,272)(247,279)(248,278)(249,277)(250,276)(251,275)(252,274)(253,280)
(254,282)(255,281)(256,288)(257,287)(258,286)(259,285)(260,284)(261,283)
(262,289)(263,291)(264,290)(265,297)(266,296)(267,295)(268,294)(269,293)
(270,292)(298,306)(299,305)(300,304)(301,303)(307,315)(308,314)(309,313)
(310,312)(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*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope