Polytope of Type {12,27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,27}*1296
if this polytope has a name.
Group : SmallGroup(1296,1781)
Rank : 3
Schlafli Type : {12,27}
Number of vertices, edges, etc : 24, 324, 54
Order of s₀s₁s₂ : 54
Order of s₀s₁s₂s₁ : 12
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) :
   3-fold quotients : {4,27}*432, {12,9}*432
   4-fold quotients : {6,27}*324
   6-fold quotients : {4,27}*216
   9-fold quotients : {4,9}*144, {12,3}*144
   12-fold quotients : {2,27}*108, {6,9}*108
   18-fold quotients : {4,9}*72
   27-fold quotients : {4,3}*48
   36-fold quotients : {2,9}*36, {6,3}*36
   54-fold quotients : {4,3}*24
   108-fold quotients : {2,3}*12
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, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 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, 87)( 50, 88)
( 51, 85)( 52, 86)( 53, 91)( 54, 92)( 55, 89)( 56, 90)( 57, 95)( 58, 96)
( 59, 93)( 60, 94)( 61, 99)( 62,100)( 63, 97)( 64, 98)( 65,103)( 66,104)
( 67,101)( 68,102)( 69,107)( 70,108)( 71,105)( 72,106)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,183)(146,184)(147,181)(148,182)(149,187)(150,188)(151,185)(152,186)
(153,191)(154,192)(155,189)(156,190)(157,195)(158,196)(159,193)(160,194)
(161,199)(162,200)(163,197)(164,198)(165,203)(166,204)(167,201)(168,202)
(169,207)(170,208)(171,205)(172,206)(173,211)(174,212)(175,209)(176,210)
(177,215)(178,216)(179,213)(180,214)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,291)(254,292)
(255,289)(256,290)(257,295)(258,296)(259,293)(260,294)(261,299)(262,300)
(263,297)(264,298)(265,303)(266,304)(267,301)(268,302)(269,307)(270,308)
(271,305)(272,306)(273,311)(274,312)(275,309)(276,310)(277,315)(278,316)
(279,313)(280,314)(281,319)(282,320)(283,317)(284,318)(285,323)(286,324)
(287,321)(288,322);;
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, 69)( 14, 70)( 15, 72)( 16, 71)
( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 61)( 22, 62)( 23, 64)( 24, 63)
( 25, 57)( 26, 58)( 27, 60)( 28, 59)( 29, 53)( 30, 54)( 31, 56)( 32, 55)
( 33, 49)( 34, 50)( 35, 52)( 36, 51)( 75, 76)( 77, 81)( 78, 82)( 79, 84)
( 80, 83)( 85,105)( 86,106)( 87,108)( 88,107)( 89,101)( 90,102)( 91,104)
( 92,103)( 93, 97)( 94, 98)( 95,100)( 96, 99)(109,285)(110,286)(111,288)
(112,287)(113,281)(114,282)(115,284)(116,283)(117,277)(118,278)(119,280)
(120,279)(121,273)(122,274)(123,276)(124,275)(125,269)(126,270)(127,272)
(128,271)(129,265)(130,266)(131,268)(132,267)(133,261)(134,262)(135,264)
(136,263)(137,257)(138,258)(139,260)(140,259)(141,253)(142,254)(143,256)
(144,255)(145,249)(146,250)(147,252)(148,251)(149,245)(150,246)(151,248)
(152,247)(153,241)(154,242)(155,244)(156,243)(157,237)(158,238)(159,240)
(160,239)(161,233)(162,234)(163,236)(164,235)(165,229)(166,230)(167,232)
(168,231)(169,225)(170,226)(171,228)(172,227)(173,221)(174,222)(175,224)
(176,223)(177,217)(178,218)(179,220)(180,219)(181,321)(182,322)(183,324)
(184,323)(185,317)(186,318)(187,320)(188,319)(189,313)(190,314)(191,316)
(192,315)(193,309)(194,310)(195,312)(196,311)(197,305)(198,306)(199,308)
(200,307)(201,301)(202,302)(203,304)(204,303)(205,297)(206,298)(207,300)
(208,299)(209,293)(210,294)(211,296)(212,295)(213,289)(214,290)(215,292)
(216,291);;
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)( 13,141)( 14,144)( 15,143)( 16,142)
( 17,137)( 18,140)( 19,139)( 20,138)( 21,133)( 22,136)( 23,135)( 24,134)
( 25,129)( 26,132)( 27,131)( 28,130)( 29,125)( 30,128)( 31,127)( 32,126)
( 33,121)( 34,124)( 35,123)( 36,122)( 37,181)( 38,184)( 39,183)( 40,182)
( 41,189)( 42,192)( 43,191)( 44,190)( 45,185)( 46,188)( 47,187)( 48,186)
( 49,213)( 50,216)( 51,215)( 52,214)( 53,209)( 54,212)( 55,211)( 56,210)
( 57,205)( 58,208)( 59,207)( 60,206)( 61,201)( 62,204)( 63,203)( 64,202)
( 65,197)( 66,200)( 67,199)( 68,198)( 69,193)( 70,196)( 71,195)( 72,194)
( 73,145)( 74,148)( 75,147)( 76,146)( 77,153)( 78,156)( 79,155)( 80,154)
( 81,149)( 82,152)( 83,151)( 84,150)( 85,177)( 86,180)( 87,179)( 88,178)
( 89,173)( 90,176)( 91,175)( 92,174)( 93,169)( 94,172)( 95,171)( 96,170)
( 97,165)( 98,168)( 99,167)(100,166)(101,161)(102,164)(103,163)(104,162)
(105,157)(106,160)(107,159)(108,158)(217,249)(218,252)(219,251)(220,250)
(221,245)(222,248)(223,247)(224,246)(225,241)(226,244)(227,243)(228,242)
(229,237)(230,240)(231,239)(232,238)(234,236)(253,321)(254,324)(255,323)
(256,322)(257,317)(258,320)(259,319)(260,318)(261,313)(262,316)(263,315)
(264,314)(265,309)(266,312)(267,311)(268,310)(269,305)(270,308)(271,307)
(272,306)(273,301)(274,304)(275,303)(276,302)(277,297)(278,300)(279,299)
(280,298)(281,293)(282,296)(283,295)(284,294)(285,289)(286,292)(287,291)
(288,290);;
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*s1*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope