Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1296b
if this polytope has a name.
Group : SmallGroup(1296,840)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 108, 324, 54
Order of s₀s₁s₂ : 36
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 : {6,6}*648b
   3-fold quotients : {12,6}*432b
   4-fold quotients : {6,6}*324a
   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 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 25)
( 14, 27)( 15, 26)( 16, 22)( 17, 24)( 18, 23)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)( 43, 49)
( 44, 51)( 45, 50)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 73)( 65, 75)
( 66, 74)( 67, 79)( 68, 81)( 69, 80)( 70, 76)( 71, 78)( 72, 77)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 91,100)( 92,102)( 93,101)( 94,106)( 95,108)
( 96,107)( 97,103)( 98,105)( 99,104)(110,111)(112,115)(113,117)(114,116)
(118,127)(119,129)(120,128)(121,133)(122,135)(123,134)(124,130)(125,132)
(126,131)(137,138)(139,142)(140,144)(141,143)(145,154)(146,156)(147,155)
(148,160)(149,162)(150,161)(151,157)(152,159)(153,158)(163,244)(164,246)
(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)(171,248)(172,262)
(173,264)(174,263)(175,268)(176,270)(177,269)(178,265)(179,267)(180,266)
(181,253)(182,255)(183,254)(184,259)(185,261)(186,260)(187,256)(188,258)
(189,257)(190,271)(191,273)(192,272)(193,277)(194,279)(195,278)(196,274)
(197,276)(198,275)(199,289)(200,291)(201,290)(202,295)(203,297)(204,296)
(205,292)(206,294)(207,293)(208,280)(209,282)(210,281)(211,286)(212,288)
(213,287)(214,283)(215,285)(216,284)(217,298)(218,300)(219,299)(220,304)
(221,306)(222,305)(223,301)(224,303)(225,302)(226,316)(227,318)(228,317)
(229,322)(230,324)(231,323)(232,319)(233,321)(234,320)(235,307)(236,309)
(237,308)(238,313)(239,315)(240,314)(241,310)(242,312)(243,311);;
s1 := (  1,172)(  2,174)(  3,173)(  4,177)(  5,176)(  6,175)(  7,179)(  8,178)
(  9,180)( 10,163)( 11,165)( 12,164)( 13,168)( 14,167)( 15,166)( 16,170)
( 17,169)( 18,171)( 19,181)( 20,183)( 21,182)( 22,186)( 23,185)( 24,184)
( 25,188)( 26,187)( 27,189)( 28,226)( 29,228)( 30,227)( 31,231)( 32,230)
( 33,229)( 34,233)( 35,232)( 36,234)( 37,217)( 38,219)( 39,218)( 40,222)
( 41,221)( 42,220)( 43,224)( 44,223)( 45,225)( 46,235)( 47,237)( 48,236)
( 49,240)( 50,239)( 51,238)( 52,242)( 53,241)( 54,243)( 55,199)( 56,201)
( 57,200)( 58,204)( 59,203)( 60,202)( 61,206)( 62,205)( 63,207)( 64,190)
( 65,192)( 66,191)( 67,195)( 68,194)( 69,193)( 70,197)( 71,196)( 72,198)
( 73,208)( 74,210)( 75,209)( 76,213)( 77,212)( 78,211)( 79,215)( 80,214)
( 81,216)( 82,253)( 83,255)( 84,254)( 85,258)( 86,257)( 87,256)( 88,260)
( 89,259)( 90,261)( 91,244)( 92,246)( 93,245)( 94,249)( 95,248)( 96,247)
( 97,251)( 98,250)( 99,252)(100,262)(101,264)(102,263)(103,267)(104,266)
(105,265)(106,269)(107,268)(108,270)(109,307)(110,309)(111,308)(112,312)
(113,311)(114,310)(115,314)(116,313)(117,315)(118,298)(119,300)(120,299)
(121,303)(122,302)(123,301)(124,305)(125,304)(126,306)(127,316)(128,318)
(129,317)(130,321)(131,320)(132,319)(133,323)(134,322)(135,324)(136,280)
(137,282)(138,281)(139,285)(140,284)(141,283)(142,287)(143,286)(144,288)
(145,271)(146,273)(147,272)(148,276)(149,275)(150,274)(151,278)(152,277)
(153,279)(154,289)(155,291)(156,290)(157,294)(158,293)(159,292)(160,296)
(161,295)(162,297);;
s2 := (  1, 28)(  2, 29)(  3, 30)(  4, 34)(  5, 35)(  6, 36)(  7, 31)(  8, 32)
(  9, 33)( 10, 40)( 11, 41)( 12, 42)( 13, 37)( 14, 38)( 15, 39)( 16, 43)
( 17, 44)( 18, 45)( 19, 52)( 20, 53)( 21, 54)( 22, 49)( 23, 50)( 24, 51)
( 25, 46)( 26, 47)( 27, 48)( 58, 61)( 59, 62)( 60, 63)( 64, 67)( 65, 68)
( 66, 69)( 73, 79)( 74, 80)( 75, 81)( 82,109)( 83,110)( 84,111)( 85,115)
( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)( 91,121)( 92,122)( 93,123)
( 94,118)( 95,119)( 96,120)( 97,124)( 98,125)( 99,126)(100,133)(101,134)
(102,135)(103,130)(104,131)(105,132)(106,127)(107,128)(108,129)(139,142)
(140,143)(141,144)(145,148)(146,149)(147,150)(154,160)(155,161)(156,162)
(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)
(171,195)(172,202)(173,203)(174,204)(175,199)(176,200)(177,201)(178,205)
(179,206)(180,207)(181,214)(182,215)(183,216)(184,211)(185,212)(186,213)
(187,208)(188,209)(189,210)(220,223)(221,224)(222,225)(226,229)(227,230)
(228,231)(235,241)(236,242)(237,243)(244,271)(245,272)(246,273)(247,277)
(248,278)(249,279)(250,274)(251,275)(252,276)(253,283)(254,284)(255,285)
(256,280)(257,281)(258,282)(259,286)(260,287)(261,288)(262,295)(263,296)
(264,297)(265,292)(266,293)(267,294)(268,289)(269,290)(270,291)(301,304)
(302,305)(303,306)(307,310)(308,311)(309,312)(316,322)(317,323)(318,324);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*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, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope