Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1296d
if this polytope has a name.
Group : SmallGroup(1296,943)
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}*648c
   3-fold quotients : {12,6}*432c
   4-fold quotients : {6,3}*324
   6-fold quotients : {6,6}*216a
   9-fold quotients : {12,6}*144c
   12-fold quotients : {6,3}*108
   18-fold quotients : {6,6}*72b
   27-fold quotients : {4,6}*48a
   36-fold quotients : {6,3}*36
   54-fold quotients : {2,6}*24
   81-fold quotients : {4,2}*16
   108-fold quotients : {2,3}*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, 33)(  5, 32)(  6, 31)(  7, 35)(  8, 34)
(  9, 36)( 10, 53)( 11, 52)( 12, 54)( 13, 46)( 14, 48)( 15, 47)( 16, 51)
( 17, 50)( 18, 49)( 19, 40)( 20, 42)( 21, 41)( 22, 45)( 23, 44)( 24, 43)
( 25, 38)( 26, 37)( 27, 39)( 55, 57)( 58, 59)( 62, 63)( 64, 79)( 65, 81)
( 66, 80)( 67, 75)( 68, 74)( 69, 73)( 70, 77)( 71, 76)( 72, 78)( 82,109)
( 83,111)( 84,110)( 85,114)( 86,113)( 87,112)( 88,116)( 89,115)( 90,117)
( 91,134)( 92,133)( 93,135)( 94,127)( 95,129)( 96,128)( 97,132)( 98,131)
( 99,130)(100,121)(101,123)(102,122)(103,126)(104,125)(105,124)(106,119)
(107,118)(108,120)(136,138)(139,140)(143,144)(145,160)(146,162)(147,161)
(148,156)(149,155)(150,154)(151,158)(152,157)(153,159)(163,190)(164,192)
(165,191)(166,195)(167,194)(168,193)(169,197)(170,196)(171,198)(172,215)
(173,214)(174,216)(175,208)(176,210)(177,209)(178,213)(179,212)(180,211)
(181,202)(182,204)(183,203)(184,207)(185,206)(186,205)(187,200)(188,199)
(189,201)(217,219)(220,221)(224,225)(226,241)(227,243)(228,242)(229,237)
(230,236)(231,235)(232,239)(233,238)(234,240)(244,271)(245,273)(246,272)
(247,276)(248,275)(249,274)(250,278)(251,277)(252,279)(253,296)(254,295)
(255,297)(256,289)(257,291)(258,290)(259,294)(260,293)(261,292)(262,283)
(263,285)(264,284)(265,288)(266,287)(267,286)(268,281)(269,280)(270,282)
(298,300)(301,302)(305,306)(307,322)(308,324)(309,323)(310,318)(311,317)
(312,316)(313,320)(314,319)(315,321);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*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)( 28,228)( 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, 33)(  5, 32)(  6, 31)(  7, 35)
(  8, 34)(  9, 36)( 10, 53)( 11, 52)( 12, 54)( 13, 46)( 14, 48)( 15, 47)
( 16, 51)( 17, 50)( 18, 49)( 19, 40)( 20, 42)( 21, 41)( 22, 45)( 23, 44)
( 24, 43)( 25, 38)( 26, 37)( 27, 39)( 55, 57)( 58, 59)( 62, 63)( 64, 79)
( 65, 81)( 66, 80)( 67, 75)( 68, 74)( 69, 73)( 70, 77)( 71, 76)( 72, 78)
( 82,109)( 83,111)( 84,110)( 85,114)( 86,113)( 87,112)( 88,116)( 89,115)
( 90,117)( 91,134)( 92,133)( 93,135)( 94,127)( 95,129)( 96,128)( 97,132)
( 98,131)( 99,130)(100,121)(101,123)(102,122)(103,126)(104,125)(105,124)
(106,119)(107,118)(108,120)(136,138)(139,140)(143,144)(145,160)(146,162)
(147,161)(148,156)(149,155)(150,154)(151,158)(152,157)(153,159)(163,190)
(164,192)(165,191)(166,195)(167,194)(168,193)(169,197)(170,196)(171,198)
(172,215)(173,214)(174,216)(175,208)(176,210)(177,209)(178,213)(179,212)
(180,211)(181,202)(182,204)(183,203)(184,207)(185,206)(186,205)(187,200)
(188,199)(189,201)(217,219)(220,221)(224,225)(226,241)(227,243)(228,242)
(229,237)(230,236)(231,235)(232,239)(233,238)(234,240)(244,271)(245,273)
(246,272)(247,276)(248,275)(249,274)(250,278)(251,277)(252,279)(253,296)
(254,295)(255,297)(256,289)(257,291)(258,290)(259,294)(260,293)(261,292)
(262,283)(263,285)(264,284)(265,288)(266,287)(267,286)(268,281)(269,280)
(270,282)(298,300)(301,302)(305,306)(307,322)(308,324)(309,323)(310,318)
(311,317)(312,316)(313,320)(314,319)(315,321);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 >;

References : None.
to this polytope