Polytope of Type {12,18}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18}*1296h
if this polytope has a name.
Group : SmallGroup(1296,930)
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}*648e
   3-fold quotients : {12,6}*432c
   4-fold quotients : {6,9}*324d
   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.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 3.
      18 facets:
         18 of {12}*24
      12 vertex figures:
         12 of {18}*36

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, 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, 56)( 59, 60)( 61, 63)( 64, 81)( 65, 80)( 66, 79)( 67, 74)( 68, 73)( 69, 75)( 70, 76)( 71, 78)( 72, 77)( 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)(136,137)(140,141)(142,144)(145,162)(146,161)(147,160)(148,155)(149,154)(150,156)(151,157)(152,159)(153,158)(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)(217,218)(221,222)(223,225)(226,243)(227,242)(228,241)(229,236)(230,235)(231,237)(232,238)(233,240)(234,239)(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)(298,299)(302,303)(304,306)(307,324)(308,323)(309,322)(310,317)(311,316)(312,318)(313,319)(314,321)(315,320);;
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*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*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, 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, 56)( 59, 60)( 61, 63)( 64, 81)( 65, 80)( 66, 79)( 67, 74)( 68, 73)( 69, 75)( 70, 76)( 71, 78)( 72, 77)( 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)(136,137)(140,141)(142,144)(145,162)(146,161)(147,160)(148,155)(149,154)(150,156)(151,157)(152,159)(153,158)(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)(217,218)(221,222)(223,225)(226,243)(227,242)(228,241)(229,236)(230,235)(231,237)(232,238)(233,240)(234,239)(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)(298,299)(302,303)(304,306)(307,324)(308,323)(309,322)(310,317)(311,316)(312,318)(313,319)(314,321)(315,320);
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*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1 >;
References : None.
to this polytope

Twisty Puzzle