Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,6}

Atlas Canonical Name {12,6}*1296a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,839)
Rank
3
Schläfli Type
{12,6}
Vertices, edges, …
108, 324, 54
Order of s0s1s2
36
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

9-fold

12-fold

18-fold

27-fold

54-fold

81-fold

108-fold

162-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 3

18 facets

60 vertex figures

P/N, where N=<((s1*s0)^2*s1*s2)^2> of order 3

18 facets

36 vertex figures

P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*s2> of order 3

18 facets

36 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 3

18 facets

36 vertex figures

P/N, where N=<(s0*s1)^4> of order 3

22 facets

36 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1> of order 9

6 facets

28 vertex figures

P/N, where N=<(s1*s2)^2, (s0*s1)^2*s2*(s1*s0)^2*s2> of order 9

6 facets

20 vertex figures

P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*s2, s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 9

6 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle