Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6,12}

Atlas Canonical Name {3,6,12}*1296b

Overview

Group
SmallGroup(1296,840)
Rank
4
Schläfli Type
{3,6,12}
Vertices, edges, …
9, 27, 108, 12
Order of s0s1s2s3
36
Order of s0s1s2s3s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

9-fold

12-fold

18-fold

27-fold

36-fold

54-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*s2)^2> of order 3

12 facets

  • 12 of 3-fold non-regular quotient of {3,6}*108

5 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  4,  8)(  5,  9)(  6,  7)( 13, 17)( 14, 18)( 15, 16)( 22, 26)( 23, 27)( 24, 25)( 28, 55)( 29, 56)( 30, 57)( 31, 62)( 32, 63)( 33, 61)( 34, 60)( 35, 58)( 36, 59)( 37, 64)( 38, 65)( 39, 66)( 40, 71)( 41, 72)( 42, 70)( 43, 69)( 44, 67)( 45, 68)( 46, 73)( 47, 74)( 48, 75)( 49, 80)( 50, 81)( 51, 79)( 52, 78)( 53, 76)( 54, 77)( 85, 89)( 86, 90)( 87, 88)( 94, 98)( 95, 99)( 96, 97)(103,107)(104,108)(105,106)(109,136)(110,137)(111,138)(112,143)(113,144)(114,142)(115,141)(116,139)(117,140)(118,145)(119,146)(120,147)(121,152)(122,153)(123,151)(124,150)(125,148)(126,149)(127,154)(128,155)(129,156)(130,161)(131,162)(132,160)(133,159)(134,157)(135,158)(166,170)(167,171)(168,169)(175,179)(176,180)(177,178)(184,188)(185,189)(186,187)(190,217)(191,218)(192,219)(193,224)(194,225)(195,223)(196,222)(197,220)(198,221)(199,226)(200,227)(201,228)(202,233)(203,234)(204,232)(205,231)(206,229)(207,230)(208,235)(209,236)(210,237)(211,242)(212,243)(213,241)(214,240)(215,238)(216,239)(247,251)(248,252)(249,250)(256,260)(257,261)(258,259)(265,269)(266,270)(267,268)(271,298)(272,299)(273,300)(274,305)(275,306)(276,304)(277,303)(278,301)(279,302)(280,307)(281,308)(282,309)(283,314)(284,315)(285,313)(286,312)(287,310)(288,311)(289,316)(290,317)(291,318)(292,323)(293,324)(294,322)(295,321)(296,319)(297,320);;
s1 := (  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);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)(  8, 15)(  9, 14)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 37)( 29, 39)( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)( 61, 67)( 62, 69)( 63, 68)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)( 90, 95)(101,102)(103,106)(104,108)(105,107)(109,118)(110,120)(111,119)(112,124)(113,126)(114,125)(115,121)(116,123)(117,122)(128,129)(130,133)(131,135)(132,134)(136,145)(137,147)(138,146)(139,151)(140,153)(141,152)(142,148)(143,150)(144,149)(155,156)(157,160)(158,162)(159,161)(163,253)(164,255)(165,254)(166,259)(167,261)(168,260)(169,256)(170,258)(171,257)(172,244)(173,246)(174,245)(175,250)(176,252)(177,251)(178,247)(179,249)(180,248)(181,262)(182,264)(183,263)(184,268)(185,270)(186,269)(187,265)(188,267)(189,266)(190,280)(191,282)(192,281)(193,286)(194,288)(195,287)(196,283)(197,285)(198,284)(199,271)(200,273)(201,272)(202,277)(203,279)(204,278)(205,274)(206,276)(207,275)(208,289)(209,291)(210,290)(211,295)(212,297)(213,296)(214,292)(215,294)(216,293)(217,307)(218,309)(219,308)(220,313)(221,315)(222,314)(223,310)(224,312)(225,311)(226,298)(227,300)(228,299)(229,304)(230,306)(231,305)(232,301)(233,303)(234,302)(235,316)(236,318)(237,317)(238,322)(239,324)(240,323)(241,319)(242,321)(243,320);;
s3 := (  1,163)(  2,165)(  3,164)(  4,169)(  5,171)(  6,170)(  7,166)(  8,168)(  9,167)( 10,181)( 11,183)( 12,182)( 13,187)( 14,189)( 15,188)( 16,184)( 17,186)( 18,185)( 19,172)( 20,174)( 21,173)( 22,178)( 23,180)( 24,179)( 25,175)( 26,177)( 27,176)( 28,190)( 29,192)( 30,191)( 31,196)( 32,198)( 33,197)( 34,193)( 35,195)( 36,194)( 37,208)( 38,210)( 39,209)( 40,214)( 41,216)( 42,215)( 43,211)( 44,213)( 45,212)( 46,199)( 47,201)( 48,200)( 49,205)( 50,207)( 51,206)( 52,202)( 53,204)( 54,203)( 55,217)( 56,219)( 57,218)( 58,223)( 59,225)( 60,224)( 61,220)( 62,222)( 63,221)( 64,235)( 65,237)( 66,236)( 67,241)( 68,243)( 69,242)( 70,238)( 71,240)( 72,239)( 73,226)( 74,228)( 75,227)( 76,232)( 77,234)( 78,233)( 79,229)( 80,231)( 81,230)( 82,244)( 83,246)( 84,245)( 85,250)( 86,252)( 87,251)( 88,247)( 89,249)( 90,248)( 91,262)( 92,264)( 93,263)( 94,268)( 95,270)( 96,269)( 97,265)( 98,267)( 99,266)(100,253)(101,255)(102,254)(103,259)(104,261)(105,260)(106,256)(107,258)(108,257)(109,271)(110,273)(111,272)(112,277)(113,279)(114,278)(115,274)(116,276)(117,275)(118,289)(119,291)(120,290)(121,295)(122,297)(123,296)(124,292)(125,294)(126,293)(127,280)(128,282)(129,281)(130,286)(131,288)(132,287)(133,283)(134,285)(135,284)(136,298)(137,300)(138,299)(139,304)(140,306)(141,305)(142,301)(143,303)(144,302)(145,316)(146,318)(147,317)(148,322)(149,324)(150,323)(151,319)(152,321)(153,320)(154,307)(155,309)(156,308)(157,313)(158,315)(159,314)(160,310)(161,312)(162,311);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(324)!(  4,  8)(  5,  9)(  6,  7)( 13, 17)( 14, 18)( 15, 16)( 22, 26)( 23, 27)( 24, 25)( 28, 55)( 29, 56)( 30, 57)( 31, 62)( 32, 63)( 33, 61)( 34, 60)( 35, 58)( 36, 59)( 37, 64)( 38, 65)( 39, 66)( 40, 71)( 41, 72)( 42, 70)( 43, 69)( 44, 67)( 45, 68)( 46, 73)( 47, 74)( 48, 75)( 49, 80)( 50, 81)( 51, 79)( 52, 78)( 53, 76)( 54, 77)( 85, 89)( 86, 90)( 87, 88)( 94, 98)( 95, 99)( 96, 97)(103,107)(104,108)(105,106)(109,136)(110,137)(111,138)(112,143)(113,144)(114,142)(115,141)(116,139)(117,140)(118,145)(119,146)(120,147)(121,152)(122,153)(123,151)(124,150)(125,148)(126,149)(127,154)(128,155)(129,156)(130,161)(131,162)(132,160)(133,159)(134,157)(135,158)(166,170)(167,171)(168,169)(175,179)(176,180)(177,178)(184,188)(185,189)(186,187)(190,217)(191,218)(192,219)(193,224)(194,225)(195,223)(196,222)(197,220)(198,221)(199,226)(200,227)(201,228)(202,233)(203,234)(204,232)(205,231)(206,229)(207,230)(208,235)(209,236)(210,237)(211,242)(212,243)(213,241)(214,240)(215,238)(216,239)(247,251)(248,252)(249,250)(256,260)(257,261)(258,259)(265,269)(266,270)(267,268)(271,298)(272,299)(273,300)(274,305)(275,306)(276,304)(277,303)(278,301)(279,302)(280,307)(281,308)(282,309)(283,314)(284,315)(285,313)(286,312)(287,310)(288,311)(289,316)(290,317)(291,318)(292,323)(293,324)(294,322)(295,321)(296,319)(297,320);
s1 := 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);
s2 := Sym(324)!(  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)(  8, 15)(  9, 14)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 37)( 29, 39)( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)( 61, 67)( 62, 69)( 63, 68)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)( 90, 95)(101,102)(103,106)(104,108)(105,107)(109,118)(110,120)(111,119)(112,124)(113,126)(114,125)(115,121)(116,123)(117,122)(128,129)(130,133)(131,135)(132,134)(136,145)(137,147)(138,146)(139,151)(140,153)(141,152)(142,148)(143,150)(144,149)(155,156)(157,160)(158,162)(159,161)(163,253)(164,255)(165,254)(166,259)(167,261)(168,260)(169,256)(170,258)(171,257)(172,244)(173,246)(174,245)(175,250)(176,252)(177,251)(178,247)(179,249)(180,248)(181,262)(182,264)(183,263)(184,268)(185,270)(186,269)(187,265)(188,267)(189,266)(190,280)(191,282)(192,281)(193,286)(194,288)(195,287)(196,283)(197,285)(198,284)(199,271)(200,273)(201,272)(202,277)(203,279)(204,278)(205,274)(206,276)(207,275)(208,289)(209,291)(210,290)(211,295)(212,297)(213,296)(214,292)(215,294)(216,293)(217,307)(218,309)(219,308)(220,313)(221,315)(222,314)(223,310)(224,312)(225,311)(226,298)(227,300)(228,299)(229,304)(230,306)(231,305)(232,301)(233,303)(234,302)(235,316)(236,318)(237,317)(238,322)(239,324)(240,323)(241,319)(242,321)(243,320);
s3 := Sym(324)!(  1,163)(  2,165)(  3,164)(  4,169)(  5,171)(  6,170)(  7,166)(  8,168)(  9,167)( 10,181)( 11,183)( 12,182)( 13,187)( 14,189)( 15,188)( 16,184)( 17,186)( 18,185)( 19,172)( 20,174)( 21,173)( 22,178)( 23,180)( 24,179)( 25,175)( 26,177)( 27,176)( 28,190)( 29,192)( 30,191)( 31,196)( 32,198)( 33,197)( 34,193)( 35,195)( 36,194)( 37,208)( 38,210)( 39,209)( 40,214)( 41,216)( 42,215)( 43,211)( 44,213)( 45,212)( 46,199)( 47,201)( 48,200)( 49,205)( 50,207)( 51,206)( 52,202)( 53,204)( 54,203)( 55,217)( 56,219)( 57,218)( 58,223)( 59,225)( 60,224)( 61,220)( 62,222)( 63,221)( 64,235)( 65,237)( 66,236)( 67,241)( 68,243)( 69,242)( 70,238)( 71,240)( 72,239)( 73,226)( 74,228)( 75,227)( 76,232)( 77,234)( 78,233)( 79,229)( 80,231)( 81,230)( 82,244)( 83,246)( 84,245)( 85,250)( 86,252)( 87,251)( 88,247)( 89,249)( 90,248)( 91,262)( 92,264)( 93,263)( 94,268)( 95,270)( 96,269)( 97,265)( 98,267)( 99,266)(100,253)(101,255)(102,254)(103,259)(104,261)(105,260)(106,256)(107,258)(108,257)(109,271)(110,273)(111,272)(112,277)(113,279)(114,278)(115,274)(116,276)(117,275)(118,289)(119,291)(120,290)(121,295)(122,297)(123,296)(124,292)(125,294)(126,293)(127,280)(128,282)(129,281)(130,286)(131,288)(132,287)(133,283)(134,285)(135,284)(136,298)(137,300)(138,299)(139,304)(140,306)(141,305)(142,301)(143,303)(144,302)(145,316)(146,318)(147,317)(148,322)(149,324)(150,323)(151,319)(152,321)(153,320)(154,307)(155,309)(156,308)(157,313)(158,315)(159,314)(160,310)(161,312)(162,311);
poly := sub<Sym(324)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References

None.

to this polytope.