Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,3,6}

Atlas Canonical Name {4,3,6}*1296a

Overview

Group
SmallGroup(1296,1786)
Rank
4
Schläfli Type
{4,3,6}
Vertices, edges, …
4, 54, 81, 54
Order of s0s1s2s3
18
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

9-fold

27-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*(s3*s2)^2)^2> of order 3

18 facets

4 vertex figures

  • 4 of 3-fold non-regular quotient of {3,6}*324
P/N, where N=<s2*s3*s2*(s1*(s3*s2)^2)^2*s3> of order 3

18 facets

4 vertex figures

  • 4 of 3-fold non-regular quotient of {3,6}*324

Representations

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

References

None.

to this polytope.