Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6,28}

Atlas Canonical Name {3,6,28}*1344

Overview

Group
SmallGroup(1344,11328)
Rank
4
Schläfli Type
{3,6,28}
Vertices, edges, …
4, 12, 112, 28
Order of s0s1s2s3
28
Order of s0s1s2s3s2s1
2
Also known as
{{3,6}4,{6,28|2}}. if this polytope has another name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

7-fold

14-fold

28-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP) 
s0 := (  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 29, 57)( 30, 58)( 31, 60)( 32, 59)( 33, 61)( 34, 62)( 35, 64)( 36, 63)( 37, 65)( 38, 66)( 39, 68)( 40, 67)( 41, 69)( 42, 70)( 43, 72)( 44, 71)( 45, 73)( 46, 74)( 47, 76)( 48, 75)( 49, 77)( 50, 78)( 51, 80)( 52, 79)( 53, 81)( 54, 82)( 55, 84)( 56, 83)( 87, 88)( 91, 92)( 95, 96)( 99,100)(103,104)(107,108)(111,112)(113,141)(114,142)(115,144)(116,143)(117,145)(118,146)(119,148)(120,147)(121,149)(122,150)(123,152)(124,151)(125,153)(126,154)(127,156)(128,155)(129,157)(130,158)(131,160)(132,159)(133,161)(134,162)(135,164)(136,163)(137,165)(138,166)(139,168)(140,167)(171,172)(175,176)(179,180)(183,184)(187,188)(191,192)(195,196)(197,225)(198,226)(199,228)(200,227)(201,229)(202,230)(203,232)(204,231)(205,233)(206,234)(207,236)(208,235)(209,237)(210,238)(211,240)(212,239)(213,241)(214,242)(215,244)(216,243)(217,245)(218,246)(219,248)(220,247)(221,249)(222,250)(223,252)(224,251)(255,256)(259,260)(263,264)(267,268)(271,272)(275,276)(279,280)(281,309)(282,310)(283,312)(284,311)(285,313)(286,314)(287,316)(288,315)(289,317)(290,318)(291,320)(292,319)(293,321)(294,322)(295,324)(296,323)(297,325)(298,326)(299,328)(300,327)(301,329)(302,330)(303,332)(304,331)(305,333)(306,334)(307,336)(308,335);;
s1 := (  1, 29)(  2, 32)(  3, 31)(  4, 30)(  5, 33)(  6, 36)(  7, 35)(  8, 34)(  9, 37)( 10, 40)( 11, 39)( 12, 38)( 13, 41)( 14, 44)( 15, 43)( 16, 42)( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 21, 49)( 22, 52)( 23, 51)( 24, 50)( 25, 53)( 26, 56)( 27, 55)( 28, 54)( 58, 60)( 62, 64)( 66, 68)( 70, 72)( 74, 76)( 78, 80)( 82, 84)( 85,113)( 86,116)( 87,115)( 88,114)( 89,117)( 90,120)( 91,119)( 92,118)( 93,121)( 94,124)( 95,123)( 96,122)( 97,125)( 98,128)( 99,127)(100,126)(101,129)(102,132)(103,131)(104,130)(105,133)(106,136)(107,135)(108,134)(109,137)(110,140)(111,139)(112,138)(142,144)(146,148)(150,152)(154,156)(158,160)(162,164)(166,168)(169,197)(170,200)(171,199)(172,198)(173,201)(174,204)(175,203)(176,202)(177,205)(178,208)(179,207)(180,206)(181,209)(182,212)(183,211)(184,210)(185,213)(186,216)(187,215)(188,214)(189,217)(190,220)(191,219)(192,218)(193,221)(194,224)(195,223)(196,222)(226,228)(230,232)(234,236)(238,240)(242,244)(246,248)(250,252)(253,281)(254,284)(255,283)(256,282)(257,285)(258,288)(259,287)(260,286)(261,289)(262,292)(263,291)(264,290)(265,293)(266,296)(267,295)(268,294)(269,297)(270,300)(271,299)(272,298)(273,301)(274,304)(275,303)(276,302)(277,305)(278,308)(279,307)(280,306)(310,312)(314,316)(318,320)(322,324)(326,328)(330,332)(334,336);;
s2 := (  1,  2)(  5, 26)(  6, 25)(  7, 27)(  8, 28)(  9, 22)( 10, 21)( 11, 23)( 12, 24)( 13, 18)( 14, 17)( 15, 19)( 16, 20)( 29, 58)( 30, 57)( 31, 59)( 32, 60)( 33, 82)( 34, 81)( 35, 83)( 36, 84)( 37, 78)( 38, 77)( 39, 79)( 40, 80)( 41, 74)( 42, 73)( 43, 75)( 44, 76)( 45, 70)( 46, 69)( 47, 71)( 48, 72)( 49, 66)( 50, 65)( 51, 67)( 52, 68)( 53, 62)( 54, 61)( 55, 63)( 56, 64)( 85, 86)( 89,110)( 90,109)( 91,111)( 92,112)( 93,106)( 94,105)( 95,107)( 96,108)( 97,102)( 98,101)( 99,103)(100,104)(113,142)(114,141)(115,143)(116,144)(117,166)(118,165)(119,167)(120,168)(121,162)(122,161)(123,163)(124,164)(125,158)(126,157)(127,159)(128,160)(129,154)(130,153)(131,155)(132,156)(133,150)(134,149)(135,151)(136,152)(137,146)(138,145)(139,147)(140,148)(169,254)(170,253)(171,255)(172,256)(173,278)(174,277)(175,279)(176,280)(177,274)(178,273)(179,275)(180,276)(181,270)(182,269)(183,271)(184,272)(185,266)(186,265)(187,267)(188,268)(189,262)(190,261)(191,263)(192,264)(193,258)(194,257)(195,259)(196,260)(197,310)(198,309)(199,311)(200,312)(201,334)(202,333)(203,335)(204,336)(205,330)(206,329)(207,331)(208,332)(209,326)(210,325)(211,327)(212,328)(213,322)(214,321)(215,323)(216,324)(217,318)(218,317)(219,319)(220,320)(221,314)(222,313)(223,315)(224,316)(225,282)(226,281)(227,283)(228,284)(229,306)(230,305)(231,307)(232,308)(233,302)(234,301)(235,303)(236,304)(237,298)(238,297)(239,299)(240,300)(241,294)(242,293)(243,295)(244,296)(245,290)(246,289)(247,291)(248,292)(249,286)(250,285)(251,287)(252,288);;
s3 := (  1,173)(  2,174)(  3,175)(  4,176)(  5,169)(  6,170)(  7,171)(  8,172)(  9,193)( 10,194)( 11,195)( 12,196)( 13,189)( 14,190)( 15,191)( 16,192)( 17,185)( 18,186)( 19,187)( 20,188)( 21,181)( 22,182)( 23,183)( 24,184)( 25,177)( 26,178)( 27,179)( 28,180)( 29,201)( 30,202)( 31,203)( 32,204)( 33,197)( 34,198)( 35,199)( 36,200)( 37,221)( 38,222)( 39,223)( 40,224)( 41,217)( 42,218)( 43,219)( 44,220)( 45,213)( 46,214)( 47,215)( 48,216)( 49,209)( 50,210)( 51,211)( 52,212)( 53,205)( 54,206)( 55,207)( 56,208)( 57,229)( 58,230)( 59,231)( 60,232)( 61,225)( 62,226)( 63,227)( 64,228)( 65,249)( 66,250)( 67,251)( 68,252)( 69,245)( 70,246)( 71,247)( 72,248)( 73,241)( 74,242)( 75,243)( 76,244)( 77,237)( 78,238)( 79,239)( 80,240)( 81,233)( 82,234)( 83,235)( 84,236)( 85,257)( 86,258)( 87,259)( 88,260)( 89,253)( 90,254)( 91,255)( 92,256)( 93,277)( 94,278)( 95,279)( 96,280)( 97,273)( 98,274)( 99,275)(100,276)(101,269)(102,270)(103,271)(104,272)(105,265)(106,266)(107,267)(108,268)(109,261)(110,262)(111,263)(112,264)(113,285)(114,286)(115,287)(116,288)(117,281)(118,282)(119,283)(120,284)(121,305)(122,306)(123,307)(124,308)(125,301)(126,302)(127,303)(128,304)(129,297)(130,298)(131,299)(132,300)(133,293)(134,294)(135,295)(136,296)(137,289)(138,290)(139,291)(140,292)(141,313)(142,314)(143,315)(144,316)(145,309)(146,310)(147,311)(148,312)(149,333)(150,334)(151,335)(152,336)(153,329)(154,330)(155,331)(156,332)(157,325)(158,326)(159,327)(160,328)(161,321)(162,322)(163,323)(164,324)(165,317)(166,318)(167,319)(168,320);;
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*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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(336)!(  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 29, 57)( 30, 58)( 31, 60)( 32, 59)( 33, 61)( 34, 62)( 35, 64)( 36, 63)( 37, 65)( 38, 66)( 39, 68)( 40, 67)( 41, 69)( 42, 70)( 43, 72)( 44, 71)( 45, 73)( 46, 74)( 47, 76)( 48, 75)( 49, 77)( 50, 78)( 51, 80)( 52, 79)( 53, 81)( 54, 82)( 55, 84)( 56, 83)( 87, 88)( 91, 92)( 95, 96)( 99,100)(103,104)(107,108)(111,112)(113,141)(114,142)(115,144)(116,143)(117,145)(118,146)(119,148)(120,147)(121,149)(122,150)(123,152)(124,151)(125,153)(126,154)(127,156)(128,155)(129,157)(130,158)(131,160)(132,159)(133,161)(134,162)(135,164)(136,163)(137,165)(138,166)(139,168)(140,167)(171,172)(175,176)(179,180)(183,184)(187,188)(191,192)(195,196)(197,225)(198,226)(199,228)(200,227)(201,229)(202,230)(203,232)(204,231)(205,233)(206,234)(207,236)(208,235)(209,237)(210,238)(211,240)(212,239)(213,241)(214,242)(215,244)(216,243)(217,245)(218,246)(219,248)(220,247)(221,249)(222,250)(223,252)(224,251)(255,256)(259,260)(263,264)(267,268)(271,272)(275,276)(279,280)(281,309)(282,310)(283,312)(284,311)(285,313)(286,314)(287,316)(288,315)(289,317)(290,318)(291,320)(292,319)(293,321)(294,322)(295,324)(296,323)(297,325)(298,326)(299,328)(300,327)(301,329)(302,330)(303,332)(304,331)(305,333)(306,334)(307,336)(308,335);
s1 := Sym(336)!(  1, 29)(  2, 32)(  3, 31)(  4, 30)(  5, 33)(  6, 36)(  7, 35)(  8, 34)(  9, 37)( 10, 40)( 11, 39)( 12, 38)( 13, 41)( 14, 44)( 15, 43)( 16, 42)( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 21, 49)( 22, 52)( 23, 51)( 24, 50)( 25, 53)( 26, 56)( 27, 55)( 28, 54)( 58, 60)( 62, 64)( 66, 68)( 70, 72)( 74, 76)( 78, 80)( 82, 84)( 85,113)( 86,116)( 87,115)( 88,114)( 89,117)( 90,120)( 91,119)( 92,118)( 93,121)( 94,124)( 95,123)( 96,122)( 97,125)( 98,128)( 99,127)(100,126)(101,129)(102,132)(103,131)(104,130)(105,133)(106,136)(107,135)(108,134)(109,137)(110,140)(111,139)(112,138)(142,144)(146,148)(150,152)(154,156)(158,160)(162,164)(166,168)(169,197)(170,200)(171,199)(172,198)(173,201)(174,204)(175,203)(176,202)(177,205)(178,208)(179,207)(180,206)(181,209)(182,212)(183,211)(184,210)(185,213)(186,216)(187,215)(188,214)(189,217)(190,220)(191,219)(192,218)(193,221)(194,224)(195,223)(196,222)(226,228)(230,232)(234,236)(238,240)(242,244)(246,248)(250,252)(253,281)(254,284)(255,283)(256,282)(257,285)(258,288)(259,287)(260,286)(261,289)(262,292)(263,291)(264,290)(265,293)(266,296)(267,295)(268,294)(269,297)(270,300)(271,299)(272,298)(273,301)(274,304)(275,303)(276,302)(277,305)(278,308)(279,307)(280,306)(310,312)(314,316)(318,320)(322,324)(326,328)(330,332)(334,336);
s2 := Sym(336)!(  1,  2)(  5, 26)(  6, 25)(  7, 27)(  8, 28)(  9, 22)( 10, 21)( 11, 23)( 12, 24)( 13, 18)( 14, 17)( 15, 19)( 16, 20)( 29, 58)( 30, 57)( 31, 59)( 32, 60)( 33, 82)( 34, 81)( 35, 83)( 36, 84)( 37, 78)( 38, 77)( 39, 79)( 40, 80)( 41, 74)( 42, 73)( 43, 75)( 44, 76)( 45, 70)( 46, 69)( 47, 71)( 48, 72)( 49, 66)( 50, 65)( 51, 67)( 52, 68)( 53, 62)( 54, 61)( 55, 63)( 56, 64)( 85, 86)( 89,110)( 90,109)( 91,111)( 92,112)( 93,106)( 94,105)( 95,107)( 96,108)( 97,102)( 98,101)( 99,103)(100,104)(113,142)(114,141)(115,143)(116,144)(117,166)(118,165)(119,167)(120,168)(121,162)(122,161)(123,163)(124,164)(125,158)(126,157)(127,159)(128,160)(129,154)(130,153)(131,155)(132,156)(133,150)(134,149)(135,151)(136,152)(137,146)(138,145)(139,147)(140,148)(169,254)(170,253)(171,255)(172,256)(173,278)(174,277)(175,279)(176,280)(177,274)(178,273)(179,275)(180,276)(181,270)(182,269)(183,271)(184,272)(185,266)(186,265)(187,267)(188,268)(189,262)(190,261)(191,263)(192,264)(193,258)(194,257)(195,259)(196,260)(197,310)(198,309)(199,311)(200,312)(201,334)(202,333)(203,335)(204,336)(205,330)(206,329)(207,331)(208,332)(209,326)(210,325)(211,327)(212,328)(213,322)(214,321)(215,323)(216,324)(217,318)(218,317)(219,319)(220,320)(221,314)(222,313)(223,315)(224,316)(225,282)(226,281)(227,283)(228,284)(229,306)(230,305)(231,307)(232,308)(233,302)(234,301)(235,303)(236,304)(237,298)(238,297)(239,299)(240,300)(241,294)(242,293)(243,295)(244,296)(245,290)(246,289)(247,291)(248,292)(249,286)(250,285)(251,287)(252,288);
s3 := Sym(336)!(  1,173)(  2,174)(  3,175)(  4,176)(  5,169)(  6,170)(  7,171)(  8,172)(  9,193)( 10,194)( 11,195)( 12,196)( 13,189)( 14,190)( 15,191)( 16,192)( 17,185)( 18,186)( 19,187)( 20,188)( 21,181)( 22,182)( 23,183)( 24,184)( 25,177)( 26,178)( 27,179)( 28,180)( 29,201)( 30,202)( 31,203)( 32,204)( 33,197)( 34,198)( 35,199)( 36,200)( 37,221)( 38,222)( 39,223)( 40,224)( 41,217)( 42,218)( 43,219)( 44,220)( 45,213)( 46,214)( 47,215)( 48,216)( 49,209)( 50,210)( 51,211)( 52,212)( 53,205)( 54,206)( 55,207)( 56,208)( 57,229)( 58,230)( 59,231)( 60,232)( 61,225)( 62,226)( 63,227)( 64,228)( 65,249)( 66,250)( 67,251)( 68,252)( 69,245)( 70,246)( 71,247)( 72,248)( 73,241)( 74,242)( 75,243)( 76,244)( 77,237)( 78,238)( 79,239)( 80,240)( 81,233)( 82,234)( 83,235)( 84,236)( 85,257)( 86,258)( 87,259)( 88,260)( 89,253)( 90,254)( 91,255)( 92,256)( 93,277)( 94,278)( 95,279)( 96,280)( 97,273)( 98,274)( 99,275)(100,276)(101,269)(102,270)(103,271)(104,272)(105,265)(106,266)(107,267)(108,268)(109,261)(110,262)(111,263)(112,264)(113,285)(114,286)(115,287)(116,288)(117,281)(118,282)(119,283)(120,284)(121,305)(122,306)(123,307)(124,308)(125,301)(126,302)(127,303)(128,304)(129,297)(130,298)(131,299)(132,300)(133,293)(134,294)(135,295)(136,296)(137,289)(138,290)(139,291)(140,292)(141,313)(142,314)(143,315)(144,316)(145,309)(146,310)(147,311)(148,312)(149,333)(150,334)(151,335)(152,336)(153,329)(154,330)(155,331)(156,332)(157,325)(158,326)(159,327)(160,328)(161,321)(162,322)(163,323)(164,324)(165,317)(166,318)(167,319)(168,320);
poly := sub<Sym(336)|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*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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.