Part of the Atlas of Small Regular Polytopes

Polytope of Type {14,4,6}

Atlas Canonical Name {14,4,6}*1344

Overview

Group
SmallGroup(1344,11695)
Rank
4
Schläfli Type
{14,4,6}
Vertices, edges, …
14, 56, 24, 12
Order of s0s1s2s3
42
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

4-fold

7-fold

8-fold

12-fold

14-fold

16-fold

24-fold

28-fold

56-fold

84-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 2

8 facets

14 vertex figures

  • 14 of 2-fold non-regular quotient of {4,6}*96

Representations

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

References

None.

to this polytope.