Polytope of Type {3,8,14}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,8,14}*1344
if this polytope has a name.
Group : SmallGroup(1344,11355)
Rank : 4
Schlafli Type : {3,8,14}
Number of vertices, edges, etc : 6, 24, 112, 14
Order of s0s1s2s3 : 84
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,4,14}*672
7-fold quotients : {3,8,2}*192
8-fold quotients : {3,2,14}*168
14-fold quotients : {3,4,2}*96
16-fold quotients : {3,2,7}*84
28-fold quotients : {3,4,2}*48
56-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 7)( 6, 8)( 11, 12)( 13, 15)( 14, 16)( 19, 20)( 21, 23)
( 22, 24)( 27, 28)( 29, 31)( 30, 32)( 35, 36)( 37, 39)( 38, 40)( 43, 44)
( 45, 47)( 46, 48)( 51, 52)( 53, 55)( 54, 56)( 57,113)( 58,114)( 59,116)
( 60,115)( 61,119)( 62,120)( 63,117)( 64,118)( 65,121)( 66,122)( 67,124)
( 68,123)( 69,127)( 70,128)( 71,125)( 72,126)( 73,129)( 74,130)( 75,132)
( 76,131)( 77,135)( 78,136)( 79,133)( 80,134)( 81,137)( 82,138)( 83,140)
( 84,139)( 85,143)( 86,144)( 87,141)( 88,142)( 89,145)( 90,146)( 91,148)
( 92,147)( 93,151)( 94,152)( 95,149)( 96,150)( 97,153)( 98,154)( 99,156)
(100,155)(101,159)(102,160)(103,157)(104,158)(105,161)(106,162)(107,164)
(108,163)(109,167)(110,168)(111,165)(112,166)(169,170)(173,176)(174,175)
(177,178)(181,184)(182,183)(185,186)(189,192)(190,191)(193,194)(197,200)
(198,199)(201,202)(205,208)(206,207)(209,210)(213,216)(214,215)(217,218)
(221,224)(222,223)(225,282)(226,281)(227,283)(228,284)(229,288)(230,287)
(231,286)(232,285)(233,290)(234,289)(235,291)(236,292)(237,296)(238,295)
(239,294)(240,293)(241,298)(242,297)(243,299)(244,300)(245,304)(246,303)
(247,302)(248,301)(249,306)(250,305)(251,307)(252,308)(253,312)(254,311)
(255,310)(256,309)(257,314)(258,313)(259,315)(260,316)(261,320)(262,319)
(263,318)(264,317)(265,322)(266,321)(267,323)(268,324)(269,328)(270,327)
(271,326)(272,325)(273,330)(274,329)(275,331)(276,332)(277,336)(278,335)
(279,334)(280,333);;
s1 := ( 1, 57)( 2, 58)( 3, 63)( 4, 64)( 5, 62)( 6, 61)( 7, 59)( 8, 60)
( 9, 65)( 10, 66)( 11, 71)( 12, 72)( 13, 70)( 14, 69)( 15, 67)( 16, 68)
( 17, 73)( 18, 74)( 19, 79)( 20, 80)( 21, 78)( 22, 77)( 23, 75)( 24, 76)
( 25, 81)( 26, 82)( 27, 87)( 28, 88)( 29, 86)( 30, 85)( 31, 83)( 32, 84)
( 33, 89)( 34, 90)( 35, 95)( 36, 96)( 37, 94)( 38, 93)( 39, 91)( 40, 92)
( 41, 97)( 42, 98)( 43,103)( 44,104)( 45,102)( 46,101)( 47, 99)( 48,100)
( 49,105)( 50,106)( 51,111)( 52,112)( 53,110)( 54,109)( 55,107)( 56,108)
(115,119)(116,120)(117,118)(123,127)(124,128)(125,126)(131,135)(132,136)
(133,134)(139,143)(140,144)(141,142)(147,151)(148,152)(149,150)(155,159)
(156,160)(157,158)(163,167)(164,168)(165,166)(169,226)(170,225)(171,232)
(172,231)(173,229)(174,230)(175,228)(176,227)(177,234)(178,233)(179,240)
(180,239)(181,237)(182,238)(183,236)(184,235)(185,242)(186,241)(187,248)
(188,247)(189,245)(190,246)(191,244)(192,243)(193,250)(194,249)(195,256)
(196,255)(197,253)(198,254)(199,252)(200,251)(201,258)(202,257)(203,264)
(204,263)(205,261)(206,262)(207,260)(208,259)(209,266)(210,265)(211,272)
(212,271)(213,269)(214,270)(215,268)(216,267)(217,274)(218,273)(219,280)
(220,279)(221,277)(222,278)(223,276)(224,275)(281,282)(283,288)(284,287)
(289,290)(291,296)(292,295)(297,298)(299,304)(300,303)(305,306)(307,312)
(308,311)(313,314)(315,320)(316,319)(321,322)(323,328)(324,327)(329,330)
(331,336)(332,335);;
s2 := ( 1,171)( 2,172)( 3,170)( 4,169)( 5,175)( 6,176)( 7,174)( 8,173)
( 9,219)( 10,220)( 11,218)( 12,217)( 13,223)( 14,224)( 15,222)( 16,221)
( 17,211)( 18,212)( 19,210)( 20,209)( 21,215)( 22,216)( 23,214)( 24,213)
( 25,203)( 26,204)( 27,202)( 28,201)( 29,207)( 30,208)( 31,206)( 32,205)
( 33,195)( 34,196)( 35,194)( 36,193)( 37,199)( 38,200)( 39,198)( 40,197)
( 41,187)( 42,188)( 43,186)( 44,185)( 45,191)( 46,192)( 47,190)( 48,189)
( 49,179)( 50,180)( 51,178)( 52,177)( 53,183)( 54,184)( 55,182)( 56,181)
( 57,227)( 58,228)( 59,226)( 60,225)( 61,231)( 62,232)( 63,230)( 64,229)
( 65,275)( 66,276)( 67,274)( 68,273)( 69,279)( 70,280)( 71,278)( 72,277)
( 73,267)( 74,268)( 75,266)( 76,265)( 77,271)( 78,272)( 79,270)( 80,269)
( 81,259)( 82,260)( 83,258)( 84,257)( 85,263)( 86,264)( 87,262)( 88,261)
( 89,251)( 90,252)( 91,250)( 92,249)( 93,255)( 94,256)( 95,254)( 96,253)
( 97,243)( 98,244)( 99,242)(100,241)(101,247)(102,248)(103,246)(104,245)
(105,235)(106,236)(107,234)(108,233)(109,239)(110,240)(111,238)(112,237)
(113,283)(114,284)(115,282)(116,281)(117,287)(118,288)(119,286)(120,285)
(121,331)(122,332)(123,330)(124,329)(125,335)(126,336)(127,334)(128,333)
(129,323)(130,324)(131,322)(132,321)(133,327)(134,328)(135,326)(136,325)
(137,315)(138,316)(139,314)(140,313)(141,319)(142,320)(143,318)(144,317)
(145,307)(146,308)(147,306)(148,305)(149,311)(150,312)(151,310)(152,309)
(153,299)(154,300)(155,298)(156,297)(157,303)(158,304)(159,302)(160,301)
(161,291)(162,292)(163,290)(164,289)(165,295)(166,296)(167,294)(168,293);;
s3 := ( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 13)( 6, 14)( 7, 15)( 8, 16)
( 17, 49)( 18, 50)( 19, 51)( 20, 52)( 21, 53)( 22, 54)( 23, 55)( 24, 56)
( 25, 41)( 26, 42)( 27, 43)( 28, 44)( 29, 45)( 30, 46)( 31, 47)( 32, 48)
( 57, 65)( 58, 66)( 59, 67)( 60, 68)( 61, 69)( 62, 70)( 63, 71)( 64, 72)
( 73,105)( 74,106)( 75,107)( 76,108)( 77,109)( 78,110)( 79,111)( 80,112)
( 81, 97)( 82, 98)( 83, 99)( 84,100)( 85,101)( 86,102)( 87,103)( 88,104)
(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)
(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)
(137,153)(138,154)(139,155)(140,156)(141,157)(142,158)(143,159)(144,160)
(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,183)(176,184)
(185,217)(186,218)(187,219)(188,220)(189,221)(190,222)(191,223)(192,224)
(193,209)(194,210)(195,211)(196,212)(197,213)(198,214)(199,215)(200,216)
(225,233)(226,234)(227,235)(228,236)(229,237)(230,238)(231,239)(232,240)
(241,273)(242,274)(243,275)(244,276)(245,277)(246,278)(247,279)(248,280)
(249,265)(250,266)(251,267)(252,268)(253,269)(254,270)(255,271)(256,272)
(281,289)(282,290)(283,291)(284,292)(285,293)(286,294)(287,295)(288,296)
(297,329)(298,330)(299,331)(300,332)(301,333)(302,334)(303,335)(304,336)
(305,321)(306,322)(307,323)(308,324)(309,325)(310,326)(311,327)(312,328);;
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*s1*s2*s0*s1*s2*s0*s1*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(336)!( 3, 4)( 5, 7)( 6, 8)( 11, 12)( 13, 15)( 14, 16)( 19, 20)
( 21, 23)( 22, 24)( 27, 28)( 29, 31)( 30, 32)( 35, 36)( 37, 39)( 38, 40)
( 43, 44)( 45, 47)( 46, 48)( 51, 52)( 53, 55)( 54, 56)( 57,113)( 58,114)
( 59,116)( 60,115)( 61,119)( 62,120)( 63,117)( 64,118)( 65,121)( 66,122)
( 67,124)( 68,123)( 69,127)( 70,128)( 71,125)( 72,126)( 73,129)( 74,130)
( 75,132)( 76,131)( 77,135)( 78,136)( 79,133)( 80,134)( 81,137)( 82,138)
( 83,140)( 84,139)( 85,143)( 86,144)( 87,141)( 88,142)( 89,145)( 90,146)
( 91,148)( 92,147)( 93,151)( 94,152)( 95,149)( 96,150)( 97,153)( 98,154)
( 99,156)(100,155)(101,159)(102,160)(103,157)(104,158)(105,161)(106,162)
(107,164)(108,163)(109,167)(110,168)(111,165)(112,166)(169,170)(173,176)
(174,175)(177,178)(181,184)(182,183)(185,186)(189,192)(190,191)(193,194)
(197,200)(198,199)(201,202)(205,208)(206,207)(209,210)(213,216)(214,215)
(217,218)(221,224)(222,223)(225,282)(226,281)(227,283)(228,284)(229,288)
(230,287)(231,286)(232,285)(233,290)(234,289)(235,291)(236,292)(237,296)
(238,295)(239,294)(240,293)(241,298)(242,297)(243,299)(244,300)(245,304)
(246,303)(247,302)(248,301)(249,306)(250,305)(251,307)(252,308)(253,312)
(254,311)(255,310)(256,309)(257,314)(258,313)(259,315)(260,316)(261,320)
(262,319)(263,318)(264,317)(265,322)(266,321)(267,323)(268,324)(269,328)
(270,327)(271,326)(272,325)(273,330)(274,329)(275,331)(276,332)(277,336)
(278,335)(279,334)(280,333);
s1 := Sym(336)!( 1, 57)( 2, 58)( 3, 63)( 4, 64)( 5, 62)( 6, 61)( 7, 59)
( 8, 60)( 9, 65)( 10, 66)( 11, 71)( 12, 72)( 13, 70)( 14, 69)( 15, 67)
( 16, 68)( 17, 73)( 18, 74)( 19, 79)( 20, 80)( 21, 78)( 22, 77)( 23, 75)
( 24, 76)( 25, 81)( 26, 82)( 27, 87)( 28, 88)( 29, 86)( 30, 85)( 31, 83)
( 32, 84)( 33, 89)( 34, 90)( 35, 95)( 36, 96)( 37, 94)( 38, 93)( 39, 91)
( 40, 92)( 41, 97)( 42, 98)( 43,103)( 44,104)( 45,102)( 46,101)( 47, 99)
( 48,100)( 49,105)( 50,106)( 51,111)( 52,112)( 53,110)( 54,109)( 55,107)
( 56,108)(115,119)(116,120)(117,118)(123,127)(124,128)(125,126)(131,135)
(132,136)(133,134)(139,143)(140,144)(141,142)(147,151)(148,152)(149,150)
(155,159)(156,160)(157,158)(163,167)(164,168)(165,166)(169,226)(170,225)
(171,232)(172,231)(173,229)(174,230)(175,228)(176,227)(177,234)(178,233)
(179,240)(180,239)(181,237)(182,238)(183,236)(184,235)(185,242)(186,241)
(187,248)(188,247)(189,245)(190,246)(191,244)(192,243)(193,250)(194,249)
(195,256)(196,255)(197,253)(198,254)(199,252)(200,251)(201,258)(202,257)
(203,264)(204,263)(205,261)(206,262)(207,260)(208,259)(209,266)(210,265)
(211,272)(212,271)(213,269)(214,270)(215,268)(216,267)(217,274)(218,273)
(219,280)(220,279)(221,277)(222,278)(223,276)(224,275)(281,282)(283,288)
(284,287)(289,290)(291,296)(292,295)(297,298)(299,304)(300,303)(305,306)
(307,312)(308,311)(313,314)(315,320)(316,319)(321,322)(323,328)(324,327)
(329,330)(331,336)(332,335);
s2 := Sym(336)!( 1,171)( 2,172)( 3,170)( 4,169)( 5,175)( 6,176)( 7,174)
( 8,173)( 9,219)( 10,220)( 11,218)( 12,217)( 13,223)( 14,224)( 15,222)
( 16,221)( 17,211)( 18,212)( 19,210)( 20,209)( 21,215)( 22,216)( 23,214)
( 24,213)( 25,203)( 26,204)( 27,202)( 28,201)( 29,207)( 30,208)( 31,206)
( 32,205)( 33,195)( 34,196)( 35,194)( 36,193)( 37,199)( 38,200)( 39,198)
( 40,197)( 41,187)( 42,188)( 43,186)( 44,185)( 45,191)( 46,192)( 47,190)
( 48,189)( 49,179)( 50,180)( 51,178)( 52,177)( 53,183)( 54,184)( 55,182)
( 56,181)( 57,227)( 58,228)( 59,226)( 60,225)( 61,231)( 62,232)( 63,230)
( 64,229)( 65,275)( 66,276)( 67,274)( 68,273)( 69,279)( 70,280)( 71,278)
( 72,277)( 73,267)( 74,268)( 75,266)( 76,265)( 77,271)( 78,272)( 79,270)
( 80,269)( 81,259)( 82,260)( 83,258)( 84,257)( 85,263)( 86,264)( 87,262)
( 88,261)( 89,251)( 90,252)( 91,250)( 92,249)( 93,255)( 94,256)( 95,254)
( 96,253)( 97,243)( 98,244)( 99,242)(100,241)(101,247)(102,248)(103,246)
(104,245)(105,235)(106,236)(107,234)(108,233)(109,239)(110,240)(111,238)
(112,237)(113,283)(114,284)(115,282)(116,281)(117,287)(118,288)(119,286)
(120,285)(121,331)(122,332)(123,330)(124,329)(125,335)(126,336)(127,334)
(128,333)(129,323)(130,324)(131,322)(132,321)(133,327)(134,328)(135,326)
(136,325)(137,315)(138,316)(139,314)(140,313)(141,319)(142,320)(143,318)
(144,317)(145,307)(146,308)(147,306)(148,305)(149,311)(150,312)(151,310)
(152,309)(153,299)(154,300)(155,298)(156,297)(157,303)(158,304)(159,302)
(160,301)(161,291)(162,292)(163,290)(164,289)(165,295)(166,296)(167,294)
(168,293);
s3 := Sym(336)!( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 13)( 6, 14)( 7, 15)
( 8, 16)( 17, 49)( 18, 50)( 19, 51)( 20, 52)( 21, 53)( 22, 54)( 23, 55)
( 24, 56)( 25, 41)( 26, 42)( 27, 43)( 28, 44)( 29, 45)( 30, 46)( 31, 47)
( 32, 48)( 57, 65)( 58, 66)( 59, 67)( 60, 68)( 61, 69)( 62, 70)( 63, 71)
( 64, 72)( 73,105)( 74,106)( 75,107)( 76,108)( 77,109)( 78,110)( 79,111)
( 80,112)( 81, 97)( 82, 98)( 83, 99)( 84,100)( 85,101)( 86,102)( 87,103)
( 88,104)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)
(120,128)(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)
(136,168)(137,153)(138,154)(139,155)(140,156)(141,157)(142,158)(143,159)
(144,160)(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,183)
(176,184)(185,217)(186,218)(187,219)(188,220)(189,221)(190,222)(191,223)
(192,224)(193,209)(194,210)(195,211)(196,212)(197,213)(198,214)(199,215)
(200,216)(225,233)(226,234)(227,235)(228,236)(229,237)(230,238)(231,239)
(232,240)(241,273)(242,274)(243,275)(244,276)(245,277)(246,278)(247,279)
(248,280)(249,265)(250,266)(251,267)(252,268)(253,269)(254,270)(255,271)
(256,272)(281,289)(282,290)(283,291)(284,292)(285,293)(286,294)(287,295)
(288,296)(297,329)(298,330)(299,331)(300,332)(301,333)(302,334)(303,335)
(304,336)(305,321)(306,322)(307,323)(308,324)(309,325)(310,326)(311,327)
(312,328);
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*s1*s2*s0*s1*s2*s0*s1*s2*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 >;
References : None.
to this polytope