Polytope of Type {3,4,34}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4,34}*1632
Also Known As : {{3,4},{4,34|2}}. if this polytope has another name.
Group : SmallGroup(1632,1195)
Rank : 4
Schlafli Type : {3,4,34}
Number of vertices, edges, etc : 6, 12, 136, 34
Order of s0s1s2s3 : 102
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) :
4-fold quotients : {3,2,34}*408
8-fold quotients : {3,2,17}*204
17-fold quotients : {3,4,2}*96
34-fold quotients : {3,4,2}*48
68-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)
( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)( 63, 64)
( 67, 68)( 69,137)( 70,138)( 71,140)( 72,139)( 73,141)( 74,142)( 75,144)
( 76,143)( 77,145)( 78,146)( 79,148)( 80,147)( 81,149)( 82,150)( 83,152)
( 84,151)( 85,153)( 86,154)( 87,156)( 88,155)( 89,157)( 90,158)( 91,160)
( 92,159)( 93,161)( 94,162)( 95,164)( 96,163)( 97,165)( 98,166)( 99,168)
(100,167)(101,169)(102,170)(103,172)(104,171)(105,173)(106,174)(107,176)
(108,175)(109,177)(110,178)(111,180)(112,179)(113,181)(114,182)(115,184)
(116,183)(117,185)(118,186)(119,188)(120,187)(121,189)(122,190)(123,192)
(124,191)(125,193)(126,194)(127,196)(128,195)(129,197)(130,198)(131,200)
(132,199)(133,201)(134,202)(135,204)(136,203)(207,208)(211,212)(215,216)
(219,220)(223,224)(227,228)(231,232)(235,236)(239,240)(243,244)(247,248)
(251,252)(255,256)(259,260)(263,264)(267,268)(271,272)(273,341)(274,342)
(275,344)(276,343)(277,345)(278,346)(279,348)(280,347)(281,349)(282,350)
(283,352)(284,351)(285,353)(286,354)(287,356)(288,355)(289,357)(290,358)
(291,360)(292,359)(293,361)(294,362)(295,364)(296,363)(297,365)(298,366)
(299,368)(300,367)(301,369)(302,370)(303,372)(304,371)(305,373)(306,374)
(307,376)(308,375)(309,377)(310,378)(311,380)(312,379)(313,381)(314,382)
(315,384)(316,383)(317,385)(318,386)(319,388)(320,387)(321,389)(322,390)
(323,392)(324,391)(325,393)(326,394)(327,396)(328,395)(329,397)(330,398)
(331,400)(332,399)(333,401)(334,402)(335,404)(336,403)(337,405)(338,406)
(339,408)(340,407);;
s1 := ( 1, 69)( 2, 72)( 3, 71)( 4, 70)( 5, 73)( 6, 76)( 7, 75)( 8, 74)
( 9, 77)( 10, 80)( 11, 79)( 12, 78)( 13, 81)( 14, 84)( 15, 83)( 16, 82)
( 17, 85)( 18, 88)( 19, 87)( 20, 86)( 21, 89)( 22, 92)( 23, 91)( 24, 90)
( 25, 93)( 26, 96)( 27, 95)( 28, 94)( 29, 97)( 30,100)( 31, 99)( 32, 98)
( 33,101)( 34,104)( 35,103)( 36,102)( 37,105)( 38,108)( 39,107)( 40,106)
( 41,109)( 42,112)( 43,111)( 44,110)( 45,113)( 46,116)( 47,115)( 48,114)
( 49,117)( 50,120)( 51,119)( 52,118)( 53,121)( 54,124)( 55,123)( 56,122)
( 57,125)( 58,128)( 59,127)( 60,126)( 61,129)( 62,132)( 63,131)( 64,130)
( 65,133)( 66,136)( 67,135)( 68,134)(138,140)(142,144)(146,148)(150,152)
(154,156)(158,160)(162,164)(166,168)(170,172)(174,176)(178,180)(182,184)
(186,188)(190,192)(194,196)(198,200)(202,204)(205,273)(206,276)(207,275)
(208,274)(209,277)(210,280)(211,279)(212,278)(213,281)(214,284)(215,283)
(216,282)(217,285)(218,288)(219,287)(220,286)(221,289)(222,292)(223,291)
(224,290)(225,293)(226,296)(227,295)(228,294)(229,297)(230,300)(231,299)
(232,298)(233,301)(234,304)(235,303)(236,302)(237,305)(238,308)(239,307)
(240,306)(241,309)(242,312)(243,311)(244,310)(245,313)(246,316)(247,315)
(248,314)(249,317)(250,320)(251,319)(252,318)(253,321)(254,324)(255,323)
(256,322)(257,325)(258,328)(259,327)(260,326)(261,329)(262,332)(263,331)
(264,330)(265,333)(266,336)(267,335)(268,334)(269,337)(270,340)(271,339)
(272,338)(342,344)(346,348)(350,352)(354,356)(358,360)(362,364)(366,368)
(370,372)(374,376)(378,380)(382,384)(386,388)(390,392)(394,396)(398,400)
(402,404)(406,408);;
s2 := ( 1, 2)( 3, 4)( 5, 66)( 6, 65)( 7, 68)( 8, 67)( 9, 62)( 10, 61)
( 11, 64)( 12, 63)( 13, 58)( 14, 57)( 15, 60)( 16, 59)( 17, 54)( 18, 53)
( 19, 56)( 20, 55)( 21, 50)( 22, 49)( 23, 52)( 24, 51)( 25, 46)( 26, 45)
( 27, 48)( 28, 47)( 29, 42)( 30, 41)( 31, 44)( 32, 43)( 33, 38)( 34, 37)
( 35, 40)( 36, 39)( 69, 70)( 71, 72)( 73,134)( 74,133)( 75,136)( 76,135)
( 77,130)( 78,129)( 79,132)( 80,131)( 81,126)( 82,125)( 83,128)( 84,127)
( 85,122)( 86,121)( 87,124)( 88,123)( 89,118)( 90,117)( 91,120)( 92,119)
( 93,114)( 94,113)( 95,116)( 96,115)( 97,110)( 98,109)( 99,112)(100,111)
(101,106)(102,105)(103,108)(104,107)(137,138)(139,140)(141,202)(142,201)
(143,204)(144,203)(145,198)(146,197)(147,200)(148,199)(149,194)(150,193)
(151,196)(152,195)(153,190)(154,189)(155,192)(156,191)(157,186)(158,185)
(159,188)(160,187)(161,182)(162,181)(163,184)(164,183)(165,178)(166,177)
(167,180)(168,179)(169,174)(170,173)(171,176)(172,175)(205,206)(207,208)
(209,270)(210,269)(211,272)(212,271)(213,266)(214,265)(215,268)(216,267)
(217,262)(218,261)(219,264)(220,263)(221,258)(222,257)(223,260)(224,259)
(225,254)(226,253)(227,256)(228,255)(229,250)(230,249)(231,252)(232,251)
(233,246)(234,245)(235,248)(236,247)(237,242)(238,241)(239,244)(240,243)
(273,274)(275,276)(277,338)(278,337)(279,340)(280,339)(281,334)(282,333)
(283,336)(284,335)(285,330)(286,329)(287,332)(288,331)(289,326)(290,325)
(291,328)(292,327)(293,322)(294,321)(295,324)(296,323)(297,318)(298,317)
(299,320)(300,319)(301,314)(302,313)(303,316)(304,315)(305,310)(306,309)
(307,312)(308,311)(341,342)(343,344)(345,406)(346,405)(347,408)(348,407)
(349,402)(350,401)(351,404)(352,403)(353,398)(354,397)(355,400)(356,399)
(357,394)(358,393)(359,396)(360,395)(361,390)(362,389)(363,392)(364,391)
(365,386)(366,385)(367,388)(368,387)(369,382)(370,381)(371,384)(372,383)
(373,378)(374,377)(375,380)(376,379);;
s3 := ( 1,209)( 2,210)( 3,211)( 4,212)( 5,205)( 6,206)( 7,207)( 8,208)
( 9,269)( 10,270)( 11,271)( 12,272)( 13,265)( 14,266)( 15,267)( 16,268)
( 17,261)( 18,262)( 19,263)( 20,264)( 21,257)( 22,258)( 23,259)( 24,260)
( 25,253)( 26,254)( 27,255)( 28,256)( 29,249)( 30,250)( 31,251)( 32,252)
( 33,245)( 34,246)( 35,247)( 36,248)( 37,241)( 38,242)( 39,243)( 40,244)
( 41,237)( 42,238)( 43,239)( 44,240)( 45,233)( 46,234)( 47,235)( 48,236)
( 49,229)( 50,230)( 51,231)( 52,232)( 53,225)( 54,226)( 55,227)( 56,228)
( 57,221)( 58,222)( 59,223)( 60,224)( 61,217)( 62,218)( 63,219)( 64,220)
( 65,213)( 66,214)( 67,215)( 68,216)( 69,277)( 70,278)( 71,279)( 72,280)
( 73,273)( 74,274)( 75,275)( 76,276)( 77,337)( 78,338)( 79,339)( 80,340)
( 81,333)( 82,334)( 83,335)( 84,336)( 85,329)( 86,330)( 87,331)( 88,332)
( 89,325)( 90,326)( 91,327)( 92,328)( 93,321)( 94,322)( 95,323)( 96,324)
( 97,317)( 98,318)( 99,319)(100,320)(101,313)(102,314)(103,315)(104,316)
(105,309)(106,310)(107,311)(108,312)(109,305)(110,306)(111,307)(112,308)
(113,301)(114,302)(115,303)(116,304)(117,297)(118,298)(119,299)(120,300)
(121,293)(122,294)(123,295)(124,296)(125,289)(126,290)(127,291)(128,292)
(129,285)(130,286)(131,287)(132,288)(133,281)(134,282)(135,283)(136,284)
(137,345)(138,346)(139,347)(140,348)(141,341)(142,342)(143,343)(144,344)
(145,405)(146,406)(147,407)(148,408)(149,401)(150,402)(151,403)(152,404)
(153,397)(154,398)(155,399)(156,400)(157,393)(158,394)(159,395)(160,396)
(161,389)(162,390)(163,391)(164,392)(165,385)(166,386)(167,387)(168,388)
(169,381)(170,382)(171,383)(172,384)(173,377)(174,378)(175,379)(176,380)
(177,373)(178,374)(179,375)(180,376)(181,369)(182,370)(183,371)(184,372)
(185,365)(186,366)(187,367)(188,368)(189,361)(190,362)(191,363)(192,364)
(193,357)(194,358)(195,359)(196,360)(197,353)(198,354)(199,355)(200,356)
(201,349)(202,350)(203,351)(204,352);;
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*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*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*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(408)!( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)
( 31, 32)( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)
( 63, 64)( 67, 68)( 69,137)( 70,138)( 71,140)( 72,139)( 73,141)( 74,142)
( 75,144)( 76,143)( 77,145)( 78,146)( 79,148)( 80,147)( 81,149)( 82,150)
( 83,152)( 84,151)( 85,153)( 86,154)( 87,156)( 88,155)( 89,157)( 90,158)
( 91,160)( 92,159)( 93,161)( 94,162)( 95,164)( 96,163)( 97,165)( 98,166)
( 99,168)(100,167)(101,169)(102,170)(103,172)(104,171)(105,173)(106,174)
(107,176)(108,175)(109,177)(110,178)(111,180)(112,179)(113,181)(114,182)
(115,184)(116,183)(117,185)(118,186)(119,188)(120,187)(121,189)(122,190)
(123,192)(124,191)(125,193)(126,194)(127,196)(128,195)(129,197)(130,198)
(131,200)(132,199)(133,201)(134,202)(135,204)(136,203)(207,208)(211,212)
(215,216)(219,220)(223,224)(227,228)(231,232)(235,236)(239,240)(243,244)
(247,248)(251,252)(255,256)(259,260)(263,264)(267,268)(271,272)(273,341)
(274,342)(275,344)(276,343)(277,345)(278,346)(279,348)(280,347)(281,349)
(282,350)(283,352)(284,351)(285,353)(286,354)(287,356)(288,355)(289,357)
(290,358)(291,360)(292,359)(293,361)(294,362)(295,364)(296,363)(297,365)
(298,366)(299,368)(300,367)(301,369)(302,370)(303,372)(304,371)(305,373)
(306,374)(307,376)(308,375)(309,377)(310,378)(311,380)(312,379)(313,381)
(314,382)(315,384)(316,383)(317,385)(318,386)(319,388)(320,387)(321,389)
(322,390)(323,392)(324,391)(325,393)(326,394)(327,396)(328,395)(329,397)
(330,398)(331,400)(332,399)(333,401)(334,402)(335,404)(336,403)(337,405)
(338,406)(339,408)(340,407);
s1 := Sym(408)!( 1, 69)( 2, 72)( 3, 71)( 4, 70)( 5, 73)( 6, 76)( 7, 75)
( 8, 74)( 9, 77)( 10, 80)( 11, 79)( 12, 78)( 13, 81)( 14, 84)( 15, 83)
( 16, 82)( 17, 85)( 18, 88)( 19, 87)( 20, 86)( 21, 89)( 22, 92)( 23, 91)
( 24, 90)( 25, 93)( 26, 96)( 27, 95)( 28, 94)( 29, 97)( 30,100)( 31, 99)
( 32, 98)( 33,101)( 34,104)( 35,103)( 36,102)( 37,105)( 38,108)( 39,107)
( 40,106)( 41,109)( 42,112)( 43,111)( 44,110)( 45,113)( 46,116)( 47,115)
( 48,114)( 49,117)( 50,120)( 51,119)( 52,118)( 53,121)( 54,124)( 55,123)
( 56,122)( 57,125)( 58,128)( 59,127)( 60,126)( 61,129)( 62,132)( 63,131)
( 64,130)( 65,133)( 66,136)( 67,135)( 68,134)(138,140)(142,144)(146,148)
(150,152)(154,156)(158,160)(162,164)(166,168)(170,172)(174,176)(178,180)
(182,184)(186,188)(190,192)(194,196)(198,200)(202,204)(205,273)(206,276)
(207,275)(208,274)(209,277)(210,280)(211,279)(212,278)(213,281)(214,284)
(215,283)(216,282)(217,285)(218,288)(219,287)(220,286)(221,289)(222,292)
(223,291)(224,290)(225,293)(226,296)(227,295)(228,294)(229,297)(230,300)
(231,299)(232,298)(233,301)(234,304)(235,303)(236,302)(237,305)(238,308)
(239,307)(240,306)(241,309)(242,312)(243,311)(244,310)(245,313)(246,316)
(247,315)(248,314)(249,317)(250,320)(251,319)(252,318)(253,321)(254,324)
(255,323)(256,322)(257,325)(258,328)(259,327)(260,326)(261,329)(262,332)
(263,331)(264,330)(265,333)(266,336)(267,335)(268,334)(269,337)(270,340)
(271,339)(272,338)(342,344)(346,348)(350,352)(354,356)(358,360)(362,364)
(366,368)(370,372)(374,376)(378,380)(382,384)(386,388)(390,392)(394,396)
(398,400)(402,404)(406,408);
s2 := Sym(408)!( 1, 2)( 3, 4)( 5, 66)( 6, 65)( 7, 68)( 8, 67)( 9, 62)
( 10, 61)( 11, 64)( 12, 63)( 13, 58)( 14, 57)( 15, 60)( 16, 59)( 17, 54)
( 18, 53)( 19, 56)( 20, 55)( 21, 50)( 22, 49)( 23, 52)( 24, 51)( 25, 46)
( 26, 45)( 27, 48)( 28, 47)( 29, 42)( 30, 41)( 31, 44)( 32, 43)( 33, 38)
( 34, 37)( 35, 40)( 36, 39)( 69, 70)( 71, 72)( 73,134)( 74,133)( 75,136)
( 76,135)( 77,130)( 78,129)( 79,132)( 80,131)( 81,126)( 82,125)( 83,128)
( 84,127)( 85,122)( 86,121)( 87,124)( 88,123)( 89,118)( 90,117)( 91,120)
( 92,119)( 93,114)( 94,113)( 95,116)( 96,115)( 97,110)( 98,109)( 99,112)
(100,111)(101,106)(102,105)(103,108)(104,107)(137,138)(139,140)(141,202)
(142,201)(143,204)(144,203)(145,198)(146,197)(147,200)(148,199)(149,194)
(150,193)(151,196)(152,195)(153,190)(154,189)(155,192)(156,191)(157,186)
(158,185)(159,188)(160,187)(161,182)(162,181)(163,184)(164,183)(165,178)
(166,177)(167,180)(168,179)(169,174)(170,173)(171,176)(172,175)(205,206)
(207,208)(209,270)(210,269)(211,272)(212,271)(213,266)(214,265)(215,268)
(216,267)(217,262)(218,261)(219,264)(220,263)(221,258)(222,257)(223,260)
(224,259)(225,254)(226,253)(227,256)(228,255)(229,250)(230,249)(231,252)
(232,251)(233,246)(234,245)(235,248)(236,247)(237,242)(238,241)(239,244)
(240,243)(273,274)(275,276)(277,338)(278,337)(279,340)(280,339)(281,334)
(282,333)(283,336)(284,335)(285,330)(286,329)(287,332)(288,331)(289,326)
(290,325)(291,328)(292,327)(293,322)(294,321)(295,324)(296,323)(297,318)
(298,317)(299,320)(300,319)(301,314)(302,313)(303,316)(304,315)(305,310)
(306,309)(307,312)(308,311)(341,342)(343,344)(345,406)(346,405)(347,408)
(348,407)(349,402)(350,401)(351,404)(352,403)(353,398)(354,397)(355,400)
(356,399)(357,394)(358,393)(359,396)(360,395)(361,390)(362,389)(363,392)
(364,391)(365,386)(366,385)(367,388)(368,387)(369,382)(370,381)(371,384)
(372,383)(373,378)(374,377)(375,380)(376,379);
s3 := Sym(408)!( 1,209)( 2,210)( 3,211)( 4,212)( 5,205)( 6,206)( 7,207)
( 8,208)( 9,269)( 10,270)( 11,271)( 12,272)( 13,265)( 14,266)( 15,267)
( 16,268)( 17,261)( 18,262)( 19,263)( 20,264)( 21,257)( 22,258)( 23,259)
( 24,260)( 25,253)( 26,254)( 27,255)( 28,256)( 29,249)( 30,250)( 31,251)
( 32,252)( 33,245)( 34,246)( 35,247)( 36,248)( 37,241)( 38,242)( 39,243)
( 40,244)( 41,237)( 42,238)( 43,239)( 44,240)( 45,233)( 46,234)( 47,235)
( 48,236)( 49,229)( 50,230)( 51,231)( 52,232)( 53,225)( 54,226)( 55,227)
( 56,228)( 57,221)( 58,222)( 59,223)( 60,224)( 61,217)( 62,218)( 63,219)
( 64,220)( 65,213)( 66,214)( 67,215)( 68,216)( 69,277)( 70,278)( 71,279)
( 72,280)( 73,273)( 74,274)( 75,275)( 76,276)( 77,337)( 78,338)( 79,339)
( 80,340)( 81,333)( 82,334)( 83,335)( 84,336)( 85,329)( 86,330)( 87,331)
( 88,332)( 89,325)( 90,326)( 91,327)( 92,328)( 93,321)( 94,322)( 95,323)
( 96,324)( 97,317)( 98,318)( 99,319)(100,320)(101,313)(102,314)(103,315)
(104,316)(105,309)(106,310)(107,311)(108,312)(109,305)(110,306)(111,307)
(112,308)(113,301)(114,302)(115,303)(116,304)(117,297)(118,298)(119,299)
(120,300)(121,293)(122,294)(123,295)(124,296)(125,289)(126,290)(127,291)
(128,292)(129,285)(130,286)(131,287)(132,288)(133,281)(134,282)(135,283)
(136,284)(137,345)(138,346)(139,347)(140,348)(141,341)(142,342)(143,343)
(144,344)(145,405)(146,406)(147,407)(148,408)(149,401)(150,402)(151,403)
(152,404)(153,397)(154,398)(155,399)(156,400)(157,393)(158,394)(159,395)
(160,396)(161,389)(162,390)(163,391)(164,392)(165,385)(166,386)(167,387)
(168,388)(169,381)(170,382)(171,383)(172,384)(173,377)(174,378)(175,379)
(176,380)(177,373)(178,374)(179,375)(180,376)(181,369)(182,370)(183,371)
(184,372)(185,365)(186,366)(187,367)(188,368)(189,361)(190,362)(191,363)
(192,364)(193,357)(194,358)(195,359)(196,360)(197,353)(198,354)(199,355)
(200,356)(201,349)(202,350)(203,351)(204,352);
poly := sub<Sym(408)|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*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*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*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