Polytope of Type {3,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,4}*1200
Also Known As : {{3,6}10,{6,4|2}}. if this polytope has another name.
Group : SmallGroup(1200,522)
Rank : 4
Schlafli Type : {3,6,4}
Number of vertices, edges, etc : 25, 75, 100, 4
Order of s0s1s2s3 : 20
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 : {3,6,2}*600
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 5)( 3, 4)( 6, 7)( 8, 10)( 11, 13)( 14, 15)( 16, 19)( 17, 18)
( 21, 25)( 22, 24)( 26, 51)( 27, 55)( 28, 54)( 29, 53)( 30, 52)( 31, 57)
( 32, 56)( 33, 60)( 34, 59)( 35, 58)( 36, 63)( 37, 62)( 38, 61)( 39, 65)
( 40, 64)( 41, 69)( 42, 68)( 43, 67)( 44, 66)( 45, 70)( 46, 75)( 47, 74)
( 48, 73)( 49, 72)( 50, 71)( 77, 80)( 78, 79)( 81, 82)( 83, 85)( 86, 88)
( 89, 90)( 91, 94)( 92, 93)( 96,100)( 97, 99)(101,126)(102,130)(103,129)
(104,128)(105,127)(106,132)(107,131)(108,135)(109,134)(110,133)(111,138)
(112,137)(113,136)(114,140)(115,139)(116,144)(117,143)(118,142)(119,141)
(120,145)(121,150)(122,149)(123,148)(124,147)(125,146)(152,155)(153,154)
(156,157)(158,160)(161,163)(164,165)(166,169)(167,168)(171,175)(172,174)
(176,201)(177,205)(178,204)(179,203)(180,202)(181,207)(182,206)(183,210)
(184,209)(185,208)(186,213)(187,212)(188,211)(189,215)(190,214)(191,219)
(192,218)(193,217)(194,216)(195,220)(196,225)(197,224)(198,223)(199,222)
(200,221)(227,230)(228,229)(231,232)(233,235)(236,238)(239,240)(241,244)
(242,243)(246,250)(247,249)(251,276)(252,280)(253,279)(254,278)(255,277)
(256,282)(257,281)(258,285)(259,284)(260,283)(261,288)(262,287)(263,286)
(264,290)(265,289)(266,294)(267,293)(268,292)(269,291)(270,295)(271,300)
(272,299)(273,298)(274,297)(275,296);;
s1 := ( 1, 26)( 2, 32)( 3, 38)( 4, 44)( 5, 50)( 6, 46)( 7, 27)( 8, 33)
( 9, 39)( 10, 45)( 11, 41)( 12, 47)( 13, 28)( 14, 34)( 15, 40)( 16, 36)
( 17, 42)( 18, 48)( 19, 29)( 20, 35)( 21, 31)( 22, 37)( 23, 43)( 24, 49)
( 25, 30)( 52, 57)( 53, 63)( 54, 69)( 55, 75)( 56, 71)( 59, 64)( 60, 70)
( 61, 66)( 62, 72)( 68, 73)( 76,101)( 77,107)( 78,113)( 79,119)( 80,125)
( 81,121)( 82,102)( 83,108)( 84,114)( 85,120)( 86,116)( 87,122)( 88,103)
( 89,109)( 90,115)( 91,111)( 92,117)( 93,123)( 94,104)( 95,110)( 96,106)
( 97,112)( 98,118)( 99,124)(100,105)(127,132)(128,138)(129,144)(130,150)
(131,146)(134,139)(135,145)(136,141)(137,147)(143,148)(151,176)(152,182)
(153,188)(154,194)(155,200)(156,196)(157,177)(158,183)(159,189)(160,195)
(161,191)(162,197)(163,178)(164,184)(165,190)(166,186)(167,192)(168,198)
(169,179)(170,185)(171,181)(172,187)(173,193)(174,199)(175,180)(202,207)
(203,213)(204,219)(205,225)(206,221)(209,214)(210,220)(211,216)(212,222)
(218,223)(226,251)(227,257)(228,263)(229,269)(230,275)(231,271)(232,252)
(233,258)(234,264)(235,270)(236,266)(237,272)(238,253)(239,259)(240,265)
(241,261)(242,267)(243,273)(244,254)(245,260)(246,256)(247,262)(248,268)
(249,274)(250,255)(277,282)(278,288)(279,294)(280,300)(281,296)(284,289)
(285,295)(286,291)(287,297)(293,298);;
s2 := ( 1, 12)( 2, 13)( 3, 14)( 4, 15)( 5, 11)( 16, 24)( 17, 25)( 18, 21)
( 19, 22)( 20, 23)( 26, 62)( 27, 63)( 28, 64)( 29, 65)( 30, 61)( 31, 56)
( 32, 57)( 33, 58)( 34, 59)( 35, 60)( 36, 55)( 37, 51)( 38, 52)( 39, 53)
( 40, 54)( 41, 74)( 42, 75)( 43, 71)( 44, 72)( 45, 73)( 46, 68)( 47, 69)
( 48, 70)( 49, 66)( 50, 67)( 76, 87)( 77, 88)( 78, 89)( 79, 90)( 80, 86)
( 91, 99)( 92,100)( 93, 96)( 94, 97)( 95, 98)(101,137)(102,138)(103,139)
(104,140)(105,136)(106,131)(107,132)(108,133)(109,134)(110,135)(111,130)
(112,126)(113,127)(114,128)(115,129)(116,149)(117,150)(118,146)(119,147)
(120,148)(121,143)(122,144)(123,145)(124,141)(125,142)(151,237)(152,238)
(153,239)(154,240)(155,236)(156,231)(157,232)(158,233)(159,234)(160,235)
(161,230)(162,226)(163,227)(164,228)(165,229)(166,249)(167,250)(168,246)
(169,247)(170,248)(171,243)(172,244)(173,245)(174,241)(175,242)(176,287)
(177,288)(178,289)(179,290)(180,286)(181,281)(182,282)(183,283)(184,284)
(185,285)(186,280)(187,276)(188,277)(189,278)(190,279)(191,299)(192,300)
(193,296)(194,297)(195,298)(196,293)(197,294)(198,295)(199,291)(200,292)
(201,262)(202,263)(203,264)(204,265)(205,261)(206,256)(207,257)(208,258)
(209,259)(210,260)(211,255)(212,251)(213,252)(214,253)(215,254)(216,274)
(217,275)(218,271)(219,272)(220,273)(221,268)(222,269)(223,270)(224,266)
(225,267);;
s3 := ( 1,151)( 2,152)( 3,153)( 4,154)( 5,155)( 6,156)( 7,157)( 8,158)
( 9,159)( 10,160)( 11,161)( 12,162)( 13,163)( 14,164)( 15,165)( 16,166)
( 17,167)( 18,168)( 19,169)( 20,170)( 21,171)( 22,172)( 23,173)( 24,174)
( 25,175)( 26,176)( 27,177)( 28,178)( 29,179)( 30,180)( 31,181)( 32,182)
( 33,183)( 34,184)( 35,185)( 36,186)( 37,187)( 38,188)( 39,189)( 40,190)
( 41,191)( 42,192)( 43,193)( 44,194)( 45,195)( 46,196)( 47,197)( 48,198)
( 49,199)( 50,200)( 51,201)( 52,202)( 53,203)( 54,204)( 55,205)( 56,206)
( 57,207)( 58,208)( 59,209)( 60,210)( 61,211)( 62,212)( 63,213)( 64,214)
( 65,215)( 66,216)( 67,217)( 68,218)( 69,219)( 70,220)( 71,221)( 72,222)
( 73,223)( 74,224)( 75,225)( 76,226)( 77,227)( 78,228)( 79,229)( 80,230)
( 81,231)( 82,232)( 83,233)( 84,234)( 85,235)( 86,236)( 87,237)( 88,238)
( 89,239)( 90,240)( 91,241)( 92,242)( 93,243)( 94,244)( 95,245)( 96,246)
( 97,247)( 98,248)( 99,249)(100,250)(101,251)(102,252)(103,253)(104,254)
(105,255)(106,256)(107,257)(108,258)(109,259)(110,260)(111,261)(112,262)
(113,263)(114,264)(115,265)(116,266)(117,267)(118,268)(119,269)(120,270)
(121,271)(122,272)(123,273)(124,274)(125,275)(126,276)(127,277)(128,278)
(129,279)(130,280)(131,281)(132,282)(133,283)(134,284)(135,285)(136,286)
(137,287)(138,288)(139,289)(140,290)(141,291)(142,292)(143,293)(144,294)
(145,295)(146,296)(147,297)(148,298)(149,299)(150,300);;
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*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(300)!( 2, 5)( 3, 4)( 6, 7)( 8, 10)( 11, 13)( 14, 15)( 16, 19)
( 17, 18)( 21, 25)( 22, 24)( 26, 51)( 27, 55)( 28, 54)( 29, 53)( 30, 52)
( 31, 57)( 32, 56)( 33, 60)( 34, 59)( 35, 58)( 36, 63)( 37, 62)( 38, 61)
( 39, 65)( 40, 64)( 41, 69)( 42, 68)( 43, 67)( 44, 66)( 45, 70)( 46, 75)
( 47, 74)( 48, 73)( 49, 72)( 50, 71)( 77, 80)( 78, 79)( 81, 82)( 83, 85)
( 86, 88)( 89, 90)( 91, 94)( 92, 93)( 96,100)( 97, 99)(101,126)(102,130)
(103,129)(104,128)(105,127)(106,132)(107,131)(108,135)(109,134)(110,133)
(111,138)(112,137)(113,136)(114,140)(115,139)(116,144)(117,143)(118,142)
(119,141)(120,145)(121,150)(122,149)(123,148)(124,147)(125,146)(152,155)
(153,154)(156,157)(158,160)(161,163)(164,165)(166,169)(167,168)(171,175)
(172,174)(176,201)(177,205)(178,204)(179,203)(180,202)(181,207)(182,206)
(183,210)(184,209)(185,208)(186,213)(187,212)(188,211)(189,215)(190,214)
(191,219)(192,218)(193,217)(194,216)(195,220)(196,225)(197,224)(198,223)
(199,222)(200,221)(227,230)(228,229)(231,232)(233,235)(236,238)(239,240)
(241,244)(242,243)(246,250)(247,249)(251,276)(252,280)(253,279)(254,278)
(255,277)(256,282)(257,281)(258,285)(259,284)(260,283)(261,288)(262,287)
(263,286)(264,290)(265,289)(266,294)(267,293)(268,292)(269,291)(270,295)
(271,300)(272,299)(273,298)(274,297)(275,296);
s1 := Sym(300)!( 1, 26)( 2, 32)( 3, 38)( 4, 44)( 5, 50)( 6, 46)( 7, 27)
( 8, 33)( 9, 39)( 10, 45)( 11, 41)( 12, 47)( 13, 28)( 14, 34)( 15, 40)
( 16, 36)( 17, 42)( 18, 48)( 19, 29)( 20, 35)( 21, 31)( 22, 37)( 23, 43)
( 24, 49)( 25, 30)( 52, 57)( 53, 63)( 54, 69)( 55, 75)( 56, 71)( 59, 64)
( 60, 70)( 61, 66)( 62, 72)( 68, 73)( 76,101)( 77,107)( 78,113)( 79,119)
( 80,125)( 81,121)( 82,102)( 83,108)( 84,114)( 85,120)( 86,116)( 87,122)
( 88,103)( 89,109)( 90,115)( 91,111)( 92,117)( 93,123)( 94,104)( 95,110)
( 96,106)( 97,112)( 98,118)( 99,124)(100,105)(127,132)(128,138)(129,144)
(130,150)(131,146)(134,139)(135,145)(136,141)(137,147)(143,148)(151,176)
(152,182)(153,188)(154,194)(155,200)(156,196)(157,177)(158,183)(159,189)
(160,195)(161,191)(162,197)(163,178)(164,184)(165,190)(166,186)(167,192)
(168,198)(169,179)(170,185)(171,181)(172,187)(173,193)(174,199)(175,180)
(202,207)(203,213)(204,219)(205,225)(206,221)(209,214)(210,220)(211,216)
(212,222)(218,223)(226,251)(227,257)(228,263)(229,269)(230,275)(231,271)
(232,252)(233,258)(234,264)(235,270)(236,266)(237,272)(238,253)(239,259)
(240,265)(241,261)(242,267)(243,273)(244,254)(245,260)(246,256)(247,262)
(248,268)(249,274)(250,255)(277,282)(278,288)(279,294)(280,300)(281,296)
(284,289)(285,295)(286,291)(287,297)(293,298);
s2 := Sym(300)!( 1, 12)( 2, 13)( 3, 14)( 4, 15)( 5, 11)( 16, 24)( 17, 25)
( 18, 21)( 19, 22)( 20, 23)( 26, 62)( 27, 63)( 28, 64)( 29, 65)( 30, 61)
( 31, 56)( 32, 57)( 33, 58)( 34, 59)( 35, 60)( 36, 55)( 37, 51)( 38, 52)
( 39, 53)( 40, 54)( 41, 74)( 42, 75)( 43, 71)( 44, 72)( 45, 73)( 46, 68)
( 47, 69)( 48, 70)( 49, 66)( 50, 67)( 76, 87)( 77, 88)( 78, 89)( 79, 90)
( 80, 86)( 91, 99)( 92,100)( 93, 96)( 94, 97)( 95, 98)(101,137)(102,138)
(103,139)(104,140)(105,136)(106,131)(107,132)(108,133)(109,134)(110,135)
(111,130)(112,126)(113,127)(114,128)(115,129)(116,149)(117,150)(118,146)
(119,147)(120,148)(121,143)(122,144)(123,145)(124,141)(125,142)(151,237)
(152,238)(153,239)(154,240)(155,236)(156,231)(157,232)(158,233)(159,234)
(160,235)(161,230)(162,226)(163,227)(164,228)(165,229)(166,249)(167,250)
(168,246)(169,247)(170,248)(171,243)(172,244)(173,245)(174,241)(175,242)
(176,287)(177,288)(178,289)(179,290)(180,286)(181,281)(182,282)(183,283)
(184,284)(185,285)(186,280)(187,276)(188,277)(189,278)(190,279)(191,299)
(192,300)(193,296)(194,297)(195,298)(196,293)(197,294)(198,295)(199,291)
(200,292)(201,262)(202,263)(203,264)(204,265)(205,261)(206,256)(207,257)
(208,258)(209,259)(210,260)(211,255)(212,251)(213,252)(214,253)(215,254)
(216,274)(217,275)(218,271)(219,272)(220,273)(221,268)(222,269)(223,270)
(224,266)(225,267);
s3 := Sym(300)!( 1,151)( 2,152)( 3,153)( 4,154)( 5,155)( 6,156)( 7,157)
( 8,158)( 9,159)( 10,160)( 11,161)( 12,162)( 13,163)( 14,164)( 15,165)
( 16,166)( 17,167)( 18,168)( 19,169)( 20,170)( 21,171)( 22,172)( 23,173)
( 24,174)( 25,175)( 26,176)( 27,177)( 28,178)( 29,179)( 30,180)( 31,181)
( 32,182)( 33,183)( 34,184)( 35,185)( 36,186)( 37,187)( 38,188)( 39,189)
( 40,190)( 41,191)( 42,192)( 43,193)( 44,194)( 45,195)( 46,196)( 47,197)
( 48,198)( 49,199)( 50,200)( 51,201)( 52,202)( 53,203)( 54,204)( 55,205)
( 56,206)( 57,207)( 58,208)( 59,209)( 60,210)( 61,211)( 62,212)( 63,213)
( 64,214)( 65,215)( 66,216)( 67,217)( 68,218)( 69,219)( 70,220)( 71,221)
( 72,222)( 73,223)( 74,224)( 75,225)( 76,226)( 77,227)( 78,228)( 79,229)
( 80,230)( 81,231)( 82,232)( 83,233)( 84,234)( 85,235)( 86,236)( 87,237)
( 88,238)( 89,239)( 90,240)( 91,241)( 92,242)( 93,243)( 94,244)( 95,245)
( 96,246)( 97,247)( 98,248)( 99,249)(100,250)(101,251)(102,252)(103,253)
(104,254)(105,255)(106,256)(107,257)(108,258)(109,259)(110,260)(111,261)
(112,262)(113,263)(114,264)(115,265)(116,266)(117,267)(118,268)(119,269)
(120,270)(121,271)(122,272)(123,273)(124,274)(125,275)(126,276)(127,277)
(128,278)(129,279)(130,280)(131,281)(132,282)(133,283)(134,284)(135,285)
(136,286)(137,287)(138,288)(139,289)(140,290)(141,291)(142,292)(143,293)
(144,294)(145,295)(146,296)(147,297)(148,298)(149,299)(150,300);
poly := sub<Sym(300)|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*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
to this polytope