Polytope of Type {2,4,8,10}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,8,10}*1280a
if this polytope has a name.
Group : SmallGroup(1280,1035859)
Rank : 5
Schlafli Type : {2,4,8,10}
Number of vertices, edges, etc : 2, 4, 16, 40, 10
Order of s0s1s2s3s4 : 40
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
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 : {2,4,4,10}*640, {2,2,8,10}*640
4-fold quotients : {2,2,4,10}*320, {2,4,2,10}*320
5-fold quotients : {2,4,8,2}*256a
8-fold quotients : {2,4,2,5}*160, {2,2,2,10}*160
10-fold quotients : {2,4,4,2}*128, {2,2,8,2}*128
16-fold quotients : {2,2,2,5}*80
20-fold quotients : {2,2,4,2}*64, {2,4,2,2}*64
40-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3, 83)( 4, 84)( 5, 85)( 6, 86)( 7, 87)( 8, 88)( 9, 89)( 10, 90)
( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)( 16, 96)( 17, 97)( 18, 98)
( 19, 99)( 20,100)( 21,101)( 22,102)( 23,103)( 24,104)( 25,105)( 26,106)
( 27,107)( 28,108)( 29,109)( 30,110)( 31,111)( 32,112)( 33,113)( 34,114)
( 35,115)( 36,116)( 37,117)( 38,118)( 39,119)( 40,120)( 41,121)( 42,122)
( 43,123)( 44,124)( 45,125)( 46,126)( 47,127)( 48,128)( 49,129)( 50,130)
( 51,131)( 52,132)( 53,133)( 54,134)( 55,135)( 56,136)( 57,137)( 58,138)
( 59,139)( 60,140)( 61,141)( 62,142)( 63,143)( 64,144)( 65,145)( 66,146)
( 67,147)( 68,148)( 69,149)( 70,150)( 71,151)( 72,152)( 73,153)( 74,154)
( 75,155)( 76,156)( 77,157)( 78,158)( 79,159)( 80,160)( 81,161)( 82,162)
(163,243)(164,244)(165,245)(166,246)(167,247)(168,248)(169,249)(170,250)
(171,251)(172,252)(173,253)(174,254)(175,255)(176,256)(177,257)(178,258)
(179,259)(180,260)(181,261)(182,262)(183,263)(184,264)(185,265)(186,266)
(187,267)(188,268)(189,269)(190,270)(191,271)(192,272)(193,273)(194,274)
(195,275)(196,276)(197,277)(198,278)(199,279)(200,280)(201,281)(202,282)
(203,283)(204,284)(205,285)(206,286)(207,287)(208,288)(209,289)(210,290)
(211,291)(212,292)(213,293)(214,294)(215,295)(216,296)(217,297)(218,298)
(219,299)(220,300)(221,301)(222,302)(223,303)(224,304)(225,305)(226,306)
(227,307)(228,308)(229,309)(230,310)(231,311)(232,312)(233,313)(234,314)
(235,315)(236,316)(237,317)(238,318)(239,319)(240,320)(241,321)(242,322);;
s2 := ( 23, 28)( 24, 29)( 25, 30)( 26, 31)( 27, 32)( 33, 38)( 34, 39)( 35, 40)
( 36, 41)( 37, 42)( 63, 68)( 64, 69)( 65, 70)( 66, 71)( 67, 72)( 73, 78)
( 74, 79)( 75, 80)( 76, 81)( 77, 82)( 83, 93)( 84, 94)( 85, 95)( 86, 96)
( 87, 97)( 88, 98)( 89, 99)( 90,100)( 91,101)( 92,102)(103,118)(104,119)
(105,120)(106,121)(107,122)(108,113)(109,114)(110,115)(111,116)(112,117)
(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)
(131,141)(132,142)(143,158)(144,159)(145,160)(146,161)(147,162)(148,153)
(149,154)(150,155)(151,156)(152,157)(163,183)(164,184)(165,185)(166,186)
(167,187)(168,188)(169,189)(170,190)(171,191)(172,192)(173,193)(174,194)
(175,195)(176,196)(177,197)(178,198)(179,199)(180,200)(181,201)(182,202)
(203,223)(204,224)(205,225)(206,226)(207,227)(208,228)(209,229)(210,230)
(211,231)(212,232)(213,233)(214,234)(215,235)(216,236)(217,237)(218,238)
(219,239)(220,240)(221,241)(222,242)(243,273)(244,274)(245,275)(246,276)
(247,277)(248,278)(249,279)(250,280)(251,281)(252,282)(253,263)(254,264)
(255,265)(256,266)(257,267)(258,268)(259,269)(260,270)(261,271)(262,272)
(283,313)(284,314)(285,315)(286,316)(287,317)(288,318)(289,319)(290,320)
(291,321)(292,322)(293,303)(294,304)(295,305)(296,306)(297,307)(298,308)
(299,309)(300,310)(301,311)(302,312);;
s3 := ( 3,203)( 4,207)( 5,206)( 6,205)( 7,204)( 8,208)( 9,212)( 10,211)
( 11,210)( 12,209)( 13,213)( 14,217)( 15,216)( 16,215)( 17,214)( 18,218)
( 19,222)( 20,221)( 21,220)( 22,219)( 23,228)( 24,232)( 25,231)( 26,230)
( 27,229)( 28,223)( 29,227)( 30,226)( 31,225)( 32,224)( 33,238)( 34,242)
( 35,241)( 36,240)( 37,239)( 38,233)( 39,237)( 40,236)( 41,235)( 42,234)
( 43,163)( 44,167)( 45,166)( 46,165)( 47,164)( 48,168)( 49,172)( 50,171)
( 51,170)( 52,169)( 53,173)( 54,177)( 55,176)( 56,175)( 57,174)( 58,178)
( 59,182)( 60,181)( 61,180)( 62,179)( 63,188)( 64,192)( 65,191)( 66,190)
( 67,189)( 68,183)( 69,187)( 70,186)( 71,185)( 72,184)( 73,198)( 74,202)
( 75,201)( 76,200)( 77,199)( 78,193)( 79,197)( 80,196)( 81,195)( 82,194)
( 83,283)( 84,287)( 85,286)( 86,285)( 87,284)( 88,288)( 89,292)( 90,291)
( 91,290)( 92,289)( 93,293)( 94,297)( 95,296)( 96,295)( 97,294)( 98,298)
( 99,302)(100,301)(101,300)(102,299)(103,308)(104,312)(105,311)(106,310)
(107,309)(108,303)(109,307)(110,306)(111,305)(112,304)(113,318)(114,322)
(115,321)(116,320)(117,319)(118,313)(119,317)(120,316)(121,315)(122,314)
(123,243)(124,247)(125,246)(126,245)(127,244)(128,248)(129,252)(130,251)
(131,250)(132,249)(133,253)(134,257)(135,256)(136,255)(137,254)(138,258)
(139,262)(140,261)(141,260)(142,259)(143,268)(144,272)(145,271)(146,270)
(147,269)(148,263)(149,267)(150,266)(151,265)(152,264)(153,278)(154,282)
(155,281)(156,280)(157,279)(158,273)(159,277)(160,276)(161,275)(162,274);;
s4 := ( 3, 44)( 4, 43)( 5, 47)( 6, 46)( 7, 45)( 8, 49)( 9, 48)( 10, 52)
( 11, 51)( 12, 50)( 13, 54)( 14, 53)( 15, 57)( 16, 56)( 17, 55)( 18, 59)
( 19, 58)( 20, 62)( 21, 61)( 22, 60)( 23, 64)( 24, 63)( 25, 67)( 26, 66)
( 27, 65)( 28, 69)( 29, 68)( 30, 72)( 31, 71)( 32, 70)( 33, 74)( 34, 73)
( 35, 77)( 36, 76)( 37, 75)( 38, 79)( 39, 78)( 40, 82)( 41, 81)( 42, 80)
( 83,124)( 84,123)( 85,127)( 86,126)( 87,125)( 88,129)( 89,128)( 90,132)
( 91,131)( 92,130)( 93,134)( 94,133)( 95,137)( 96,136)( 97,135)( 98,139)
( 99,138)(100,142)(101,141)(102,140)(103,144)(104,143)(105,147)(106,146)
(107,145)(108,149)(109,148)(110,152)(111,151)(112,150)(113,154)(114,153)
(115,157)(116,156)(117,155)(118,159)(119,158)(120,162)(121,161)(122,160)
(163,204)(164,203)(165,207)(166,206)(167,205)(168,209)(169,208)(170,212)
(171,211)(172,210)(173,214)(174,213)(175,217)(176,216)(177,215)(178,219)
(179,218)(180,222)(181,221)(182,220)(183,224)(184,223)(185,227)(186,226)
(187,225)(188,229)(189,228)(190,232)(191,231)(192,230)(193,234)(194,233)
(195,237)(196,236)(197,235)(198,239)(199,238)(200,242)(201,241)(202,240)
(243,284)(244,283)(245,287)(246,286)(247,285)(248,289)(249,288)(250,292)
(251,291)(252,290)(253,294)(254,293)(255,297)(256,296)(257,295)(258,299)
(259,298)(260,302)(261,301)(262,300)(263,304)(264,303)(265,307)(266,306)
(267,305)(268,309)(269,308)(270,312)(271,311)(272,310)(273,314)(274,313)
(275,317)(276,316)(277,315)(278,319)(279,318)(280,322)(281,321)(282,320);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s4*s3*s2*s3*s4*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(322)!(1,2);
s1 := Sym(322)!( 3, 83)( 4, 84)( 5, 85)( 6, 86)( 7, 87)( 8, 88)( 9, 89)
( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)( 16, 96)( 17, 97)
( 18, 98)( 19, 99)( 20,100)( 21,101)( 22,102)( 23,103)( 24,104)( 25,105)
( 26,106)( 27,107)( 28,108)( 29,109)( 30,110)( 31,111)( 32,112)( 33,113)
( 34,114)( 35,115)( 36,116)( 37,117)( 38,118)( 39,119)( 40,120)( 41,121)
( 42,122)( 43,123)( 44,124)( 45,125)( 46,126)( 47,127)( 48,128)( 49,129)
( 50,130)( 51,131)( 52,132)( 53,133)( 54,134)( 55,135)( 56,136)( 57,137)
( 58,138)( 59,139)( 60,140)( 61,141)( 62,142)( 63,143)( 64,144)( 65,145)
( 66,146)( 67,147)( 68,148)( 69,149)( 70,150)( 71,151)( 72,152)( 73,153)
( 74,154)( 75,155)( 76,156)( 77,157)( 78,158)( 79,159)( 80,160)( 81,161)
( 82,162)(163,243)(164,244)(165,245)(166,246)(167,247)(168,248)(169,249)
(170,250)(171,251)(172,252)(173,253)(174,254)(175,255)(176,256)(177,257)
(178,258)(179,259)(180,260)(181,261)(182,262)(183,263)(184,264)(185,265)
(186,266)(187,267)(188,268)(189,269)(190,270)(191,271)(192,272)(193,273)
(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)(200,280)(201,281)
(202,282)(203,283)(204,284)(205,285)(206,286)(207,287)(208,288)(209,289)
(210,290)(211,291)(212,292)(213,293)(214,294)(215,295)(216,296)(217,297)
(218,298)(219,299)(220,300)(221,301)(222,302)(223,303)(224,304)(225,305)
(226,306)(227,307)(228,308)(229,309)(230,310)(231,311)(232,312)(233,313)
(234,314)(235,315)(236,316)(237,317)(238,318)(239,319)(240,320)(241,321)
(242,322);
s2 := Sym(322)!( 23, 28)( 24, 29)( 25, 30)( 26, 31)( 27, 32)( 33, 38)( 34, 39)
( 35, 40)( 36, 41)( 37, 42)( 63, 68)( 64, 69)( 65, 70)( 66, 71)( 67, 72)
( 73, 78)( 74, 79)( 75, 80)( 76, 81)( 77, 82)( 83, 93)( 84, 94)( 85, 95)
( 86, 96)( 87, 97)( 88, 98)( 89, 99)( 90,100)( 91,101)( 92,102)(103,118)
(104,119)(105,120)(106,121)(107,122)(108,113)(109,114)(110,115)(111,116)
(112,117)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)
(130,140)(131,141)(132,142)(143,158)(144,159)(145,160)(146,161)(147,162)
(148,153)(149,154)(150,155)(151,156)(152,157)(163,183)(164,184)(165,185)
(166,186)(167,187)(168,188)(169,189)(170,190)(171,191)(172,192)(173,193)
(174,194)(175,195)(176,196)(177,197)(178,198)(179,199)(180,200)(181,201)
(182,202)(203,223)(204,224)(205,225)(206,226)(207,227)(208,228)(209,229)
(210,230)(211,231)(212,232)(213,233)(214,234)(215,235)(216,236)(217,237)
(218,238)(219,239)(220,240)(221,241)(222,242)(243,273)(244,274)(245,275)
(246,276)(247,277)(248,278)(249,279)(250,280)(251,281)(252,282)(253,263)
(254,264)(255,265)(256,266)(257,267)(258,268)(259,269)(260,270)(261,271)
(262,272)(283,313)(284,314)(285,315)(286,316)(287,317)(288,318)(289,319)
(290,320)(291,321)(292,322)(293,303)(294,304)(295,305)(296,306)(297,307)
(298,308)(299,309)(300,310)(301,311)(302,312);
s3 := Sym(322)!( 3,203)( 4,207)( 5,206)( 6,205)( 7,204)( 8,208)( 9,212)
( 10,211)( 11,210)( 12,209)( 13,213)( 14,217)( 15,216)( 16,215)( 17,214)
( 18,218)( 19,222)( 20,221)( 21,220)( 22,219)( 23,228)( 24,232)( 25,231)
( 26,230)( 27,229)( 28,223)( 29,227)( 30,226)( 31,225)( 32,224)( 33,238)
( 34,242)( 35,241)( 36,240)( 37,239)( 38,233)( 39,237)( 40,236)( 41,235)
( 42,234)( 43,163)( 44,167)( 45,166)( 46,165)( 47,164)( 48,168)( 49,172)
( 50,171)( 51,170)( 52,169)( 53,173)( 54,177)( 55,176)( 56,175)( 57,174)
( 58,178)( 59,182)( 60,181)( 61,180)( 62,179)( 63,188)( 64,192)( 65,191)
( 66,190)( 67,189)( 68,183)( 69,187)( 70,186)( 71,185)( 72,184)( 73,198)
( 74,202)( 75,201)( 76,200)( 77,199)( 78,193)( 79,197)( 80,196)( 81,195)
( 82,194)( 83,283)( 84,287)( 85,286)( 86,285)( 87,284)( 88,288)( 89,292)
( 90,291)( 91,290)( 92,289)( 93,293)( 94,297)( 95,296)( 96,295)( 97,294)
( 98,298)( 99,302)(100,301)(101,300)(102,299)(103,308)(104,312)(105,311)
(106,310)(107,309)(108,303)(109,307)(110,306)(111,305)(112,304)(113,318)
(114,322)(115,321)(116,320)(117,319)(118,313)(119,317)(120,316)(121,315)
(122,314)(123,243)(124,247)(125,246)(126,245)(127,244)(128,248)(129,252)
(130,251)(131,250)(132,249)(133,253)(134,257)(135,256)(136,255)(137,254)
(138,258)(139,262)(140,261)(141,260)(142,259)(143,268)(144,272)(145,271)
(146,270)(147,269)(148,263)(149,267)(150,266)(151,265)(152,264)(153,278)
(154,282)(155,281)(156,280)(157,279)(158,273)(159,277)(160,276)(161,275)
(162,274);
s4 := Sym(322)!( 3, 44)( 4, 43)( 5, 47)( 6, 46)( 7, 45)( 8, 49)( 9, 48)
( 10, 52)( 11, 51)( 12, 50)( 13, 54)( 14, 53)( 15, 57)( 16, 56)( 17, 55)
( 18, 59)( 19, 58)( 20, 62)( 21, 61)( 22, 60)( 23, 64)( 24, 63)( 25, 67)
( 26, 66)( 27, 65)( 28, 69)( 29, 68)( 30, 72)( 31, 71)( 32, 70)( 33, 74)
( 34, 73)( 35, 77)( 36, 76)( 37, 75)( 38, 79)( 39, 78)( 40, 82)( 41, 81)
( 42, 80)( 83,124)( 84,123)( 85,127)( 86,126)( 87,125)( 88,129)( 89,128)
( 90,132)( 91,131)( 92,130)( 93,134)( 94,133)( 95,137)( 96,136)( 97,135)
( 98,139)( 99,138)(100,142)(101,141)(102,140)(103,144)(104,143)(105,147)
(106,146)(107,145)(108,149)(109,148)(110,152)(111,151)(112,150)(113,154)
(114,153)(115,157)(116,156)(117,155)(118,159)(119,158)(120,162)(121,161)
(122,160)(163,204)(164,203)(165,207)(166,206)(167,205)(168,209)(169,208)
(170,212)(171,211)(172,210)(173,214)(174,213)(175,217)(176,216)(177,215)
(178,219)(179,218)(180,222)(181,221)(182,220)(183,224)(184,223)(185,227)
(186,226)(187,225)(188,229)(189,228)(190,232)(191,231)(192,230)(193,234)
(194,233)(195,237)(196,236)(197,235)(198,239)(199,238)(200,242)(201,241)
(202,240)(243,284)(244,283)(245,287)(246,286)(247,285)(248,289)(249,288)
(250,292)(251,291)(252,290)(253,294)(254,293)(255,297)(256,296)(257,295)
(258,299)(259,298)(260,302)(261,301)(262,300)(263,304)(264,303)(265,307)
(266,306)(267,305)(268,309)(269,308)(270,312)(271,311)(272,310)(273,314)
(274,313)(275,317)(276,316)(277,315)(278,319)(279,318)(280,322)(281,321)
(282,320);
poly := sub<Sym(322)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s4*s3*s2*s3*s4*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope