Polytope of Type {2,4,10}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,10}*1280b
if this polytope has a name.
Group : SmallGroup(1280,1116461)
Rank : 4
Schlafli Type : {2,4,10}
Number of vertices, edges, etc : 2, 32, 160, 80
Order of s0s1s2s3 : 10
Order of s0s1s2s3s2s1 : 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 : {2,4,5}*640, {2,4,10}*640a, {2,4,10}*640b
4-fold quotients : {2,4,5}*320
16-fold quotients : {2,2,10}*80
32-fold quotients : {2,2,5}*40
80-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,171)( 4,172)( 5,173)( 6,174)( 7,175)( 8,176)( 9,177)( 10,178)
( 11,163)( 12,164)( 13,165)( 14,166)( 15,167)( 16,168)( 17,169)( 18,170)
( 19,187)( 20,188)( 21,189)( 22,190)( 23,191)( 24,192)( 25,193)( 26,194)
( 27,179)( 28,180)( 29,181)( 30,182)( 31,183)( 32,184)( 33,185)( 34,186)
( 35,203)( 36,204)( 37,205)( 38,206)( 39,207)( 40,208)( 41,209)( 42,210)
( 43,195)( 44,196)( 45,197)( 46,198)( 47,199)( 48,200)( 49,201)( 50,202)
( 51,219)( 52,220)( 53,221)( 54,222)( 55,223)( 56,224)( 57,225)( 58,226)
( 59,211)( 60,212)( 61,213)( 62,214)( 63,215)( 64,216)( 65,217)( 66,218)
( 67,235)( 68,236)( 69,237)( 70,238)( 71,239)( 72,240)( 73,241)( 74,242)
( 75,227)( 76,228)( 77,229)( 78,230)( 79,231)( 80,232)( 81,233)( 82,234)
( 83,251)( 84,252)( 85,253)( 86,254)( 87,255)( 88,256)( 89,257)( 90,258)
( 91,243)( 92,244)( 93,245)( 94,246)( 95,247)( 96,248)( 97,249)( 98,250)
( 99,267)(100,268)(101,269)(102,270)(103,271)(104,272)(105,273)(106,274)
(107,259)(108,260)(109,261)(110,262)(111,263)(112,264)(113,265)(114,266)
(115,283)(116,284)(117,285)(118,286)(119,287)(120,288)(121,289)(122,290)
(123,275)(124,276)(125,277)(126,278)(127,279)(128,280)(129,281)(130,282)
(131,299)(132,300)(133,301)(134,302)(135,303)(136,304)(137,305)(138,306)
(139,291)(140,292)(141,293)(142,294)(143,295)(144,296)(145,297)(146,298)
(147,315)(148,316)(149,317)(150,318)(151,319)(152,320)(153,321)(154,322)
(155,307)(156,308)(157,309)(158,310)(159,311)(160,312)(161,313)(162,314);;
s2 := ( 4, 13)( 5, 16)( 6, 10)( 8, 17)( 9, 12)( 11, 15)( 19, 67)( 20, 77)
( 21, 80)( 22, 74)( 23, 71)( 24, 81)( 25, 76)( 26, 70)( 27, 79)( 28, 73)
( 29, 68)( 30, 78)( 31, 75)( 32, 69)( 33, 72)( 34, 82)( 35, 51)( 36, 61)
( 37, 64)( 38, 58)( 39, 55)( 40, 65)( 41, 60)( 42, 54)( 43, 63)( 44, 57)
( 45, 52)( 46, 62)( 47, 59)( 48, 53)( 49, 56)( 50, 66)( 84, 93)( 85, 96)
( 86, 90)( 88, 97)( 89, 92)( 91, 95)( 99,147)(100,157)(101,160)(102,154)
(103,151)(104,161)(105,156)(106,150)(107,159)(108,153)(109,148)(110,158)
(111,155)(112,149)(113,152)(114,162)(115,131)(116,141)(117,144)(118,138)
(119,135)(120,145)(121,140)(122,134)(123,143)(124,137)(125,132)(126,142)
(127,139)(128,133)(129,136)(130,146)(164,173)(165,176)(166,170)(168,177)
(169,172)(171,175)(179,227)(180,237)(181,240)(182,234)(183,231)(184,241)
(185,236)(186,230)(187,239)(188,233)(189,228)(190,238)(191,235)(192,229)
(193,232)(194,242)(195,211)(196,221)(197,224)(198,218)(199,215)(200,225)
(201,220)(202,214)(203,223)(204,217)(205,212)(206,222)(207,219)(208,213)
(209,216)(210,226)(244,253)(245,256)(246,250)(248,257)(249,252)(251,255)
(259,307)(260,317)(261,320)(262,314)(263,311)(264,321)(265,316)(266,310)
(267,319)(268,313)(269,308)(270,318)(271,315)(272,309)(273,312)(274,322)
(275,291)(276,301)(277,304)(278,298)(279,295)(280,305)(281,300)(282,294)
(283,303)(284,297)(285,292)(286,302)(287,299)(288,293)(289,296)(290,306);;
s3 := ( 3,147)( 4,159)( 5,154)( 6,158)( 7,156)( 8,152)( 9,161)( 10,149)
( 11,155)( 12,151)( 13,162)( 14,150)( 15,148)( 16,160)( 17,153)( 18,157)
( 19,131)( 20,143)( 21,138)( 22,142)( 23,140)( 24,136)( 25,145)( 26,133)
( 27,139)( 28,135)( 29,146)( 30,134)( 31,132)( 32,144)( 33,137)( 34,141)
( 35,115)( 36,127)( 37,122)( 38,126)( 39,124)( 40,120)( 41,129)( 42,117)
( 43,123)( 44,119)( 45,130)( 46,118)( 47,116)( 48,128)( 49,121)( 50,125)
( 51, 99)( 52,111)( 53,106)( 54,110)( 55,108)( 56,104)( 57,113)( 58,101)
( 59,107)( 60,103)( 61,114)( 62,102)( 63,100)( 64,112)( 65,105)( 66,109)
( 67, 83)( 68, 95)( 69, 90)( 70, 94)( 71, 92)( 72, 88)( 73, 97)( 74, 85)
( 75, 91)( 76, 87)( 77, 98)( 78, 86)( 79, 84)( 80, 96)( 81, 89)( 82, 93)
(163,307)(164,319)(165,314)(166,318)(167,316)(168,312)(169,321)(170,309)
(171,315)(172,311)(173,322)(174,310)(175,308)(176,320)(177,313)(178,317)
(179,291)(180,303)(181,298)(182,302)(183,300)(184,296)(185,305)(186,293)
(187,299)(188,295)(189,306)(190,294)(191,292)(192,304)(193,297)(194,301)
(195,275)(196,287)(197,282)(198,286)(199,284)(200,280)(201,289)(202,277)
(203,283)(204,279)(205,290)(206,278)(207,276)(208,288)(209,281)(210,285)
(211,259)(212,271)(213,266)(214,270)(215,268)(216,264)(217,273)(218,261)
(219,267)(220,263)(221,274)(222,262)(223,260)(224,272)(225,265)(226,269)
(227,243)(228,255)(229,250)(230,254)(231,252)(232,248)(233,257)(234,245)
(235,251)(236,247)(237,258)(238,246)(239,244)(240,256)(241,249)(242,253);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2,
s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2,
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(322)!(1,2);
s1 := Sym(322)!( 3,171)( 4,172)( 5,173)( 6,174)( 7,175)( 8,176)( 9,177)
( 10,178)( 11,163)( 12,164)( 13,165)( 14,166)( 15,167)( 16,168)( 17,169)
( 18,170)( 19,187)( 20,188)( 21,189)( 22,190)( 23,191)( 24,192)( 25,193)
( 26,194)( 27,179)( 28,180)( 29,181)( 30,182)( 31,183)( 32,184)( 33,185)
( 34,186)( 35,203)( 36,204)( 37,205)( 38,206)( 39,207)( 40,208)( 41,209)
( 42,210)( 43,195)( 44,196)( 45,197)( 46,198)( 47,199)( 48,200)( 49,201)
( 50,202)( 51,219)( 52,220)( 53,221)( 54,222)( 55,223)( 56,224)( 57,225)
( 58,226)( 59,211)( 60,212)( 61,213)( 62,214)( 63,215)( 64,216)( 65,217)
( 66,218)( 67,235)( 68,236)( 69,237)( 70,238)( 71,239)( 72,240)( 73,241)
( 74,242)( 75,227)( 76,228)( 77,229)( 78,230)( 79,231)( 80,232)( 81,233)
( 82,234)( 83,251)( 84,252)( 85,253)( 86,254)( 87,255)( 88,256)( 89,257)
( 90,258)( 91,243)( 92,244)( 93,245)( 94,246)( 95,247)( 96,248)( 97,249)
( 98,250)( 99,267)(100,268)(101,269)(102,270)(103,271)(104,272)(105,273)
(106,274)(107,259)(108,260)(109,261)(110,262)(111,263)(112,264)(113,265)
(114,266)(115,283)(116,284)(117,285)(118,286)(119,287)(120,288)(121,289)
(122,290)(123,275)(124,276)(125,277)(126,278)(127,279)(128,280)(129,281)
(130,282)(131,299)(132,300)(133,301)(134,302)(135,303)(136,304)(137,305)
(138,306)(139,291)(140,292)(141,293)(142,294)(143,295)(144,296)(145,297)
(146,298)(147,315)(148,316)(149,317)(150,318)(151,319)(152,320)(153,321)
(154,322)(155,307)(156,308)(157,309)(158,310)(159,311)(160,312)(161,313)
(162,314);
s2 := Sym(322)!( 4, 13)( 5, 16)( 6, 10)( 8, 17)( 9, 12)( 11, 15)( 19, 67)
( 20, 77)( 21, 80)( 22, 74)( 23, 71)( 24, 81)( 25, 76)( 26, 70)( 27, 79)
( 28, 73)( 29, 68)( 30, 78)( 31, 75)( 32, 69)( 33, 72)( 34, 82)( 35, 51)
( 36, 61)( 37, 64)( 38, 58)( 39, 55)( 40, 65)( 41, 60)( 42, 54)( 43, 63)
( 44, 57)( 45, 52)( 46, 62)( 47, 59)( 48, 53)( 49, 56)( 50, 66)( 84, 93)
( 85, 96)( 86, 90)( 88, 97)( 89, 92)( 91, 95)( 99,147)(100,157)(101,160)
(102,154)(103,151)(104,161)(105,156)(106,150)(107,159)(108,153)(109,148)
(110,158)(111,155)(112,149)(113,152)(114,162)(115,131)(116,141)(117,144)
(118,138)(119,135)(120,145)(121,140)(122,134)(123,143)(124,137)(125,132)
(126,142)(127,139)(128,133)(129,136)(130,146)(164,173)(165,176)(166,170)
(168,177)(169,172)(171,175)(179,227)(180,237)(181,240)(182,234)(183,231)
(184,241)(185,236)(186,230)(187,239)(188,233)(189,228)(190,238)(191,235)
(192,229)(193,232)(194,242)(195,211)(196,221)(197,224)(198,218)(199,215)
(200,225)(201,220)(202,214)(203,223)(204,217)(205,212)(206,222)(207,219)
(208,213)(209,216)(210,226)(244,253)(245,256)(246,250)(248,257)(249,252)
(251,255)(259,307)(260,317)(261,320)(262,314)(263,311)(264,321)(265,316)
(266,310)(267,319)(268,313)(269,308)(270,318)(271,315)(272,309)(273,312)
(274,322)(275,291)(276,301)(277,304)(278,298)(279,295)(280,305)(281,300)
(282,294)(283,303)(284,297)(285,292)(286,302)(287,299)(288,293)(289,296)
(290,306);
s3 := Sym(322)!( 3,147)( 4,159)( 5,154)( 6,158)( 7,156)( 8,152)( 9,161)
( 10,149)( 11,155)( 12,151)( 13,162)( 14,150)( 15,148)( 16,160)( 17,153)
( 18,157)( 19,131)( 20,143)( 21,138)( 22,142)( 23,140)( 24,136)( 25,145)
( 26,133)( 27,139)( 28,135)( 29,146)( 30,134)( 31,132)( 32,144)( 33,137)
( 34,141)( 35,115)( 36,127)( 37,122)( 38,126)( 39,124)( 40,120)( 41,129)
( 42,117)( 43,123)( 44,119)( 45,130)( 46,118)( 47,116)( 48,128)( 49,121)
( 50,125)( 51, 99)( 52,111)( 53,106)( 54,110)( 55,108)( 56,104)( 57,113)
( 58,101)( 59,107)( 60,103)( 61,114)( 62,102)( 63,100)( 64,112)( 65,105)
( 66,109)( 67, 83)( 68, 95)( 69, 90)( 70, 94)( 71, 92)( 72, 88)( 73, 97)
( 74, 85)( 75, 91)( 76, 87)( 77, 98)( 78, 86)( 79, 84)( 80, 96)( 81, 89)
( 82, 93)(163,307)(164,319)(165,314)(166,318)(167,316)(168,312)(169,321)
(170,309)(171,315)(172,311)(173,322)(174,310)(175,308)(176,320)(177,313)
(178,317)(179,291)(180,303)(181,298)(182,302)(183,300)(184,296)(185,305)
(186,293)(187,299)(188,295)(189,306)(190,294)(191,292)(192,304)(193,297)
(194,301)(195,275)(196,287)(197,282)(198,286)(199,284)(200,280)(201,289)
(202,277)(203,283)(204,279)(205,290)(206,278)(207,276)(208,288)(209,281)
(210,285)(211,259)(212,271)(213,266)(214,270)(215,268)(216,264)(217,273)
(218,261)(219,267)(220,263)(221,274)(222,262)(223,260)(224,272)(225,265)
(226,269)(227,243)(228,255)(229,250)(230,254)(231,252)(232,248)(233,257)
(234,245)(235,251)(236,247)(237,258)(238,246)(239,244)(240,256)(241,249)
(242,253);
poly := sub<Sym(322)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2,
s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope