Polytope of Type {2,4,10}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,10}*640b
if this polytope has a name.
Group : SmallGroup(640,21537)
Rank : 4
Schlafli Type : {2,4,10}
Number of vertices, edges, etc : 2, 16, 80, 40
Order of s0s1s2s3 : 10
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,4,10,2} of size 1280
Vertex Figure Of :
{2,2,4,10} of size 1280
{3,2,4,10} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,4,5}*320
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,8,10}*1280c, {2,8,10}*1280d, {2,4,20}*1280d, {2,4,20}*1280e, {2,4,10}*1280b
3-fold covers : {2,12,10}*1920a, {2,4,30}*1920d
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3, 84)( 4, 83)( 5, 86)( 6, 85)( 7, 88)( 8, 87)( 9, 90)( 10, 89)
( 11, 92)( 12, 91)( 13, 94)( 14, 93)( 15, 96)( 16, 95)( 17, 98)( 18, 97)
( 19,100)( 20, 99)( 21,102)( 22,101)( 23,104)( 24,103)( 25,106)( 26,105)
( 27,108)( 28,107)( 29,110)( 30,109)( 31,112)( 32,111)( 33,114)( 34,113)
( 35,116)( 36,115)( 37,118)( 38,117)( 39,120)( 40,119)( 41,122)( 42,121)
( 43,124)( 44,123)( 45,126)( 46,125)( 47,128)( 48,127)( 49,130)( 50,129)
( 51,132)( 52,131)( 53,134)( 54,133)( 55,136)( 56,135)( 57,138)( 58,137)
( 59,140)( 60,139)( 61,142)( 62,141)( 63,144)( 64,143)( 65,146)( 66,145)
( 67,148)( 68,147)( 69,150)( 70,149)( 71,152)( 72,151)( 73,154)( 74,153)
( 75,156)( 76,155)( 77,158)( 78,157)( 79,160)( 80,159)( 81,162)( 82,161)
(163,244)(164,243)(165,246)(166,245)(167,248)(168,247)(169,250)(170,249)
(171,252)(172,251)(173,254)(174,253)(175,256)(176,255)(177,258)(178,257)
(179,260)(180,259)(181,262)(182,261)(183,264)(184,263)(185,266)(186,265)
(187,268)(188,267)(189,270)(190,269)(191,272)(192,271)(193,274)(194,273)
(195,276)(196,275)(197,278)(198,277)(199,280)(200,279)(201,282)(202,281)
(203,284)(204,283)(205,286)(206,285)(207,288)(208,287)(209,290)(210,289)
(211,292)(212,291)(213,294)(214,293)(215,296)(216,295)(217,298)(218,297)
(219,300)(220,299)(221,302)(222,301)(223,304)(224,303)(225,306)(226,305)
(227,308)(228,307)(229,310)(230,309)(231,312)(232,311)(233,314)(234,313)
(235,316)(236,315)(237,318)(238,317)(239,320)(240,319)(241,322)(242,321);;
s2 := ( 3,163)( 4,173)( 5,176)( 6,170)( 7,167)( 8,177)( 9,172)( 10,166)
( 11,175)( 12,169)( 13,164)( 14,174)( 15,171)( 16,165)( 17,168)( 18,178)
( 19,227)( 20,237)( 21,240)( 22,234)( 23,231)( 24,241)( 25,236)( 26,230)
( 27,239)( 28,233)( 29,228)( 30,238)( 31,235)( 32,229)( 33,232)( 34,242)
( 35,211)( 36,221)( 37,224)( 38,218)( 39,215)( 40,225)( 41,220)( 42,214)
( 43,223)( 44,217)( 45,212)( 46,222)( 47,219)( 48,213)( 49,216)( 50,226)
( 51,195)( 52,205)( 53,208)( 54,202)( 55,199)( 56,209)( 57,204)( 58,198)
( 59,207)( 60,201)( 61,196)( 62,206)( 63,203)( 64,197)( 65,200)( 66,210)
( 67,179)( 68,189)( 69,192)( 70,186)( 71,183)( 72,193)( 73,188)( 74,182)
( 75,191)( 76,185)( 77,180)( 78,190)( 79,187)( 80,181)( 81,184)( 82,194)
( 83,243)( 84,253)( 85,256)( 86,250)( 87,247)( 88,257)( 89,252)( 90,246)
( 91,255)( 92,249)( 93,244)( 94,254)( 95,251)( 96,245)( 97,248)( 98,258)
( 99,307)(100,317)(101,320)(102,314)(103,311)(104,321)(105,316)(106,310)
(107,319)(108,313)(109,308)(110,318)(111,315)(112,309)(113,312)(114,322)
(115,291)(116,301)(117,304)(118,298)(119,295)(120,305)(121,300)(122,294)
(123,303)(124,297)(125,292)(126,302)(127,299)(128,293)(129,296)(130,306)
(131,275)(132,285)(133,288)(134,282)(135,279)(136,289)(137,284)(138,278)
(139,287)(140,281)(141,276)(142,286)(143,283)(144,277)(145,280)(146,290)
(147,259)(148,269)(149,272)(150,266)(151,263)(152,273)(153,268)(154,262)
(155,271)(156,265)(157,260)(158,270)(159,267)(160,261)(161,264)(162,274);;
s3 := ( 3,275)( 4,276)( 5,278)( 6,277)( 7,280)( 8,279)( 9,281)( 10,282)
( 11,290)( 12,289)( 13,287)( 14,288)( 15,285)( 16,286)( 17,284)( 18,283)
( 19,259)( 20,260)( 21,262)( 22,261)( 23,264)( 24,263)( 25,265)( 26,266)
( 27,274)( 28,273)( 29,271)( 30,272)( 31,269)( 32,270)( 33,268)( 34,267)
( 35,243)( 36,244)( 37,246)( 38,245)( 39,248)( 40,247)( 41,249)( 42,250)
( 43,258)( 44,257)( 45,255)( 46,256)( 47,253)( 48,254)( 49,252)( 50,251)
( 51,307)( 52,308)( 53,310)( 54,309)( 55,312)( 56,311)( 57,313)( 58,314)
( 59,322)( 60,321)( 61,319)( 62,320)( 63,317)( 64,318)( 65,316)( 66,315)
( 67,291)( 68,292)( 69,294)( 70,293)( 71,296)( 72,295)( 73,297)( 74,298)
( 75,306)( 76,305)( 77,303)( 78,304)( 79,301)( 80,302)( 81,300)( 82,299)
( 83,195)( 84,196)( 85,198)( 86,197)( 87,200)( 88,199)( 89,201)( 90,202)
( 91,210)( 92,209)( 93,207)( 94,208)( 95,205)( 96,206)( 97,204)( 98,203)
( 99,179)(100,180)(101,182)(102,181)(103,184)(104,183)(105,185)(106,186)
(107,194)(108,193)(109,191)(110,192)(111,189)(112,190)(113,188)(114,187)
(115,163)(116,164)(117,166)(118,165)(119,168)(120,167)(121,169)(122,170)
(123,178)(124,177)(125,175)(126,176)(127,173)(128,174)(129,172)(130,171)
(131,227)(132,228)(133,230)(134,229)(135,232)(136,231)(137,233)(138,234)
(139,242)(140,241)(141,239)(142,240)(143,237)(144,238)(145,236)(146,235)
(147,211)(148,212)(149,214)(150,213)(151,216)(152,215)(153,217)(154,218)
(155,226)(156,225)(157,223)(158,224)(159,221)(160,222)(161,220)(162,219);;
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*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(322)!(1,2);
s1 := Sym(322)!( 3, 84)( 4, 83)( 5, 86)( 6, 85)( 7, 88)( 8, 87)( 9, 90)
( 10, 89)( 11, 92)( 12, 91)( 13, 94)( 14, 93)( 15, 96)( 16, 95)( 17, 98)
( 18, 97)( 19,100)( 20, 99)( 21,102)( 22,101)( 23,104)( 24,103)( 25,106)
( 26,105)( 27,108)( 28,107)( 29,110)( 30,109)( 31,112)( 32,111)( 33,114)
( 34,113)( 35,116)( 36,115)( 37,118)( 38,117)( 39,120)( 40,119)( 41,122)
( 42,121)( 43,124)( 44,123)( 45,126)( 46,125)( 47,128)( 48,127)( 49,130)
( 50,129)( 51,132)( 52,131)( 53,134)( 54,133)( 55,136)( 56,135)( 57,138)
( 58,137)( 59,140)( 60,139)( 61,142)( 62,141)( 63,144)( 64,143)( 65,146)
( 66,145)( 67,148)( 68,147)( 69,150)( 70,149)( 71,152)( 72,151)( 73,154)
( 74,153)( 75,156)( 76,155)( 77,158)( 78,157)( 79,160)( 80,159)( 81,162)
( 82,161)(163,244)(164,243)(165,246)(166,245)(167,248)(168,247)(169,250)
(170,249)(171,252)(172,251)(173,254)(174,253)(175,256)(176,255)(177,258)
(178,257)(179,260)(180,259)(181,262)(182,261)(183,264)(184,263)(185,266)
(186,265)(187,268)(188,267)(189,270)(190,269)(191,272)(192,271)(193,274)
(194,273)(195,276)(196,275)(197,278)(198,277)(199,280)(200,279)(201,282)
(202,281)(203,284)(204,283)(205,286)(206,285)(207,288)(208,287)(209,290)
(210,289)(211,292)(212,291)(213,294)(214,293)(215,296)(216,295)(217,298)
(218,297)(219,300)(220,299)(221,302)(222,301)(223,304)(224,303)(225,306)
(226,305)(227,308)(228,307)(229,310)(230,309)(231,312)(232,311)(233,314)
(234,313)(235,316)(236,315)(237,318)(238,317)(239,320)(240,319)(241,322)
(242,321);
s2 := Sym(322)!( 3,163)( 4,173)( 5,176)( 6,170)( 7,167)( 8,177)( 9,172)
( 10,166)( 11,175)( 12,169)( 13,164)( 14,174)( 15,171)( 16,165)( 17,168)
( 18,178)( 19,227)( 20,237)( 21,240)( 22,234)( 23,231)( 24,241)( 25,236)
( 26,230)( 27,239)( 28,233)( 29,228)( 30,238)( 31,235)( 32,229)( 33,232)
( 34,242)( 35,211)( 36,221)( 37,224)( 38,218)( 39,215)( 40,225)( 41,220)
( 42,214)( 43,223)( 44,217)( 45,212)( 46,222)( 47,219)( 48,213)( 49,216)
( 50,226)( 51,195)( 52,205)( 53,208)( 54,202)( 55,199)( 56,209)( 57,204)
( 58,198)( 59,207)( 60,201)( 61,196)( 62,206)( 63,203)( 64,197)( 65,200)
( 66,210)( 67,179)( 68,189)( 69,192)( 70,186)( 71,183)( 72,193)( 73,188)
( 74,182)( 75,191)( 76,185)( 77,180)( 78,190)( 79,187)( 80,181)( 81,184)
( 82,194)( 83,243)( 84,253)( 85,256)( 86,250)( 87,247)( 88,257)( 89,252)
( 90,246)( 91,255)( 92,249)( 93,244)( 94,254)( 95,251)( 96,245)( 97,248)
( 98,258)( 99,307)(100,317)(101,320)(102,314)(103,311)(104,321)(105,316)
(106,310)(107,319)(108,313)(109,308)(110,318)(111,315)(112,309)(113,312)
(114,322)(115,291)(116,301)(117,304)(118,298)(119,295)(120,305)(121,300)
(122,294)(123,303)(124,297)(125,292)(126,302)(127,299)(128,293)(129,296)
(130,306)(131,275)(132,285)(133,288)(134,282)(135,279)(136,289)(137,284)
(138,278)(139,287)(140,281)(141,276)(142,286)(143,283)(144,277)(145,280)
(146,290)(147,259)(148,269)(149,272)(150,266)(151,263)(152,273)(153,268)
(154,262)(155,271)(156,265)(157,260)(158,270)(159,267)(160,261)(161,264)
(162,274);
s3 := Sym(322)!( 3,275)( 4,276)( 5,278)( 6,277)( 7,280)( 8,279)( 9,281)
( 10,282)( 11,290)( 12,289)( 13,287)( 14,288)( 15,285)( 16,286)( 17,284)
( 18,283)( 19,259)( 20,260)( 21,262)( 22,261)( 23,264)( 24,263)( 25,265)
( 26,266)( 27,274)( 28,273)( 29,271)( 30,272)( 31,269)( 32,270)( 33,268)
( 34,267)( 35,243)( 36,244)( 37,246)( 38,245)( 39,248)( 40,247)( 41,249)
( 42,250)( 43,258)( 44,257)( 45,255)( 46,256)( 47,253)( 48,254)( 49,252)
( 50,251)( 51,307)( 52,308)( 53,310)( 54,309)( 55,312)( 56,311)( 57,313)
( 58,314)( 59,322)( 60,321)( 61,319)( 62,320)( 63,317)( 64,318)( 65,316)
( 66,315)( 67,291)( 68,292)( 69,294)( 70,293)( 71,296)( 72,295)( 73,297)
( 74,298)( 75,306)( 76,305)( 77,303)( 78,304)( 79,301)( 80,302)( 81,300)
( 82,299)( 83,195)( 84,196)( 85,198)( 86,197)( 87,200)( 88,199)( 89,201)
( 90,202)( 91,210)( 92,209)( 93,207)( 94,208)( 95,205)( 96,206)( 97,204)
( 98,203)( 99,179)(100,180)(101,182)(102,181)(103,184)(104,183)(105,185)
(106,186)(107,194)(108,193)(109,191)(110,192)(111,189)(112,190)(113,188)
(114,187)(115,163)(116,164)(117,166)(118,165)(119,168)(120,167)(121,169)
(122,170)(123,178)(124,177)(125,175)(126,176)(127,173)(128,174)(129,172)
(130,171)(131,227)(132,228)(133,230)(134,229)(135,232)(136,231)(137,233)
(138,234)(139,242)(140,241)(141,239)(142,240)(143,237)(144,238)(145,236)
(146,235)(147,211)(148,212)(149,214)(150,213)(151,216)(152,215)(153,217)
(154,218)(155,226)(156,225)(157,223)(158,224)(159,221)(160,222)(161,220)
(162,219);
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*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 >;
to this polytope