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