Polytope of Type {2,10,4}

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

to this polytope