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