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# Polytope of Type {10,4,2}

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

```

to this polytope