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# Polytope of Type {3,8,10}

Atlas Canonical Name : {3,8,10}*960
if this polytope has a name.
Group : SmallGroup(960,10979)
Rank : 4
Schlafli Type : {3,8,10}
Number of vertices, edges, etc : 6, 24, 80, 10
Order of s0s1s2s3 : 60
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,8,10,2} of size 1920
Vertex Figure Of :
{2,3,8,10} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,4,10}*480
5-fold quotients : {3,8,2}*192
8-fold quotients : {3,2,10}*120
10-fold quotients : {3,4,2}*96
16-fold quotients : {3,2,5}*60
20-fold quotients : {3,4,2}*48
40-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,8,20}*1920, {6,8,10}*1920a
Permutation Representation (GAP) :
```s0 := (  3,  6)(  4,  5)(  7,  8)( 11, 14)( 12, 13)( 15, 16)( 19, 22)( 20, 21)
( 23, 24)( 27, 30)( 28, 29)( 31, 32)( 35, 38)( 36, 37)( 39, 40)( 41, 81)
( 42, 82)( 43, 86)( 44, 85)( 45, 84)( 46, 83)( 47, 88)( 48, 87)( 49, 89)
( 50, 90)( 51, 94)( 52, 93)( 53, 92)( 54, 91)( 55, 96)( 56, 95)( 57, 97)
( 58, 98)( 59,102)( 60,101)( 61,100)( 62, 99)( 63,104)( 64,103)( 65,105)
( 66,106)( 67,110)( 68,109)( 69,108)( 70,107)( 71,112)( 72,111)( 73,113)
( 74,114)( 75,118)( 76,117)( 77,116)( 78,115)( 79,120)( 80,119)(121,122)
(123,125)(124,126)(129,130)(131,133)(132,134)(137,138)(139,141)(140,142)
(145,146)(147,149)(148,150)(153,154)(155,157)(156,158)(161,202)(162,201)
(163,205)(164,206)(165,203)(166,204)(167,207)(168,208)(169,210)(170,209)
(171,213)(172,214)(173,211)(174,212)(175,215)(176,216)(177,218)(178,217)
(179,221)(180,222)(181,219)(182,220)(183,223)(184,224)(185,226)(186,225)
(187,229)(188,230)(189,227)(190,228)(191,231)(192,232)(193,234)(194,233)
(195,237)(196,238)(197,235)(198,236)(199,239)(200,240);;
s1 := (  1, 41)(  2, 42)(  3, 44)(  4, 43)(  5, 47)(  6, 48)(  7, 45)(  8, 46)
(  9, 49)( 10, 50)( 11, 52)( 12, 51)( 13, 55)( 14, 56)( 15, 53)( 16, 54)
( 17, 57)( 18, 58)( 19, 60)( 20, 59)( 21, 63)( 22, 64)( 23, 61)( 24, 62)
( 25, 65)( 26, 66)( 27, 68)( 28, 67)( 29, 71)( 30, 72)( 31, 69)( 32, 70)
( 33, 73)( 34, 74)( 35, 76)( 36, 75)( 37, 79)( 38, 80)( 39, 77)( 40, 78)
( 83, 84)( 85, 87)( 86, 88)( 91, 92)( 93, 95)( 94, 96)( 99,100)(101,103)
(102,104)(107,108)(109,111)(110,112)(115,116)(117,119)(118,120)(121,162)
(122,161)(123,163)(124,164)(125,168)(126,167)(127,166)(128,165)(129,170)
(130,169)(131,171)(132,172)(133,176)(134,175)(135,174)(136,173)(137,178)
(138,177)(139,179)(140,180)(141,184)(142,183)(143,182)(144,181)(145,186)
(146,185)(147,187)(148,188)(149,192)(150,191)(151,190)(152,189)(153,194)
(154,193)(155,195)(156,196)(157,200)(158,199)(159,198)(160,197)(201,202)
(205,208)(206,207)(209,210)(213,216)(214,215)(217,218)(221,224)(222,223)
(225,226)(229,232)(230,231)(233,234)(237,240)(238,239);;
s2 := (  1,127)(  2,128)(  3,125)(  4,126)(  5,124)(  6,123)(  7,122)(  8,121)
(  9,159)( 10,160)( 11,157)( 12,158)( 13,156)( 14,155)( 15,154)( 16,153)
( 17,151)( 18,152)( 19,149)( 20,150)( 21,148)( 22,147)( 23,146)( 24,145)
( 25,143)( 26,144)( 27,141)( 28,142)( 29,140)( 30,139)( 31,138)( 32,137)
( 33,135)( 34,136)( 35,133)( 36,134)( 37,132)( 38,131)( 39,130)( 40,129)
( 41,167)( 42,168)( 43,165)( 44,166)( 45,164)( 46,163)( 47,162)( 48,161)
( 49,199)( 50,200)( 51,197)( 52,198)( 53,196)( 54,195)( 55,194)( 56,193)
( 57,191)( 58,192)( 59,189)( 60,190)( 61,188)( 62,187)( 63,186)( 64,185)
( 65,183)( 66,184)( 67,181)( 68,182)( 69,180)( 70,179)( 71,178)( 72,177)
( 73,175)( 74,176)( 75,173)( 76,174)( 77,172)( 78,171)( 79,170)( 80,169)
( 81,207)( 82,208)( 83,205)( 84,206)( 85,204)( 86,203)( 87,202)( 88,201)
( 89,239)( 90,240)( 91,237)( 92,238)( 93,236)( 94,235)( 95,234)( 96,233)
( 97,231)( 98,232)( 99,229)(100,230)(101,228)(102,227)(103,226)(104,225)
(105,223)(106,224)(107,221)(108,222)(109,220)(110,219)(111,218)(112,217)
(113,215)(114,216)(115,213)(116,214)(117,212)(118,211)(119,210)(120,209);;
s3 := (  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)(  8, 16)
( 17, 33)( 18, 34)( 19, 35)( 20, 36)( 21, 37)( 22, 38)( 23, 39)( 24, 40)
( 41, 49)( 42, 50)( 43, 51)( 44, 52)( 45, 53)( 46, 54)( 47, 55)( 48, 56)
( 57, 73)( 58, 74)( 59, 75)( 60, 76)( 61, 77)( 62, 78)( 63, 79)( 64, 80)
( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)( 88, 96)
( 97,113)( 98,114)( 99,115)(100,116)(101,117)(102,118)(103,119)(104,120)
(121,129)(122,130)(123,131)(124,132)(125,133)(126,134)(127,135)(128,136)
(137,153)(138,154)(139,155)(140,156)(141,157)(142,158)(143,159)(144,160)
(161,169)(162,170)(163,171)(164,172)(165,173)(166,174)(167,175)(168,176)
(177,193)(178,194)(179,195)(180,196)(181,197)(182,198)(183,199)(184,200)
(201,209)(202,210)(203,211)(204,212)(205,213)(206,214)(207,215)(208,216)
(217,233)(218,234)(219,235)(220,236)(221,237)(222,238)(223,239)(224,240);;
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, s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(240)!(  3,  6)(  4,  5)(  7,  8)( 11, 14)( 12, 13)( 15, 16)( 19, 22)
( 20, 21)( 23, 24)( 27, 30)( 28, 29)( 31, 32)( 35, 38)( 36, 37)( 39, 40)
( 41, 81)( 42, 82)( 43, 86)( 44, 85)( 45, 84)( 46, 83)( 47, 88)( 48, 87)
( 49, 89)( 50, 90)( 51, 94)( 52, 93)( 53, 92)( 54, 91)( 55, 96)( 56, 95)
( 57, 97)( 58, 98)( 59,102)( 60,101)( 61,100)( 62, 99)( 63,104)( 64,103)
( 65,105)( 66,106)( 67,110)( 68,109)( 69,108)( 70,107)( 71,112)( 72,111)
( 73,113)( 74,114)( 75,118)( 76,117)( 77,116)( 78,115)( 79,120)( 80,119)
(121,122)(123,125)(124,126)(129,130)(131,133)(132,134)(137,138)(139,141)
(140,142)(145,146)(147,149)(148,150)(153,154)(155,157)(156,158)(161,202)
(162,201)(163,205)(164,206)(165,203)(166,204)(167,207)(168,208)(169,210)
(170,209)(171,213)(172,214)(173,211)(174,212)(175,215)(176,216)(177,218)
(178,217)(179,221)(180,222)(181,219)(182,220)(183,223)(184,224)(185,226)
(186,225)(187,229)(188,230)(189,227)(190,228)(191,231)(192,232)(193,234)
(194,233)(195,237)(196,238)(197,235)(198,236)(199,239)(200,240);
s1 := Sym(240)!(  1, 41)(  2, 42)(  3, 44)(  4, 43)(  5, 47)(  6, 48)(  7, 45)
(  8, 46)(  9, 49)( 10, 50)( 11, 52)( 12, 51)( 13, 55)( 14, 56)( 15, 53)
( 16, 54)( 17, 57)( 18, 58)( 19, 60)( 20, 59)( 21, 63)( 22, 64)( 23, 61)
( 24, 62)( 25, 65)( 26, 66)( 27, 68)( 28, 67)( 29, 71)( 30, 72)( 31, 69)
( 32, 70)( 33, 73)( 34, 74)( 35, 76)( 36, 75)( 37, 79)( 38, 80)( 39, 77)
( 40, 78)( 83, 84)( 85, 87)( 86, 88)( 91, 92)( 93, 95)( 94, 96)( 99,100)
(101,103)(102,104)(107,108)(109,111)(110,112)(115,116)(117,119)(118,120)
(121,162)(122,161)(123,163)(124,164)(125,168)(126,167)(127,166)(128,165)
(129,170)(130,169)(131,171)(132,172)(133,176)(134,175)(135,174)(136,173)
(137,178)(138,177)(139,179)(140,180)(141,184)(142,183)(143,182)(144,181)
(145,186)(146,185)(147,187)(148,188)(149,192)(150,191)(151,190)(152,189)
(153,194)(154,193)(155,195)(156,196)(157,200)(158,199)(159,198)(160,197)
(201,202)(205,208)(206,207)(209,210)(213,216)(214,215)(217,218)(221,224)
(222,223)(225,226)(229,232)(230,231)(233,234)(237,240)(238,239);
s2 := Sym(240)!(  1,127)(  2,128)(  3,125)(  4,126)(  5,124)(  6,123)(  7,122)
(  8,121)(  9,159)( 10,160)( 11,157)( 12,158)( 13,156)( 14,155)( 15,154)
( 16,153)( 17,151)( 18,152)( 19,149)( 20,150)( 21,148)( 22,147)( 23,146)
( 24,145)( 25,143)( 26,144)( 27,141)( 28,142)( 29,140)( 30,139)( 31,138)
( 32,137)( 33,135)( 34,136)( 35,133)( 36,134)( 37,132)( 38,131)( 39,130)
( 40,129)( 41,167)( 42,168)( 43,165)( 44,166)( 45,164)( 46,163)( 47,162)
( 48,161)( 49,199)( 50,200)( 51,197)( 52,198)( 53,196)( 54,195)( 55,194)
( 56,193)( 57,191)( 58,192)( 59,189)( 60,190)( 61,188)( 62,187)( 63,186)
( 64,185)( 65,183)( 66,184)( 67,181)( 68,182)( 69,180)( 70,179)( 71,178)
( 72,177)( 73,175)( 74,176)( 75,173)( 76,174)( 77,172)( 78,171)( 79,170)
( 80,169)( 81,207)( 82,208)( 83,205)( 84,206)( 85,204)( 86,203)( 87,202)
( 88,201)( 89,239)( 90,240)( 91,237)( 92,238)( 93,236)( 94,235)( 95,234)
( 96,233)( 97,231)( 98,232)( 99,229)(100,230)(101,228)(102,227)(103,226)
(104,225)(105,223)(106,224)(107,221)(108,222)(109,220)(110,219)(111,218)
(112,217)(113,215)(114,216)(115,213)(116,214)(117,212)(118,211)(119,210)
(120,209);
s3 := Sym(240)!(  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)
(  8, 16)( 17, 33)( 18, 34)( 19, 35)( 20, 36)( 21, 37)( 22, 38)( 23, 39)
( 24, 40)( 41, 49)( 42, 50)( 43, 51)( 44, 52)( 45, 53)( 46, 54)( 47, 55)
( 48, 56)( 57, 73)( 58, 74)( 59, 75)( 60, 76)( 61, 77)( 62, 78)( 63, 79)
( 64, 80)( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)
( 88, 96)( 97,113)( 98,114)( 99,115)(100,116)(101,117)(102,118)(103,119)
(104,120)(121,129)(122,130)(123,131)(124,132)(125,133)(126,134)(127,135)
(128,136)(137,153)(138,154)(139,155)(140,156)(141,157)(142,158)(143,159)
(144,160)(161,169)(162,170)(163,171)(164,172)(165,173)(166,174)(167,175)
(168,176)(177,193)(178,194)(179,195)(180,196)(181,197)(182,198)(183,199)
(184,200)(201,209)(202,210)(203,211)(204,212)(205,213)(206,214)(207,215)
(208,216)(217,233)(218,234)(219,235)(220,236)(221,237)(222,238)(223,239)
(224,240);
poly := sub<Sym(240)|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,
s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope