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# Polytope of Type {4,6,10}

Atlas Canonical Name : {4,6,10}*960e
if this polytope has a name.
Group : SmallGroup(960,11372)
Rank : 4
Schlafli Type : {4,6,10}
Number of vertices, edges, etc : 8, 24, 60, 10
Order of s0s1s2s3 : 30
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,6,10,2} of size 1920
Vertex Figure Of :
{2,4,6,10} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,6,10}*480b
4-fold quotients : {2,6,10}*240
5-fold quotients : {4,6,2}*192
10-fold quotients : {4,3,2}*96, {4,6,2}*96b, {4,6,2}*96c
12-fold quotients : {2,2,10}*80
20-fold quotients : {4,3,2}*48, {2,6,2}*48
24-fold quotients : {2,2,5}*40
40-fold quotients : {2,3,2}*24
60-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,12,10}*1920b, {4,6,20}*1920a, {4,6,10}*1920b, {4,12,10}*1920c, {8,6,10}*1920a, {8,6,10}*1920b
Permutation Representation (GAP) :
```s0 := (  1,123)(  2,124)(  3,121)(  4,122)(  5,127)(  6,128)(  7,125)(  8,126)
(  9,131)( 10,132)( 11,129)( 12,130)( 13,135)( 14,136)( 15,133)( 16,134)
( 17,139)( 18,140)( 19,137)( 20,138)( 21,143)( 22,144)( 23,141)( 24,142)
( 25,147)( 26,148)( 27,145)( 28,146)( 29,151)( 30,152)( 31,149)( 32,150)
( 33,155)( 34,156)( 35,153)( 36,154)( 37,159)( 38,160)( 39,157)( 40,158)
( 41,163)( 42,164)( 43,161)( 44,162)( 45,167)( 46,168)( 47,165)( 48,166)
( 49,171)( 50,172)( 51,169)( 52,170)( 53,175)( 54,176)( 55,173)( 56,174)
( 57,179)( 58,180)( 59,177)( 60,178)( 61,183)( 62,184)( 63,181)( 64,182)
( 65,187)( 66,188)( 67,185)( 68,186)( 69,191)( 70,192)( 71,189)( 72,190)
( 73,195)( 74,196)( 75,193)( 76,194)( 77,199)( 78,200)( 79,197)( 80,198)
( 81,203)( 82,204)( 83,201)( 84,202)( 85,207)( 86,208)( 87,205)( 88,206)
( 89,211)( 90,212)( 91,209)( 92,210)( 93,215)( 94,216)( 95,213)( 96,214)
( 97,219)( 98,220)( 99,217)(100,218)(101,223)(102,224)(103,221)(104,222)
(105,227)(106,228)(107,225)(108,226)(109,231)(110,232)(111,229)(112,230)
(113,235)(114,236)(115,233)(116,234)(117,239)(118,240)(119,237)(120,238);;
s1 := (  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 21, 41)( 22, 43)( 23, 42)
( 24, 44)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 49)( 30, 51)( 31, 50)
( 32, 52)( 33, 53)( 34, 55)( 35, 54)( 36, 56)( 37, 57)( 38, 59)( 39, 58)
( 40, 60)( 62, 63)( 66, 67)( 70, 71)( 74, 75)( 78, 79)( 81,101)( 82,103)
( 83,102)( 84,104)( 85,105)( 86,107)( 87,106)( 88,108)( 89,109)( 90,111)
( 91,110)( 92,112)( 93,113)( 94,115)( 95,114)( 96,116)( 97,117)( 98,119)
( 99,118)(100,120)(122,123)(126,127)(130,131)(134,135)(138,139)(141,161)
(142,163)(143,162)(144,164)(145,165)(146,167)(147,166)(148,168)(149,169)
(150,171)(151,170)(152,172)(153,173)(154,175)(155,174)(156,176)(157,177)
(158,179)(159,178)(160,180)(182,183)(186,187)(190,191)(194,195)(198,199)
(201,221)(202,223)(203,222)(204,224)(205,225)(206,227)(207,226)(208,228)
(209,229)(210,231)(211,230)(212,232)(213,233)(214,235)(215,234)(216,236)
(217,237)(218,239)(219,238)(220,240);;
s2 := (  1, 41)(  2, 44)(  3, 43)(  4, 42)(  5, 57)(  6, 60)(  7, 59)(  8, 58)
(  9, 53)( 10, 56)( 11, 55)( 12, 54)( 13, 49)( 14, 52)( 15, 51)( 16, 50)
( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 22, 24)( 25, 37)( 26, 40)( 27, 39)
( 28, 38)( 29, 33)( 30, 36)( 31, 35)( 32, 34)( 61,101)( 62,104)( 63,103)
( 64,102)( 65,117)( 66,120)( 67,119)( 68,118)( 69,113)( 70,116)( 71,115)
( 72,114)( 73,109)( 74,112)( 75,111)( 76,110)( 77,105)( 78,108)( 79,107)
( 80,106)( 82, 84)( 85, 97)( 86,100)( 87, 99)( 88, 98)( 89, 93)( 90, 96)
( 91, 95)( 92, 94)(121,161)(122,164)(123,163)(124,162)(125,177)(126,180)
(127,179)(128,178)(129,173)(130,176)(131,175)(132,174)(133,169)(134,172)
(135,171)(136,170)(137,165)(138,168)(139,167)(140,166)(142,144)(145,157)
(146,160)(147,159)(148,158)(149,153)(150,156)(151,155)(152,154)(181,221)
(182,224)(183,223)(184,222)(185,237)(186,240)(187,239)(188,238)(189,233)
(190,236)(191,235)(192,234)(193,229)(194,232)(195,231)(196,230)(197,225)
(198,228)(199,227)(200,226)(202,204)(205,217)(206,220)(207,219)(208,218)
(209,213)(210,216)(211,215)(212,214);;
s3 := (  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 61)(  6, 62)(  7, 63)(  8, 64)
(  9, 77)( 10, 78)( 11, 79)( 12, 80)( 13, 73)( 14, 74)( 15, 75)( 16, 76)
( 17, 69)( 18, 70)( 19, 71)( 20, 72)( 21, 85)( 22, 86)( 23, 87)( 24, 88)
( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29, 97)( 30, 98)( 31, 99)( 32,100)
( 33, 93)( 34, 94)( 35, 95)( 36, 96)( 37, 89)( 38, 90)( 39, 91)( 40, 92)
( 41,105)( 42,106)( 43,107)( 44,108)( 45,101)( 46,102)( 47,103)( 48,104)
( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)( 56,116)
( 57,109)( 58,110)( 59,111)( 60,112)(121,185)(122,186)(123,187)(124,188)
(125,181)(126,182)(127,183)(128,184)(129,197)(130,198)(131,199)(132,200)
(133,193)(134,194)(135,195)(136,196)(137,189)(138,190)(139,191)(140,192)
(141,205)(142,206)(143,207)(144,208)(145,201)(146,202)(147,203)(148,204)
(149,217)(150,218)(151,219)(152,220)(153,213)(154,214)(155,215)(156,216)
(157,209)(158,210)(159,211)(160,212)(161,225)(162,226)(163,227)(164,228)
(165,221)(166,222)(167,223)(168,224)(169,237)(170,238)(171,239)(172,240)
(173,233)(174,234)(175,235)(176,236)(177,229)(178,230)(179,231)(180,232);;
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*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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)!(  1,123)(  2,124)(  3,121)(  4,122)(  5,127)(  6,128)(  7,125)
(  8,126)(  9,131)( 10,132)( 11,129)( 12,130)( 13,135)( 14,136)( 15,133)
( 16,134)( 17,139)( 18,140)( 19,137)( 20,138)( 21,143)( 22,144)( 23,141)
( 24,142)( 25,147)( 26,148)( 27,145)( 28,146)( 29,151)( 30,152)( 31,149)
( 32,150)( 33,155)( 34,156)( 35,153)( 36,154)( 37,159)( 38,160)( 39,157)
( 40,158)( 41,163)( 42,164)( 43,161)( 44,162)( 45,167)( 46,168)( 47,165)
( 48,166)( 49,171)( 50,172)( 51,169)( 52,170)( 53,175)( 54,176)( 55,173)
( 56,174)( 57,179)( 58,180)( 59,177)( 60,178)( 61,183)( 62,184)( 63,181)
( 64,182)( 65,187)( 66,188)( 67,185)( 68,186)( 69,191)( 70,192)( 71,189)
( 72,190)( 73,195)( 74,196)( 75,193)( 76,194)( 77,199)( 78,200)( 79,197)
( 80,198)( 81,203)( 82,204)( 83,201)( 84,202)( 85,207)( 86,208)( 87,205)
( 88,206)( 89,211)( 90,212)( 91,209)( 92,210)( 93,215)( 94,216)( 95,213)
( 96,214)( 97,219)( 98,220)( 99,217)(100,218)(101,223)(102,224)(103,221)
(104,222)(105,227)(106,228)(107,225)(108,226)(109,231)(110,232)(111,229)
(112,230)(113,235)(114,236)(115,233)(116,234)(117,239)(118,240)(119,237)
(120,238);
s1 := Sym(240)!(  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 21, 41)( 22, 43)
( 23, 42)( 24, 44)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 49)( 30, 51)
( 31, 50)( 32, 52)( 33, 53)( 34, 55)( 35, 54)( 36, 56)( 37, 57)( 38, 59)
( 39, 58)( 40, 60)( 62, 63)( 66, 67)( 70, 71)( 74, 75)( 78, 79)( 81,101)
( 82,103)( 83,102)( 84,104)( 85,105)( 86,107)( 87,106)( 88,108)( 89,109)
( 90,111)( 91,110)( 92,112)( 93,113)( 94,115)( 95,114)( 96,116)( 97,117)
( 98,119)( 99,118)(100,120)(122,123)(126,127)(130,131)(134,135)(138,139)
(141,161)(142,163)(143,162)(144,164)(145,165)(146,167)(147,166)(148,168)
(149,169)(150,171)(151,170)(152,172)(153,173)(154,175)(155,174)(156,176)
(157,177)(158,179)(159,178)(160,180)(182,183)(186,187)(190,191)(194,195)
(198,199)(201,221)(202,223)(203,222)(204,224)(205,225)(206,227)(207,226)
(208,228)(209,229)(210,231)(211,230)(212,232)(213,233)(214,235)(215,234)
(216,236)(217,237)(218,239)(219,238)(220,240);
s2 := Sym(240)!(  1, 41)(  2, 44)(  3, 43)(  4, 42)(  5, 57)(  6, 60)(  7, 59)
(  8, 58)(  9, 53)( 10, 56)( 11, 55)( 12, 54)( 13, 49)( 14, 52)( 15, 51)
( 16, 50)( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 22, 24)( 25, 37)( 26, 40)
( 27, 39)( 28, 38)( 29, 33)( 30, 36)( 31, 35)( 32, 34)( 61,101)( 62,104)
( 63,103)( 64,102)( 65,117)( 66,120)( 67,119)( 68,118)( 69,113)( 70,116)
( 71,115)( 72,114)( 73,109)( 74,112)( 75,111)( 76,110)( 77,105)( 78,108)
( 79,107)( 80,106)( 82, 84)( 85, 97)( 86,100)( 87, 99)( 88, 98)( 89, 93)
( 90, 96)( 91, 95)( 92, 94)(121,161)(122,164)(123,163)(124,162)(125,177)
(126,180)(127,179)(128,178)(129,173)(130,176)(131,175)(132,174)(133,169)
(134,172)(135,171)(136,170)(137,165)(138,168)(139,167)(140,166)(142,144)
(145,157)(146,160)(147,159)(148,158)(149,153)(150,156)(151,155)(152,154)
(181,221)(182,224)(183,223)(184,222)(185,237)(186,240)(187,239)(188,238)
(189,233)(190,236)(191,235)(192,234)(193,229)(194,232)(195,231)(196,230)
(197,225)(198,228)(199,227)(200,226)(202,204)(205,217)(206,220)(207,219)
(208,218)(209,213)(210,216)(211,215)(212,214);
s3 := Sym(240)!(  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 61)(  6, 62)(  7, 63)
(  8, 64)(  9, 77)( 10, 78)( 11, 79)( 12, 80)( 13, 73)( 14, 74)( 15, 75)
( 16, 76)( 17, 69)( 18, 70)( 19, 71)( 20, 72)( 21, 85)( 22, 86)( 23, 87)
( 24, 88)( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29, 97)( 30, 98)( 31, 99)
( 32,100)( 33, 93)( 34, 94)( 35, 95)( 36, 96)( 37, 89)( 38, 90)( 39, 91)
( 40, 92)( 41,105)( 42,106)( 43,107)( 44,108)( 45,101)( 46,102)( 47,103)
( 48,104)( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)
( 56,116)( 57,109)( 58,110)( 59,111)( 60,112)(121,185)(122,186)(123,187)
(124,188)(125,181)(126,182)(127,183)(128,184)(129,197)(130,198)(131,199)
(132,200)(133,193)(134,194)(135,195)(136,196)(137,189)(138,190)(139,191)
(140,192)(141,205)(142,206)(143,207)(144,208)(145,201)(146,202)(147,203)
(148,204)(149,217)(150,218)(151,219)(152,220)(153,213)(154,214)(155,215)
(156,216)(157,209)(158,210)(159,211)(160,212)(161,225)(162,226)(163,227)
(164,228)(165,221)(166,222)(167,223)(168,224)(169,237)(170,238)(171,239)
(172,240)(173,233)(174,234)(175,235)(176,236)(177,229)(178,230)(179,231)
(180,232);
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*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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