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# Polytope of Type {15,4,2}

Atlas Canonical Name : {15,4,2}*960
if this polytope has a name.
Group : SmallGroup(960,11375)
Rank : 4
Schlafli Type : {15,4,2}
Number of vertices, edges, etc : 60, 120, 16, 2
Order of s0s1s2s3 : 30
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{15,4,2,2} of size 1920
Vertex Figure Of :
{2,15,4,2} of size 1920
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {5,4,2}*320
Covers (Minimal Covers in Boldface) :
2-fold covers : {15,8,2}*1920b, {15,8,2}*1920c, {15,4,2}*1920, {30,4,2}*1920c, {30,4,2}*1920d
Permutation Representation (GAP) :
```s0 := (  2, 11)(  3, 14)(  4,  8)(  6, 15)(  7, 10)(  9, 13)( 17, 65)( 18, 75)
( 19, 78)( 20, 72)( 21, 69)( 22, 79)( 23, 74)( 24, 68)( 25, 77)( 26, 71)
( 27, 66)( 28, 76)( 29, 73)( 30, 67)( 31, 70)( 32, 80)( 33, 49)( 34, 59)
( 35, 62)( 36, 56)( 37, 53)( 38, 63)( 39, 58)( 40, 52)( 41, 61)( 42, 55)
( 43, 50)( 44, 60)( 45, 57)( 46, 51)( 47, 54)( 48, 64)( 81,161)( 82,171)
( 83,174)( 84,168)( 85,165)( 86,175)( 87,170)( 88,164)( 89,173)( 90,167)
( 91,162)( 92,172)( 93,169)( 94,163)( 95,166)( 96,176)( 97,225)( 98,235)
( 99,238)(100,232)(101,229)(102,239)(103,234)(104,228)(105,237)(106,231)
(107,226)(108,236)(109,233)(110,227)(111,230)(112,240)(113,209)(114,219)
(115,222)(116,216)(117,213)(118,223)(119,218)(120,212)(121,221)(122,215)
(123,210)(124,220)(125,217)(126,211)(127,214)(128,224)(129,193)(130,203)
(131,206)(132,200)(133,197)(134,207)(135,202)(136,196)(137,205)(138,199)
(139,194)(140,204)(141,201)(142,195)(143,198)(144,208)(145,177)(146,187)
(147,190)(148,184)(149,181)(150,191)(151,186)(152,180)(153,189)(154,183)
(155,178)(156,188)(157,185)(158,179)(159,182)(160,192);;
s1 := (  1, 97)(  2, 98)(  3,100)(  4, 99)(  5,102)(  6,101)(  7,103)(  8,104)
(  9,112)( 10,111)( 11,109)( 12,110)( 13,107)( 14,108)( 15,106)( 16,105)
( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 86)( 22, 85)( 23, 87)( 24, 88)
( 25, 96)( 26, 95)( 27, 93)( 28, 94)( 29, 91)( 30, 92)( 31, 90)( 32, 89)
( 33,145)( 34,146)( 35,148)( 36,147)( 37,150)( 38,149)( 39,151)( 40,152)
( 41,160)( 42,159)( 43,157)( 44,158)( 45,155)( 46,156)( 47,154)( 48,153)
( 49,129)( 50,130)( 51,132)( 52,131)( 53,134)( 54,133)( 55,135)( 56,136)
( 57,144)( 58,143)( 59,141)( 60,142)( 61,139)( 62,140)( 63,138)( 64,137)
( 65,113)( 66,114)( 67,116)( 68,115)( 69,118)( 70,117)( 71,119)( 72,120)
( 73,128)( 74,127)( 75,125)( 76,126)( 77,123)( 78,124)( 79,122)( 80,121)
(161,177)(162,178)(163,180)(164,179)(165,182)(166,181)(167,183)(168,184)
(169,192)(170,191)(171,189)(172,190)(173,187)(174,188)(175,186)(176,185)
(193,225)(194,226)(195,228)(196,227)(197,230)(198,229)(199,231)(200,232)
(201,240)(202,239)(203,237)(204,238)(205,235)(206,236)(207,234)(208,233)
(211,212)(213,214)(217,224)(218,223)(219,221)(220,222);;
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);;
s3 := (241,242);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(242)!(  2, 11)(  3, 14)(  4,  8)(  6, 15)(  7, 10)(  9, 13)( 17, 65)
( 18, 75)( 19, 78)( 20, 72)( 21, 69)( 22, 79)( 23, 74)( 24, 68)( 25, 77)
( 26, 71)( 27, 66)( 28, 76)( 29, 73)( 30, 67)( 31, 70)( 32, 80)( 33, 49)
( 34, 59)( 35, 62)( 36, 56)( 37, 53)( 38, 63)( 39, 58)( 40, 52)( 41, 61)
( 42, 55)( 43, 50)( 44, 60)( 45, 57)( 46, 51)( 47, 54)( 48, 64)( 81,161)
( 82,171)( 83,174)( 84,168)( 85,165)( 86,175)( 87,170)( 88,164)( 89,173)
( 90,167)( 91,162)( 92,172)( 93,169)( 94,163)( 95,166)( 96,176)( 97,225)
( 98,235)( 99,238)(100,232)(101,229)(102,239)(103,234)(104,228)(105,237)
(106,231)(107,226)(108,236)(109,233)(110,227)(111,230)(112,240)(113,209)
(114,219)(115,222)(116,216)(117,213)(118,223)(119,218)(120,212)(121,221)
(122,215)(123,210)(124,220)(125,217)(126,211)(127,214)(128,224)(129,193)
(130,203)(131,206)(132,200)(133,197)(134,207)(135,202)(136,196)(137,205)
(138,199)(139,194)(140,204)(141,201)(142,195)(143,198)(144,208)(145,177)
(146,187)(147,190)(148,184)(149,181)(150,191)(151,186)(152,180)(153,189)
(154,183)(155,178)(156,188)(157,185)(158,179)(159,182)(160,192);
s1 := Sym(242)!(  1, 97)(  2, 98)(  3,100)(  4, 99)(  5,102)(  6,101)(  7,103)
(  8,104)(  9,112)( 10,111)( 11,109)( 12,110)( 13,107)( 14,108)( 15,106)
( 16,105)( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 86)( 22, 85)( 23, 87)
( 24, 88)( 25, 96)( 26, 95)( 27, 93)( 28, 94)( 29, 91)( 30, 92)( 31, 90)
( 32, 89)( 33,145)( 34,146)( 35,148)( 36,147)( 37,150)( 38,149)( 39,151)
( 40,152)( 41,160)( 42,159)( 43,157)( 44,158)( 45,155)( 46,156)( 47,154)
( 48,153)( 49,129)( 50,130)( 51,132)( 52,131)( 53,134)( 54,133)( 55,135)
( 56,136)( 57,144)( 58,143)( 59,141)( 60,142)( 61,139)( 62,140)( 63,138)
( 64,137)( 65,113)( 66,114)( 67,116)( 68,115)( 69,118)( 70,117)( 71,119)
( 72,120)( 73,128)( 74,127)( 75,125)( 76,126)( 77,123)( 78,124)( 79,122)
( 80,121)(161,177)(162,178)(163,180)(164,179)(165,182)(166,181)(167,183)
(168,184)(169,192)(170,191)(171,189)(172,190)(173,187)(174,188)(175,186)
(176,185)(193,225)(194,226)(195,228)(196,227)(197,230)(198,229)(199,231)
(200,232)(201,240)(202,239)(203,237)(204,238)(205,235)(206,236)(207,234)
(208,233)(211,212)(213,214)(217,224)(218,223)(219,221)(220,222);
s2 := Sym(242)!(  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);
s3 := Sym(242)!(241,242);
poly := sub<Sym(242)|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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```

to this polytope