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# Polytope of Type {2,4,15}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,15}*960
if this polytope has a name.
Group : SmallGroup(960,11375)
Rank : 4
Schlafli Type : {2,4,15}
Number of vertices, edges, etc : 2, 16, 120, 60
Order of s0s1s2s3 : 30
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,4,15,2} of size 1920
Vertex Figure Of :
{2,2,4,15} of size 1920
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,4,5}*320
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,8,15}*1920b, {2,8,15}*1920c, {2,4,15}*1920, {2,4,30}*1920c, {2,4,30}*1920d
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)(  8, 16)(  9, 17)( 10, 18)
( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)( 24, 32)( 25, 33)( 26, 34)
( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)( 41, 49)( 42, 50)
( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)( 56, 64)( 57, 65)( 58, 66)
( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)( 72, 80)( 73, 81)( 74, 82)
( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)( 88, 96)( 89, 97)( 90, 98)
( 99,107)(100,108)(101,109)(102,110)(103,111)(104,112)(105,113)(106,114)
(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)(121,129)(122,130)
(131,139)(132,140)(133,141)(134,142)(135,143)(136,144)(137,145)(138,146)
(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)(153,161)(154,162)
(163,171)(164,172)(165,173)(166,174)(167,175)(168,176)(169,177)(170,178)
(179,187)(180,188)(181,189)(182,190)(183,191)(184,192)(185,193)(186,194)
(195,203)(196,204)(197,205)(198,206)(199,207)(200,208)(201,209)(202,210)
(211,219)(212,220)(213,221)(214,222)(215,223)(216,224)(217,225)(218,226)
(227,235)(228,236)(229,237)(230,238)(231,239)(232,240)(233,241)(234,242);;
s2 := (  4, 13)(  5, 16)(  6, 10)(  8, 17)(  9, 12)( 11, 15)( 19, 67)( 20, 77)
( 21, 80)( 22, 74)( 23, 71)( 24, 81)( 25, 76)( 26, 70)( 27, 79)( 28, 73)
( 29, 68)( 30, 78)( 31, 75)( 32, 69)( 33, 72)( 34, 82)( 35, 51)( 36, 61)
( 37, 64)( 38, 58)( 39, 55)( 40, 65)( 41, 60)( 42, 54)( 43, 63)( 44, 57)
( 45, 52)( 46, 62)( 47, 59)( 48, 53)( 49, 56)( 50, 66)( 83,163)( 84,173)
( 85,176)( 86,170)( 87,167)( 88,177)( 89,172)( 90,166)( 91,175)( 92,169)
( 93,164)( 94,174)( 95,171)( 96,165)( 97,168)( 98,178)( 99,227)(100,237)
(101,240)(102,234)(103,231)(104,241)(105,236)(106,230)(107,239)(108,233)
(109,228)(110,238)(111,235)(112,229)(113,232)(114,242)(115,211)(116,221)
(117,224)(118,218)(119,215)(120,225)(121,220)(122,214)(123,223)(124,217)
(125,212)(126,222)(127,219)(128,213)(129,216)(130,226)(131,195)(132,205)
(133,208)(134,202)(135,199)(136,209)(137,204)(138,198)(139,207)(140,201)
(141,196)(142,206)(143,203)(144,197)(145,200)(146,210)(147,179)(148,189)
(149,192)(150,186)(151,183)(152,193)(153,188)(154,182)(155,191)(156,185)
(157,180)(158,190)(159,187)(160,181)(161,184)(162,194);;
s3 := (  3,115)(  4,127)(  5,122)(  6,126)(  7,124)(  8,120)(  9,129)( 10,117)
( 11,123)( 12,119)( 13,130)( 14,118)( 15,116)( 16,128)( 17,121)( 18,125)
( 19, 99)( 20,111)( 21,106)( 22,110)( 23,108)( 24,104)( 25,113)( 26,101)
( 27,107)( 28,103)( 29,114)( 30,102)( 31,100)( 32,112)( 33,105)( 34,109)
( 35, 83)( 36, 95)( 37, 90)( 38, 94)( 39, 92)( 40, 88)( 41, 97)( 42, 85)
( 43, 91)( 44, 87)( 45, 98)( 46, 86)( 47, 84)( 48, 96)( 49, 89)( 50, 93)
( 51,147)( 52,159)( 53,154)( 54,158)( 55,156)( 56,152)( 57,161)( 58,149)
( 59,155)( 60,151)( 61,162)( 62,150)( 63,148)( 64,160)( 65,153)( 66,157)
( 67,131)( 68,143)( 69,138)( 70,142)( 71,140)( 72,136)( 73,145)( 74,133)
( 75,139)( 76,135)( 77,146)( 78,134)( 79,132)( 80,144)( 81,137)( 82,141)
(163,195)(164,207)(165,202)(166,206)(167,204)(168,200)(169,209)(170,197)
(171,203)(172,199)(173,210)(174,198)(175,196)(176,208)(177,201)(178,205)
(180,191)(181,186)(182,190)(183,188)(185,193)(189,194)(211,227)(212,239)
(213,234)(214,238)(215,236)(216,232)(217,241)(218,229)(219,235)(220,231)
(221,242)(222,230)(223,228)(224,240)(225,233)(226,237);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2,
s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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(242)!(1,2);
s1 := Sym(242)!(  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)(  8, 16)(  9, 17)
( 10, 18)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)( 24, 32)( 25, 33)
( 26, 34)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)( 41, 49)
( 42, 50)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)( 56, 64)( 57, 65)
( 58, 66)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)( 72, 80)( 73, 81)
( 74, 82)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)( 88, 96)( 89, 97)
( 90, 98)( 99,107)(100,108)(101,109)(102,110)(103,111)(104,112)(105,113)
(106,114)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)(121,129)
(122,130)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144)(137,145)
(138,146)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)(153,161)
(154,162)(163,171)(164,172)(165,173)(166,174)(167,175)(168,176)(169,177)
(170,178)(179,187)(180,188)(181,189)(182,190)(183,191)(184,192)(185,193)
(186,194)(195,203)(196,204)(197,205)(198,206)(199,207)(200,208)(201,209)
(202,210)(211,219)(212,220)(213,221)(214,222)(215,223)(216,224)(217,225)
(218,226)(227,235)(228,236)(229,237)(230,238)(231,239)(232,240)(233,241)
(234,242);
s2 := Sym(242)!(  4, 13)(  5, 16)(  6, 10)(  8, 17)(  9, 12)( 11, 15)( 19, 67)
( 20, 77)( 21, 80)( 22, 74)( 23, 71)( 24, 81)( 25, 76)( 26, 70)( 27, 79)
( 28, 73)( 29, 68)( 30, 78)( 31, 75)( 32, 69)( 33, 72)( 34, 82)( 35, 51)
( 36, 61)( 37, 64)( 38, 58)( 39, 55)( 40, 65)( 41, 60)( 42, 54)( 43, 63)
( 44, 57)( 45, 52)( 46, 62)( 47, 59)( 48, 53)( 49, 56)( 50, 66)( 83,163)
( 84,173)( 85,176)( 86,170)( 87,167)( 88,177)( 89,172)( 90,166)( 91,175)
( 92,169)( 93,164)( 94,174)( 95,171)( 96,165)( 97,168)( 98,178)( 99,227)
(100,237)(101,240)(102,234)(103,231)(104,241)(105,236)(106,230)(107,239)
(108,233)(109,228)(110,238)(111,235)(112,229)(113,232)(114,242)(115,211)
(116,221)(117,224)(118,218)(119,215)(120,225)(121,220)(122,214)(123,223)
(124,217)(125,212)(126,222)(127,219)(128,213)(129,216)(130,226)(131,195)
(132,205)(133,208)(134,202)(135,199)(136,209)(137,204)(138,198)(139,207)
(140,201)(141,196)(142,206)(143,203)(144,197)(145,200)(146,210)(147,179)
(148,189)(149,192)(150,186)(151,183)(152,193)(153,188)(154,182)(155,191)
(156,185)(157,180)(158,190)(159,187)(160,181)(161,184)(162,194);
s3 := Sym(242)!(  3,115)(  4,127)(  5,122)(  6,126)(  7,124)(  8,120)(  9,129)
( 10,117)( 11,123)( 12,119)( 13,130)( 14,118)( 15,116)( 16,128)( 17,121)
( 18,125)( 19, 99)( 20,111)( 21,106)( 22,110)( 23,108)( 24,104)( 25,113)
( 26,101)( 27,107)( 28,103)( 29,114)( 30,102)( 31,100)( 32,112)( 33,105)
( 34,109)( 35, 83)( 36, 95)( 37, 90)( 38, 94)( 39, 92)( 40, 88)( 41, 97)
( 42, 85)( 43, 91)( 44, 87)( 45, 98)( 46, 86)( 47, 84)( 48, 96)( 49, 89)
( 50, 93)( 51,147)( 52,159)( 53,154)( 54,158)( 55,156)( 56,152)( 57,161)
( 58,149)( 59,155)( 60,151)( 61,162)( 62,150)( 63,148)( 64,160)( 65,153)
( 66,157)( 67,131)( 68,143)( 69,138)( 70,142)( 71,140)( 72,136)( 73,145)
( 74,133)( 75,139)( 76,135)( 77,146)( 78,134)( 79,132)( 80,144)( 81,137)
( 82,141)(163,195)(164,207)(165,202)(166,206)(167,204)(168,200)(169,209)
(170,197)(171,203)(172,199)(173,210)(174,198)(175,196)(176,208)(177,201)
(178,205)(180,191)(181,186)(182,190)(183,188)(185,193)(189,194)(211,227)
(212,239)(213,234)(214,238)(215,236)(216,232)(217,241)(218,229)(219,235)
(220,231)(221,242)(222,230)(223,228)(224,240)(225,233)(226,237);
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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2,
s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```

to this polytope