Polytope of Type {15,10,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {15,10,6}*1800
if this polytope has a name.
Group : SmallGroup(1800,678)
Rank : 4
Schlafli Type : {15,10,6}
Number of vertices, edges, etc : 15, 75, 30, 6
Order of s₀s₁s₂s₃ : 30
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {5,10,6}*600, {15,10,2}*600
   5-fold quotients : {15,2,6}*360
   9-fold quotients : {5,10,2}*200
   10-fold quotients : {15,2,3}*180
   15-fold quotients : {5,2,6}*120, {15,2,2}*120
   25-fold quotients : {3,2,6}*72
   30-fold quotients : {5,2,3}*60
   45-fold quotients : {5,2,2}*40
   50-fold quotients : {3,2,3}*36
   75-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope