Polytope of Type {6,10,15}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope