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Polytope of Type {10,6,15}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,6,15}*1800
if this polytope has a name.
Group : SmallGroup(1800,678)
Rank : 4
Schlafli Type : {10,6,15}
Number of vertices, edges, etc : 10, 30, 45, 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 : {10,2,15}*600
   5-fold quotients : {10,6,3}*360, {2,6,15}*360
   6-fold quotients : {5,2,15}*300
   9-fold quotients : {10,2,5}*200
   15-fold quotients : {10,2,3}*120, {2,2,15}*120
   18-fold quotients : {5,2,5}*100
   25-fold quotients : {2,6,3}*72
   30-fold quotients : {5,2,3}*60
   45-fold quotients : {2,2,5}*40
   75-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)( 38, 39)
( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)( 58, 59)
( 62, 65)( 63, 64)( 67, 70)( 68, 69)( 72, 75)( 73, 74)( 77, 80)( 78, 79)
( 82, 85)( 83, 84)( 87, 90)( 88, 89)( 92, 95)( 93, 94)( 97,100)( 98, 99)
(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119)
(122,125)(123,124)(127,130)(128,129)(132,135)(133,134)(137,140)(138,139)
(142,145)(143,144)(147,150)(148,149)(152,155)(153,154)(157,160)(158,159)
(162,165)(163,164)(167,170)(168,169)(172,175)(173,174)(177,180)(178,179)
(182,185)(183,184)(187,190)(188,189)(192,195)(193,194)(197,200)(198,199)
(202,205)(203,204)(207,210)(208,209)(212,215)(213,214)(217,220)(218,219)
(222,225)(223,224);;
s1 := (  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)( 18, 20)
( 21, 22)( 23, 25)( 26, 52)( 27, 51)( 28, 55)( 29, 54)( 30, 53)( 31, 57)
( 32, 56)( 33, 60)( 34, 59)( 35, 58)( 36, 62)( 37, 61)( 38, 65)( 39, 64)
( 40, 63)( 41, 67)( 42, 66)( 43, 70)( 44, 69)( 45, 68)( 46, 72)( 47, 71)
( 48, 75)( 49, 74)( 50, 73)( 76, 77)( 78, 80)( 81, 82)( 83, 85)( 86, 87)
( 88, 90)( 91, 92)( 93, 95)( 96, 97)( 98,100)(101,127)(102,126)(103,130)
(104,129)(105,128)(106,132)(107,131)(108,135)(109,134)(110,133)(111,137)
(112,136)(113,140)(114,139)(115,138)(116,142)(117,141)(118,145)(119,144)
(120,143)(121,147)(122,146)(123,150)(124,149)(125,148)(151,152)(153,155)
(156,157)(158,160)(161,162)(163,165)(166,167)(168,170)(171,172)(173,175)
(176,202)(177,201)(178,205)(179,204)(180,203)(181,207)(182,206)(183,210)
(184,209)(185,208)(186,212)(187,211)(188,215)(189,214)(190,213)(191,217)
(192,216)(193,220)(194,219)(195,218)(196,222)(197,221)(198,225)(199,224)
(200,223);;
s2 := (  1, 26)(  2, 27)(  3, 28)(  4, 29)(  5, 30)(  6, 46)(  7, 47)(  8, 48)
(  9, 49)( 10, 50)( 11, 41)( 12, 42)( 13, 43)( 14, 44)( 15, 45)( 16, 36)
( 17, 37)( 18, 38)( 19, 39)( 20, 40)( 21, 31)( 22, 32)( 23, 33)( 24, 34)
( 25, 35)( 56, 71)( 57, 72)( 58, 73)( 59, 74)( 60, 75)( 61, 66)( 62, 67)
( 63, 68)( 64, 69)( 65, 70)( 76,176)( 77,177)( 78,178)( 79,179)( 80,180)
( 81,196)( 82,197)( 83,198)( 84,199)( 85,200)( 86,191)( 87,192)( 88,193)
( 89,194)( 90,195)( 91,186)( 92,187)( 93,188)( 94,189)( 95,190)( 96,181)
( 97,182)( 98,183)( 99,184)(100,185)(101,151)(102,152)(103,153)(104,154)
(105,155)(106,171)(107,172)(108,173)(109,174)(110,175)(111,166)(112,167)
(113,168)(114,169)(115,170)(116,161)(117,162)(118,163)(119,164)(120,165)
(121,156)(122,157)(123,158)(124,159)(125,160)(126,201)(127,202)(128,203)
(129,204)(130,205)(131,221)(132,222)(133,223)(134,224)(135,225)(136,216)
(137,217)(138,218)(139,219)(140,220)(141,211)(142,212)(143,213)(144,214)
(145,215)(146,206)(147,207)(148,208)(149,209)(150,210);;
s3 := (  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 76)(  7, 77)(  8, 78)
(  9, 79)( 10, 80)( 11, 96)( 12, 97)( 13, 98)( 14, 99)( 15,100)( 16, 91)
( 17, 92)( 18, 93)( 19, 94)( 20, 95)( 21, 86)( 22, 87)( 23, 88)( 24, 89)
( 25, 90)( 26,131)( 27,132)( 28,133)( 29,134)( 30,135)( 31,126)( 32,127)
( 33,128)( 34,129)( 35,130)( 36,146)( 37,147)( 38,148)( 39,149)( 40,150)
( 41,141)( 42,142)( 43,143)( 44,144)( 45,145)( 46,136)( 47,137)( 48,138)
( 49,139)( 50,140)( 51,106)( 52,107)( 53,108)( 54,109)( 55,110)( 56,101)
( 57,102)( 58,103)( 59,104)( 60,105)( 61,121)( 62,122)( 63,123)( 64,124)
( 65,125)( 66,116)( 67,117)( 68,118)( 69,119)( 70,120)( 71,111)( 72,112)
( 73,113)( 74,114)( 75,115)(151,156)(152,157)(153,158)(154,159)(155,160)
(161,171)(162,172)(163,173)(164,174)(165,175)(176,206)(177,207)(178,208)
(179,209)(180,210)(181,201)(182,202)(183,203)(184,204)(185,205)(186,221)
(187,222)(188,223)(189,224)(190,225)(191,216)(192,217)(193,218)(194,219)
(195,220)(196,211)(197,212)(198,213)(199,214)(200,215);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(225)!(  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)
( 18, 19)( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)
( 38, 39)( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)
( 58, 59)( 62, 65)( 63, 64)( 67, 70)( 68, 69)( 72, 75)( 73, 74)( 77, 80)
( 78, 79)( 82, 85)( 83, 84)( 87, 90)( 88, 89)( 92, 95)( 93, 94)( 97,100)
( 98, 99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)
(118,119)(122,125)(123,124)(127,130)(128,129)(132,135)(133,134)(137,140)
(138,139)(142,145)(143,144)(147,150)(148,149)(152,155)(153,154)(157,160)
(158,159)(162,165)(163,164)(167,170)(168,169)(172,175)(173,174)(177,180)
(178,179)(182,185)(183,184)(187,190)(188,189)(192,195)(193,194)(197,200)
(198,199)(202,205)(203,204)(207,210)(208,209)(212,215)(213,214)(217,220)
(218,219)(222,225)(223,224);
s1 := Sym(225)!(  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)
( 18, 20)( 21, 22)( 23, 25)( 26, 52)( 27, 51)( 28, 55)( 29, 54)( 30, 53)
( 31, 57)( 32, 56)( 33, 60)( 34, 59)( 35, 58)( 36, 62)( 37, 61)( 38, 65)
( 39, 64)( 40, 63)( 41, 67)( 42, 66)( 43, 70)( 44, 69)( 45, 68)( 46, 72)
( 47, 71)( 48, 75)( 49, 74)( 50, 73)( 76, 77)( 78, 80)( 81, 82)( 83, 85)
( 86, 87)( 88, 90)( 91, 92)( 93, 95)( 96, 97)( 98,100)(101,127)(102,126)
(103,130)(104,129)(105,128)(106,132)(107,131)(108,135)(109,134)(110,133)
(111,137)(112,136)(113,140)(114,139)(115,138)(116,142)(117,141)(118,145)
(119,144)(120,143)(121,147)(122,146)(123,150)(124,149)(125,148)(151,152)
(153,155)(156,157)(158,160)(161,162)(163,165)(166,167)(168,170)(171,172)
(173,175)(176,202)(177,201)(178,205)(179,204)(180,203)(181,207)(182,206)
(183,210)(184,209)(185,208)(186,212)(187,211)(188,215)(189,214)(190,213)
(191,217)(192,216)(193,220)(194,219)(195,218)(196,222)(197,221)(198,225)
(199,224)(200,223);
s2 := Sym(225)!(  1, 26)(  2, 27)(  3, 28)(  4, 29)(  5, 30)(  6, 46)(  7, 47)
(  8, 48)(  9, 49)( 10, 50)( 11, 41)( 12, 42)( 13, 43)( 14, 44)( 15, 45)
( 16, 36)( 17, 37)( 18, 38)( 19, 39)( 20, 40)( 21, 31)( 22, 32)( 23, 33)
( 24, 34)( 25, 35)( 56, 71)( 57, 72)( 58, 73)( 59, 74)( 60, 75)( 61, 66)
( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 76,176)( 77,177)( 78,178)( 79,179)
( 80,180)( 81,196)( 82,197)( 83,198)( 84,199)( 85,200)( 86,191)( 87,192)
( 88,193)( 89,194)( 90,195)( 91,186)( 92,187)( 93,188)( 94,189)( 95,190)
( 96,181)( 97,182)( 98,183)( 99,184)(100,185)(101,151)(102,152)(103,153)
(104,154)(105,155)(106,171)(107,172)(108,173)(109,174)(110,175)(111,166)
(112,167)(113,168)(114,169)(115,170)(116,161)(117,162)(118,163)(119,164)
(120,165)(121,156)(122,157)(123,158)(124,159)(125,160)(126,201)(127,202)
(128,203)(129,204)(130,205)(131,221)(132,222)(133,223)(134,224)(135,225)
(136,216)(137,217)(138,218)(139,219)(140,220)(141,211)(142,212)(143,213)
(144,214)(145,215)(146,206)(147,207)(148,208)(149,209)(150,210);
s3 := Sym(225)!(  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 76)(  7, 77)
(  8, 78)(  9, 79)( 10, 80)( 11, 96)( 12, 97)( 13, 98)( 14, 99)( 15,100)
( 16, 91)( 17, 92)( 18, 93)( 19, 94)( 20, 95)( 21, 86)( 22, 87)( 23, 88)
( 24, 89)( 25, 90)( 26,131)( 27,132)( 28,133)( 29,134)( 30,135)( 31,126)
( 32,127)( 33,128)( 34,129)( 35,130)( 36,146)( 37,147)( 38,148)( 39,149)
( 40,150)( 41,141)( 42,142)( 43,143)( 44,144)( 45,145)( 46,136)( 47,137)
( 48,138)( 49,139)( 50,140)( 51,106)( 52,107)( 53,108)( 54,109)( 55,110)
( 56,101)( 57,102)( 58,103)( 59,104)( 60,105)( 61,121)( 62,122)( 63,123)
( 64,124)( 65,125)( 66,116)( 67,117)( 68,118)( 69,119)( 70,120)( 71,111)
( 72,112)( 73,113)( 74,114)( 75,115)(151,156)(152,157)(153,158)(154,159)
(155,160)(161,171)(162,172)(163,173)(164,174)(165,175)(176,206)(177,207)
(178,208)(179,209)(180,210)(181,201)(182,202)(183,203)(184,204)(185,205)
(186,221)(187,222)(188,223)(189,224)(190,225)(191,216)(192,217)(193,218)
(194,219)(195,220)(196,211)(197,212)(198,213)(199,214)(200,215);
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 >; 
 
References : None.
to this polytope