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