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