Polytope of Type {10,6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,6,12}*1440a
Also Known As : {{10,6|2},{6,12|2}}. if this polytope has another name.
Group : SmallGroup(1440,5282)
Rank : 4
Schlafli Type : {10,6,12}
Number of vertices, edges, etc : 10, 30, 36, 12
Order of s0s1s2s3 : 60
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) :
2-fold quotients : {10,6,6}*720a
3-fold quotients : {10,2,12}*480, {10,6,4}*480a
5-fold quotients : {2,6,12}*288a
6-fold quotients : {5,2,12}*240, {10,2,6}*240, {10,6,2}*240
9-fold quotients : {10,2,4}*160
10-fold quotients : {2,6,6}*144a
12-fold quotients : {5,2,6}*120, {10,2,3}*120
15-fold quotients : {2,2,12}*96, {2,6,4}*96a
18-fold quotients : {5,2,4}*80, {10,2,2}*80
24-fold quotients : {5,2,3}*60
30-fold quotients : {2,2,6}*48, {2,6,2}*48
36-fold quotients : {5,2,2}*40
45-fold quotients : {2,2,4}*32
60-fold quotients : {2,2,3}*24, {2,3,2}*24
90-fold quotients : {2,2,2}*16
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);;
s1 := ( 1, 2)( 3, 5)( 6, 12)( 7, 11)( 8, 15)( 9, 14)( 10, 13)( 16, 17)
( 18, 20)( 21, 27)( 22, 26)( 23, 30)( 24, 29)( 25, 28)( 31, 32)( 33, 35)
( 36, 42)( 37, 41)( 38, 45)( 39, 44)( 40, 43)( 46, 47)( 48, 50)( 51, 57)
( 52, 56)( 53, 60)( 54, 59)( 55, 58)( 61, 62)( 63, 65)( 66, 72)( 67, 71)
( 68, 75)( 69, 74)( 70, 73)( 76, 77)( 78, 80)( 81, 87)( 82, 86)( 83, 90)
( 84, 89)( 85, 88)( 91, 92)( 93, 95)( 96,102)( 97,101)( 98,105)( 99,104)
(100,103)(106,107)(108,110)(111,117)(112,116)(113,120)(114,119)(115,118)
(121,122)(123,125)(126,132)(127,131)(128,135)(129,134)(130,133)(136,137)
(138,140)(141,147)(142,146)(143,150)(144,149)(145,148)(151,152)(153,155)
(156,162)(157,161)(158,165)(159,164)(160,163)(166,167)(168,170)(171,177)
(172,176)(173,180)(174,179)(175,178);;
s2 := ( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5, 10)( 16, 36)( 17, 37)( 18, 38)
( 19, 39)( 20, 40)( 21, 31)( 22, 32)( 23, 33)( 24, 34)( 25, 35)( 26, 41)
( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 46, 51)( 47, 52)( 48, 53)( 49, 54)
( 50, 55)( 61, 81)( 62, 82)( 63, 83)( 64, 84)( 65, 85)( 66, 76)( 67, 77)
( 68, 78)( 69, 79)( 70, 80)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)
( 91,141)( 92,142)( 93,143)( 94,144)( 95,145)( 96,136)( 97,137)( 98,138)
( 99,139)(100,140)(101,146)(102,147)(103,148)(104,149)(105,150)(106,171)
(107,172)(108,173)(109,174)(110,175)(111,166)(112,167)(113,168)(114,169)
(115,170)(116,176)(117,177)(118,178)(119,179)(120,180)(121,156)(122,157)
(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)(130,155)
(131,161)(132,162)(133,163)(134,164)(135,165);;
s3 := ( 1,106)( 2,107)( 3,108)( 4,109)( 5,110)( 6,111)( 7,112)( 8,113)
( 9,114)( 10,115)( 11,116)( 12,117)( 13,118)( 14,119)( 15,120)( 16, 91)
( 17, 92)( 18, 93)( 19, 94)( 20, 95)( 21, 96)( 22, 97)( 23, 98)( 24, 99)
( 25,100)( 26,101)( 27,102)( 28,103)( 29,104)( 30,105)( 31,121)( 32,122)
( 33,123)( 34,124)( 35,125)( 36,126)( 37,127)( 38,128)( 39,129)( 40,130)
( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,151)( 47,152)( 48,153)
( 49,154)( 50,155)( 51,156)( 52,157)( 53,158)( 54,159)( 55,160)( 56,161)
( 57,162)( 58,163)( 59,164)( 60,165)( 61,136)( 62,137)( 63,138)( 64,139)
( 65,140)( 66,141)( 67,142)( 68,143)( 69,144)( 70,145)( 71,146)( 72,147)
( 73,148)( 74,149)( 75,150)( 76,166)( 77,167)( 78,168)( 79,169)( 80,170)
( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)( 88,178)
( 89,179)( 90,180);;
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,
s1*s2*s3*s2*s1*s2*s3*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(180)!( 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);
s1 := Sym(180)!( 1, 2)( 3, 5)( 6, 12)( 7, 11)( 8, 15)( 9, 14)( 10, 13)
( 16, 17)( 18, 20)( 21, 27)( 22, 26)( 23, 30)( 24, 29)( 25, 28)( 31, 32)
( 33, 35)( 36, 42)( 37, 41)( 38, 45)( 39, 44)( 40, 43)( 46, 47)( 48, 50)
( 51, 57)( 52, 56)( 53, 60)( 54, 59)( 55, 58)( 61, 62)( 63, 65)( 66, 72)
( 67, 71)( 68, 75)( 69, 74)( 70, 73)( 76, 77)( 78, 80)( 81, 87)( 82, 86)
( 83, 90)( 84, 89)( 85, 88)( 91, 92)( 93, 95)( 96,102)( 97,101)( 98,105)
( 99,104)(100,103)(106,107)(108,110)(111,117)(112,116)(113,120)(114,119)
(115,118)(121,122)(123,125)(126,132)(127,131)(128,135)(129,134)(130,133)
(136,137)(138,140)(141,147)(142,146)(143,150)(144,149)(145,148)(151,152)
(153,155)(156,162)(157,161)(158,165)(159,164)(160,163)(166,167)(168,170)
(171,177)(172,176)(173,180)(174,179)(175,178);
s2 := Sym(180)!( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5, 10)( 16, 36)( 17, 37)
( 18, 38)( 19, 39)( 20, 40)( 21, 31)( 22, 32)( 23, 33)( 24, 34)( 25, 35)
( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 46, 51)( 47, 52)( 48, 53)
( 49, 54)( 50, 55)( 61, 81)( 62, 82)( 63, 83)( 64, 84)( 65, 85)( 66, 76)
( 67, 77)( 68, 78)( 69, 79)( 70, 80)( 71, 86)( 72, 87)( 73, 88)( 74, 89)
( 75, 90)( 91,141)( 92,142)( 93,143)( 94,144)( 95,145)( 96,136)( 97,137)
( 98,138)( 99,139)(100,140)(101,146)(102,147)(103,148)(104,149)(105,150)
(106,171)(107,172)(108,173)(109,174)(110,175)(111,166)(112,167)(113,168)
(114,169)(115,170)(116,176)(117,177)(118,178)(119,179)(120,180)(121,156)
(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)
(130,155)(131,161)(132,162)(133,163)(134,164)(135,165);
s3 := Sym(180)!( 1,106)( 2,107)( 3,108)( 4,109)( 5,110)( 6,111)( 7,112)
( 8,113)( 9,114)( 10,115)( 11,116)( 12,117)( 13,118)( 14,119)( 15,120)
( 16, 91)( 17, 92)( 18, 93)( 19, 94)( 20, 95)( 21, 96)( 22, 97)( 23, 98)
( 24, 99)( 25,100)( 26,101)( 27,102)( 28,103)( 29,104)( 30,105)( 31,121)
( 32,122)( 33,123)( 34,124)( 35,125)( 36,126)( 37,127)( 38,128)( 39,129)
( 40,130)( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,151)( 47,152)
( 48,153)( 49,154)( 50,155)( 51,156)( 52,157)( 53,158)( 54,159)( 55,160)
( 56,161)( 57,162)( 58,163)( 59,164)( 60,165)( 61,136)( 62,137)( 63,138)
( 64,139)( 65,140)( 66,141)( 67,142)( 68,143)( 69,144)( 70,145)( 71,146)
( 72,147)( 73,148)( 74,149)( 75,150)( 76,166)( 77,167)( 78,168)( 79,169)
( 80,170)( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)
( 88,178)( 89,179)( 90,180);
poly := sub<Sym(180)|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, s1*s2*s3*s2*s1*s2*s3*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 >;
References : None.
to this polytope