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# Polytope of Type {10,6,12}

Atlas Canonical Name : {10,6,12}*1440c
if this polytope has a name.
Group : SmallGroup(1440,5358)
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}*720c
3-fold quotients : {10,6,4}*480a
5-fold quotients : {2,6,12}*288c
6-fold quotients : {10,6,2}*240
9-fold quotients : {10,2,4}*160
10-fold quotients : {2,6,6}*144c
15-fold quotients : {2,6,4}*96a
18-fold quotients : {5,2,4}*80, {10,2,2}*80
20-fold quotients : {2,3,6}*72
30-fold quotients : {2,6,2}*48
36-fold quotients : {5,2,2}*40
45-fold quotients : {2,2,4}*32
60-fold quotients : {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, 32)
( 17, 31)( 18, 35)( 19, 34)( 20, 33)( 21, 42)( 22, 41)( 23, 45)( 24, 44)
( 25, 43)( 26, 37)( 27, 36)( 28, 40)( 29, 39)( 30, 38)( 46, 47)( 48, 50)
( 51, 57)( 52, 56)( 53, 60)( 54, 59)( 55, 58)( 61, 77)( 62, 76)( 63, 80)
( 64, 79)( 65, 78)( 66, 87)( 67, 86)( 68, 90)( 69, 89)( 70, 88)( 71, 82)
( 72, 81)( 73, 85)( 74, 84)( 75, 83)( 91, 92)( 93, 95)( 96,102)( 97,101)
( 98,105)( 99,104)(100,103)(106,122)(107,121)(108,125)(109,124)(110,123)
(111,132)(112,131)(113,135)(114,134)(115,133)(116,127)(117,126)(118,130)
(119,129)(120,128)(136,137)(138,140)(141,147)(142,146)(143,150)(144,149)
(145,148)(151,167)(152,166)(153,170)(154,169)(155,168)(156,177)(157,176)
(158,180)(159,179)(160,178)(161,172)(162,171)(163,175)(164,174)(165,173);;
s2 := (  1,111)(  2,112)(  3,113)(  4,114)(  5,115)(  6,106)(  7,107)(  8,108)
(  9,109)( 10,110)( 11,116)( 12,117)( 13,118)( 14,119)( 15,120)( 16, 96)
( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21, 91)( 22, 92)( 23, 93)( 24, 94)
( 25, 95)( 26,101)( 27,102)( 28,103)( 29,104)( 30,105)( 31,126)( 32,127)
( 33,128)( 34,129)( 35,130)( 36,121)( 37,122)( 38,123)( 39,124)( 40,125)
( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,156)( 47,157)( 48,158)
( 49,159)( 50,160)( 51,151)( 52,152)( 53,153)( 54,154)( 55,155)( 56,161)
( 57,162)( 58,163)( 59,164)( 60,165)( 61,141)( 62,142)( 63,143)( 64,144)
( 65,145)( 66,136)( 67,137)( 68,138)( 69,139)( 70,140)( 71,146)( 72,147)
( 73,148)( 74,149)( 75,150)( 76,171)( 77,172)( 78,173)( 79,174)( 80,175)
( 81,166)( 82,167)( 83,168)( 84,169)( 85,170)( 86,176)( 87,177)( 88,178)
( 89,179)( 90,180);;
s3 := (  6, 11)(  7, 12)(  8, 13)(  9, 14)( 10, 15)( 21, 26)( 22, 27)( 23, 28)
( 24, 29)( 25, 30)( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 51, 56)
( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 66, 71)( 67, 72)( 68, 73)( 69, 74)
( 70, 75)( 81, 86)( 82, 87)( 83, 88)( 84, 89)( 85, 90)( 91,136)( 92,137)
( 93,138)( 94,139)( 95,140)( 96,146)( 97,147)( 98,148)( 99,149)(100,150)
(101,141)(102,142)(103,143)(104,144)(105,145)(106,151)(107,152)(108,153)
(109,154)(110,155)(111,161)(112,162)(113,163)(114,164)(115,165)(116,156)
(117,157)(118,158)(119,159)(120,160)(121,166)(122,167)(123,168)(124,169)
(125,170)(126,176)(127,177)(128,178)(129,179)(130,180)(131,171)(132,172)
(133,173)(134,174)(135,175);;
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*s3*s2*s3*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 ];;
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, 32)( 17, 31)( 18, 35)( 19, 34)( 20, 33)( 21, 42)( 22, 41)( 23, 45)
( 24, 44)( 25, 43)( 26, 37)( 27, 36)( 28, 40)( 29, 39)( 30, 38)( 46, 47)
( 48, 50)( 51, 57)( 52, 56)( 53, 60)( 54, 59)( 55, 58)( 61, 77)( 62, 76)
( 63, 80)( 64, 79)( 65, 78)( 66, 87)( 67, 86)( 68, 90)( 69, 89)( 70, 88)
( 71, 82)( 72, 81)( 73, 85)( 74, 84)( 75, 83)( 91, 92)( 93, 95)( 96,102)
( 97,101)( 98,105)( 99,104)(100,103)(106,122)(107,121)(108,125)(109,124)
(110,123)(111,132)(112,131)(113,135)(114,134)(115,133)(116,127)(117,126)
(118,130)(119,129)(120,128)(136,137)(138,140)(141,147)(142,146)(143,150)
(144,149)(145,148)(151,167)(152,166)(153,170)(154,169)(155,168)(156,177)
(157,176)(158,180)(159,179)(160,178)(161,172)(162,171)(163,175)(164,174)
(165,173);
s2 := Sym(180)!(  1,111)(  2,112)(  3,113)(  4,114)(  5,115)(  6,106)(  7,107)
(  8,108)(  9,109)( 10,110)( 11,116)( 12,117)( 13,118)( 14,119)( 15,120)
( 16, 96)( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21, 91)( 22, 92)( 23, 93)
( 24, 94)( 25, 95)( 26,101)( 27,102)( 28,103)( 29,104)( 30,105)( 31,126)
( 32,127)( 33,128)( 34,129)( 35,130)( 36,121)( 37,122)( 38,123)( 39,124)
( 40,125)( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,156)( 47,157)
( 48,158)( 49,159)( 50,160)( 51,151)( 52,152)( 53,153)( 54,154)( 55,155)
( 56,161)( 57,162)( 58,163)( 59,164)( 60,165)( 61,141)( 62,142)( 63,143)
( 64,144)( 65,145)( 66,136)( 67,137)( 68,138)( 69,139)( 70,140)( 71,146)
( 72,147)( 73,148)( 74,149)( 75,150)( 76,171)( 77,172)( 78,173)( 79,174)
( 80,175)( 81,166)( 82,167)( 83,168)( 84,169)( 85,170)( 86,176)( 87,177)
( 88,178)( 89,179)( 90,180);
s3 := Sym(180)!(  6, 11)(  7, 12)(  8, 13)(  9, 14)( 10, 15)( 21, 26)( 22, 27)
( 23, 28)( 24, 29)( 25, 30)( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)
( 51, 56)( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 66, 71)( 67, 72)( 68, 73)
( 69, 74)( 70, 75)( 81, 86)( 82, 87)( 83, 88)( 84, 89)( 85, 90)( 91,136)
( 92,137)( 93,138)( 94,139)( 95,140)( 96,146)( 97,147)( 98,148)( 99,149)
(100,150)(101,141)(102,142)(103,143)(104,144)(105,145)(106,151)(107,152)
(108,153)(109,154)(110,155)(111,161)(112,162)(113,163)(114,164)(115,165)
(116,156)(117,157)(118,158)(119,159)(120,160)(121,166)(122,167)(123,168)
(124,169)(125,170)(126,176)(127,177)(128,178)(129,179)(130,180)(131,171)
(132,172)(133,173)(134,174)(135,175);
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, s3*s1*s2*s3*s2*s3*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 >;

References : None.
to this polytope