Polytope of Type {14,6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,6,9}*1512
if this polytope has a name.
Group : SmallGroup(1512,485)
Rank : 4
Schlafli Type : {14,6,9}
Number of vertices, edges, etc : 14, 42, 27, 9
Order of s₀s₁s₂s₃ : 126
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 : {14,2,9}*504, {14,6,3}*504
   6-fold quotients : {7,2,9}*252
   7-fold quotients : {2,6,9}*216
   9-fold quotients : {14,2,3}*168
   18-fold quotients : {7,2,3}*84
   21-fold quotients : {2,2,9}*72, {2,6,3}*72
   63-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  4, 19)(  5, 20)(  6, 21)(  7, 16)(  8, 17)(  9, 18)( 10, 13)( 11, 14)
( 12, 15)( 25, 40)( 26, 41)( 27, 42)( 28, 37)( 29, 38)( 30, 39)( 31, 34)
( 32, 35)( 33, 36)( 46, 61)( 47, 62)( 48, 63)( 49, 58)( 50, 59)( 51, 60)
( 52, 55)( 53, 56)( 54, 57)( 67, 82)( 68, 83)( 69, 84)( 70, 79)( 71, 80)
( 72, 81)( 73, 76)( 74, 77)( 75, 78)( 88,103)( 89,104)( 90,105)( 91,100)
( 92,101)( 93,102)( 94, 97)( 95, 98)( 96, 99)(109,124)(110,125)(111,126)
(112,121)(113,122)(114,123)(115,118)(116,119)(117,120)(130,145)(131,146)
(132,147)(133,142)(134,143)(135,144)(136,139)(137,140)(138,141)(151,166)
(152,167)(153,168)(154,163)(155,164)(156,165)(157,160)(158,161)(159,162)
(172,187)(173,188)(174,189)(175,184)(176,185)(177,186)(178,181)(179,182)
(180,183);;
s1 := (  1,  4)(  2,  5)(  3,  6)(  7, 19)(  8, 20)(  9, 21)( 10, 16)( 11, 17)
( 12, 18)( 22, 46)( 23, 47)( 24, 48)( 25, 43)( 26, 44)( 27, 45)( 28, 61)
( 29, 62)( 30, 63)( 31, 58)( 32, 59)( 33, 60)( 34, 55)( 35, 56)( 36, 57)
( 37, 52)( 38, 53)( 39, 54)( 40, 49)( 41, 50)( 42, 51)( 64, 67)( 65, 68)
( 66, 69)( 70, 82)( 71, 83)( 72, 84)( 73, 79)( 74, 80)( 75, 81)( 85,109)
( 86,110)( 87,111)( 88,106)( 89,107)( 90,108)( 91,124)( 92,125)( 93,126)
( 94,121)( 95,122)( 96,123)( 97,118)( 98,119)( 99,120)(100,115)(101,116)
(102,117)(103,112)(104,113)(105,114)(127,130)(128,131)(129,132)(133,145)
(134,146)(135,147)(136,142)(137,143)(138,144)(148,172)(149,173)(150,174)
(151,169)(152,170)(153,171)(154,187)(155,188)(156,189)(157,184)(158,185)
(159,186)(160,181)(161,182)(162,183)(163,178)(164,179)(165,180)(166,175)
(167,176)(168,177);;
s2 := (  1, 22)(  2, 24)(  3, 23)(  4, 25)(  5, 27)(  6, 26)(  7, 28)(  8, 30)
(  9, 29)( 10, 31)( 11, 33)( 12, 32)( 13, 34)( 14, 36)( 15, 35)( 16, 37)
( 17, 39)( 18, 38)( 19, 40)( 20, 42)( 21, 41)( 44, 45)( 47, 48)( 50, 51)
( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 64,149)( 65,148)( 66,150)( 67,152)
( 68,151)( 69,153)( 70,155)( 71,154)( 72,156)( 73,158)( 74,157)( 75,159)
( 76,161)( 77,160)( 78,162)( 79,164)( 80,163)( 81,165)( 82,167)( 83,166)
( 84,168)( 85,128)( 86,127)( 87,129)( 88,131)( 89,130)( 90,132)( 91,134)
( 92,133)( 93,135)( 94,137)( 95,136)( 96,138)( 97,140)( 98,139)( 99,141)
(100,143)(101,142)(102,144)(103,146)(104,145)(105,147)(106,170)(107,169)
(108,171)(109,173)(110,172)(111,174)(112,176)(113,175)(114,177)(115,179)
(116,178)(117,180)(118,182)(119,181)(120,183)(121,185)(122,184)(123,186)
(124,188)(125,187)(126,189);;
s3 := (  1, 64)(  2, 66)(  3, 65)(  4, 67)(  5, 69)(  6, 68)(  7, 70)(  8, 72)
(  9, 71)( 10, 73)( 11, 75)( 12, 74)( 13, 76)( 14, 78)( 15, 77)( 16, 79)
( 17, 81)( 18, 80)( 19, 82)( 20, 84)( 21, 83)( 22,106)( 23,108)( 24,107)
( 25,109)( 26,111)( 27,110)( 28,112)( 29,114)( 30,113)( 31,115)( 32,117)
( 33,116)( 34,118)( 35,120)( 36,119)( 37,121)( 38,123)( 39,122)( 40,124)
( 41,126)( 42,125)( 43, 85)( 44, 87)( 45, 86)( 46, 88)( 47, 90)( 48, 89)
( 49, 91)( 50, 93)( 51, 92)( 52, 94)( 53, 96)( 54, 95)( 55, 97)( 56, 99)
( 57, 98)( 58,100)( 59,102)( 60,101)( 61,103)( 62,105)( 63,104)(127,128)
(130,131)(133,134)(136,137)(139,140)(142,143)(145,146)(148,170)(149,169)
(150,171)(151,173)(152,172)(153,174)(154,176)(155,175)(156,177)(157,179)
(158,178)(159,180)(160,182)(161,181)(162,183)(163,185)(164,184)(165,186)
(166,188)(167,187)(168,189);;
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, 
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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(189)!(  4, 19)(  5, 20)(  6, 21)(  7, 16)(  8, 17)(  9, 18)( 10, 13)
( 11, 14)( 12, 15)( 25, 40)( 26, 41)( 27, 42)( 28, 37)( 29, 38)( 30, 39)
( 31, 34)( 32, 35)( 33, 36)( 46, 61)( 47, 62)( 48, 63)( 49, 58)( 50, 59)
( 51, 60)( 52, 55)( 53, 56)( 54, 57)( 67, 82)( 68, 83)( 69, 84)( 70, 79)
( 71, 80)( 72, 81)( 73, 76)( 74, 77)( 75, 78)( 88,103)( 89,104)( 90,105)
( 91,100)( 92,101)( 93,102)( 94, 97)( 95, 98)( 96, 99)(109,124)(110,125)
(111,126)(112,121)(113,122)(114,123)(115,118)(116,119)(117,120)(130,145)
(131,146)(132,147)(133,142)(134,143)(135,144)(136,139)(137,140)(138,141)
(151,166)(152,167)(153,168)(154,163)(155,164)(156,165)(157,160)(158,161)
(159,162)(172,187)(173,188)(174,189)(175,184)(176,185)(177,186)(178,181)
(179,182)(180,183);
s1 := Sym(189)!(  1,  4)(  2,  5)(  3,  6)(  7, 19)(  8, 20)(  9, 21)( 10, 16)
( 11, 17)( 12, 18)( 22, 46)( 23, 47)( 24, 48)( 25, 43)( 26, 44)( 27, 45)
( 28, 61)( 29, 62)( 30, 63)( 31, 58)( 32, 59)( 33, 60)( 34, 55)( 35, 56)
( 36, 57)( 37, 52)( 38, 53)( 39, 54)( 40, 49)( 41, 50)( 42, 51)( 64, 67)
( 65, 68)( 66, 69)( 70, 82)( 71, 83)( 72, 84)( 73, 79)( 74, 80)( 75, 81)
( 85,109)( 86,110)( 87,111)( 88,106)( 89,107)( 90,108)( 91,124)( 92,125)
( 93,126)( 94,121)( 95,122)( 96,123)( 97,118)( 98,119)( 99,120)(100,115)
(101,116)(102,117)(103,112)(104,113)(105,114)(127,130)(128,131)(129,132)
(133,145)(134,146)(135,147)(136,142)(137,143)(138,144)(148,172)(149,173)
(150,174)(151,169)(152,170)(153,171)(154,187)(155,188)(156,189)(157,184)
(158,185)(159,186)(160,181)(161,182)(162,183)(163,178)(164,179)(165,180)
(166,175)(167,176)(168,177);
s2 := Sym(189)!(  1, 22)(  2, 24)(  3, 23)(  4, 25)(  5, 27)(  6, 26)(  7, 28)
(  8, 30)(  9, 29)( 10, 31)( 11, 33)( 12, 32)( 13, 34)( 14, 36)( 15, 35)
( 16, 37)( 17, 39)( 18, 38)( 19, 40)( 20, 42)( 21, 41)( 44, 45)( 47, 48)
( 50, 51)( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 64,149)( 65,148)( 66,150)
( 67,152)( 68,151)( 69,153)( 70,155)( 71,154)( 72,156)( 73,158)( 74,157)
( 75,159)( 76,161)( 77,160)( 78,162)( 79,164)( 80,163)( 81,165)( 82,167)
( 83,166)( 84,168)( 85,128)( 86,127)( 87,129)( 88,131)( 89,130)( 90,132)
( 91,134)( 92,133)( 93,135)( 94,137)( 95,136)( 96,138)( 97,140)( 98,139)
( 99,141)(100,143)(101,142)(102,144)(103,146)(104,145)(105,147)(106,170)
(107,169)(108,171)(109,173)(110,172)(111,174)(112,176)(113,175)(114,177)
(115,179)(116,178)(117,180)(118,182)(119,181)(120,183)(121,185)(122,184)
(123,186)(124,188)(125,187)(126,189);
s3 := Sym(189)!(  1, 64)(  2, 66)(  3, 65)(  4, 67)(  5, 69)(  6, 68)(  7, 70)
(  8, 72)(  9, 71)( 10, 73)( 11, 75)( 12, 74)( 13, 76)( 14, 78)( 15, 77)
( 16, 79)( 17, 81)( 18, 80)( 19, 82)( 20, 84)( 21, 83)( 22,106)( 23,108)
( 24,107)( 25,109)( 26,111)( 27,110)( 28,112)( 29,114)( 30,113)( 31,115)
( 32,117)( 33,116)( 34,118)( 35,120)( 36,119)( 37,121)( 38,123)( 39,122)
( 40,124)( 41,126)( 42,125)( 43, 85)( 44, 87)( 45, 86)( 46, 88)( 47, 90)
( 48, 89)( 49, 91)( 50, 93)( 51, 92)( 52, 94)( 53, 96)( 54, 95)( 55, 97)
( 56, 99)( 57, 98)( 58,100)( 59,102)( 60,101)( 61,103)( 62,105)( 63,104)
(127,128)(130,131)(133,134)(136,137)(139,140)(142,143)(145,146)(148,170)
(149,169)(150,171)(151,173)(152,172)(153,174)(154,176)(155,175)(156,177)
(157,179)(158,178)(159,180)(160,182)(161,181)(162,183)(163,185)(164,184)
(165,186)(166,188)(167,187)(168,189);
poly := sub<Sym(189)|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, 
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 >;

References : None.
to this polytope