Polytope of Type {6,6,21}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,21}*1512b
if this polytope has a name.
Group : SmallGroup(1512,838)
Rank : 4
Schlafli Type : {6,6,21}
Number of vertices, edges, etc : 6, 18, 63, 21
Order of s0s1s2s3 : 42
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 : {2,6,21}*504, {6,2,21}*504
6-fold quotients : {3,2,21}*252
7-fold quotients : {6,6,3}*216b
9-fold quotients : {6,2,7}*168, {2,2,21}*168
18-fold quotients : {3,2,7}*84
21-fold quotients : {2,6,3}*72, {6,2,3}*72
27-fold quotients : {2,2,7}*56
42-fold quotients : {3,2,3}*36
63-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 22, 43)( 23, 44)( 24, 45)( 25, 46)( 26, 47)( 27, 48)( 28, 49)( 29, 50)
( 30, 51)( 31, 52)( 32, 53)( 33, 54)( 34, 55)( 35, 56)( 36, 57)( 37, 58)
( 38, 59)( 39, 60)( 40, 61)( 41, 62)( 42, 63)( 85,106)( 86,107)( 87,108)
( 88,109)( 89,110)( 90,111)( 91,112)( 92,113)( 93,114)( 94,115)( 95,116)
( 96,117)( 97,118)( 98,119)( 99,120)(100,121)(101,122)(102,123)(103,124)
(104,125)(105,126)(148,169)(149,170)(150,171)(151,172)(152,173)(153,174)
(154,175)(155,176)(156,177)(157,178)(158,179)(159,180)(160,181)(161,182)
(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189);;
s1 := ( 1, 22)( 2, 23)( 3, 24)( 4, 25)( 5, 26)( 6, 27)( 7, 28)( 8, 29)
( 9, 30)( 10, 31)( 11, 32)( 12, 33)( 13, 34)( 14, 35)( 15, 36)( 16, 37)
( 17, 38)( 18, 39)( 19, 40)( 20, 41)( 21, 42)( 64,148)( 65,149)( 66,150)
( 67,151)( 68,152)( 69,153)( 70,154)( 71,155)( 72,156)( 73,157)( 74,158)
( 75,159)( 76,160)( 77,161)( 78,162)( 79,163)( 80,164)( 81,165)( 82,166)
( 83,167)( 84,168)( 85,127)( 86,128)( 87,129)( 88,130)( 89,131)( 90,132)
( 91,133)( 92,134)( 93,135)( 94,136)( 95,137)( 96,138)( 97,139)( 98,140)
( 99,141)(100,142)(101,143)(102,144)(103,145)(104,146)(105,147)(106,169)
(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)(114,177)
(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,184)(122,185)
(123,186)(124,187)(125,188)(126,189);;
s2 := ( 1, 64)( 2, 70)( 3, 69)( 4, 68)( 5, 67)( 6, 66)( 7, 65)( 8, 78)
( 9, 84)( 10, 83)( 11, 82)( 12, 81)( 13, 80)( 14, 79)( 15, 71)( 16, 77)
( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 85)( 23, 91)( 24, 90)
( 25, 89)( 26, 88)( 27, 87)( 28, 86)( 29, 99)( 30,105)( 31,104)( 32,103)
( 33,102)( 34,101)( 35,100)( 36, 92)( 37, 98)( 38, 97)( 39, 96)( 40, 95)
( 41, 94)( 42, 93)( 43,106)( 44,112)( 45,111)( 46,110)( 47,109)( 48,108)
( 49,107)( 50,120)( 51,126)( 52,125)( 53,124)( 54,123)( 55,122)( 56,121)
( 57,113)( 58,119)( 59,118)( 60,117)( 61,116)( 62,115)( 63,114)(128,133)
(129,132)(130,131)(134,141)(135,147)(136,146)(137,145)(138,144)(139,143)
(140,142)(149,154)(150,153)(151,152)(155,162)(156,168)(157,167)(158,166)
(159,165)(160,164)(161,163)(170,175)(171,174)(172,173)(176,183)(177,189)
(178,188)(179,187)(180,186)(181,185)(182,184);;
s3 := ( 1, 9)( 2, 8)( 3, 14)( 4, 13)( 5, 12)( 6, 11)( 7, 10)( 15, 16)
( 17, 21)( 18, 20)( 22, 30)( 23, 29)( 24, 35)( 25, 34)( 26, 33)( 27, 32)
( 28, 31)( 36, 37)( 38, 42)( 39, 41)( 43, 51)( 44, 50)( 45, 56)( 46, 55)
( 47, 54)( 48, 53)( 49, 52)( 57, 58)( 59, 63)( 60, 62)( 64,135)( 65,134)
( 66,140)( 67,139)( 68,138)( 69,137)( 70,136)( 71,128)( 72,127)( 73,133)
( 74,132)( 75,131)( 76,130)( 77,129)( 78,142)( 79,141)( 80,147)( 81,146)
( 82,145)( 83,144)( 84,143)( 85,156)( 86,155)( 87,161)( 88,160)( 89,159)
( 90,158)( 91,157)( 92,149)( 93,148)( 94,154)( 95,153)( 96,152)( 97,151)
( 98,150)( 99,163)(100,162)(101,168)(102,167)(103,166)(104,165)(105,164)
(106,177)(107,176)(108,182)(109,181)(110,180)(111,179)(112,178)(113,170)
(114,169)(115,175)(116,174)(117,173)(118,172)(119,171)(120,184)(121,183)
(122,189)(123,188)(124,187)(125,186)(126,185);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
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*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(189)!( 22, 43)( 23, 44)( 24, 45)( 25, 46)( 26, 47)( 27, 48)( 28, 49)
( 29, 50)( 30, 51)( 31, 52)( 32, 53)( 33, 54)( 34, 55)( 35, 56)( 36, 57)
( 37, 58)( 38, 59)( 39, 60)( 40, 61)( 41, 62)( 42, 63)( 85,106)( 86,107)
( 87,108)( 88,109)( 89,110)( 90,111)( 91,112)( 92,113)( 93,114)( 94,115)
( 95,116)( 96,117)( 97,118)( 98,119)( 99,120)(100,121)(101,122)(102,123)
(103,124)(104,125)(105,126)(148,169)(149,170)(150,171)(151,172)(152,173)
(153,174)(154,175)(155,176)(156,177)(157,178)(158,179)(159,180)(160,181)
(161,182)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189);
s1 := Sym(189)!( 1, 22)( 2, 23)( 3, 24)( 4, 25)( 5, 26)( 6, 27)( 7, 28)
( 8, 29)( 9, 30)( 10, 31)( 11, 32)( 12, 33)( 13, 34)( 14, 35)( 15, 36)
( 16, 37)( 17, 38)( 18, 39)( 19, 40)( 20, 41)( 21, 42)( 64,148)( 65,149)
( 66,150)( 67,151)( 68,152)( 69,153)( 70,154)( 71,155)( 72,156)( 73,157)
( 74,158)( 75,159)( 76,160)( 77,161)( 78,162)( 79,163)( 80,164)( 81,165)
( 82,166)( 83,167)( 84,168)( 85,127)( 86,128)( 87,129)( 88,130)( 89,131)
( 90,132)( 91,133)( 92,134)( 93,135)( 94,136)( 95,137)( 96,138)( 97,139)
( 98,140)( 99,141)(100,142)(101,143)(102,144)(103,145)(104,146)(105,147)
(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)
(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,184)
(122,185)(123,186)(124,187)(125,188)(126,189);
s2 := Sym(189)!( 1, 64)( 2, 70)( 3, 69)( 4, 68)( 5, 67)( 6, 66)( 7, 65)
( 8, 78)( 9, 84)( 10, 83)( 11, 82)( 12, 81)( 13, 80)( 14, 79)( 15, 71)
( 16, 77)( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 85)( 23, 91)
( 24, 90)( 25, 89)( 26, 88)( 27, 87)( 28, 86)( 29, 99)( 30,105)( 31,104)
( 32,103)( 33,102)( 34,101)( 35,100)( 36, 92)( 37, 98)( 38, 97)( 39, 96)
( 40, 95)( 41, 94)( 42, 93)( 43,106)( 44,112)( 45,111)( 46,110)( 47,109)
( 48,108)( 49,107)( 50,120)( 51,126)( 52,125)( 53,124)( 54,123)( 55,122)
( 56,121)( 57,113)( 58,119)( 59,118)( 60,117)( 61,116)( 62,115)( 63,114)
(128,133)(129,132)(130,131)(134,141)(135,147)(136,146)(137,145)(138,144)
(139,143)(140,142)(149,154)(150,153)(151,152)(155,162)(156,168)(157,167)
(158,166)(159,165)(160,164)(161,163)(170,175)(171,174)(172,173)(176,183)
(177,189)(178,188)(179,187)(180,186)(181,185)(182,184);
s3 := Sym(189)!( 1, 9)( 2, 8)( 3, 14)( 4, 13)( 5, 12)( 6, 11)( 7, 10)
( 15, 16)( 17, 21)( 18, 20)( 22, 30)( 23, 29)( 24, 35)( 25, 34)( 26, 33)
( 27, 32)( 28, 31)( 36, 37)( 38, 42)( 39, 41)( 43, 51)( 44, 50)( 45, 56)
( 46, 55)( 47, 54)( 48, 53)( 49, 52)( 57, 58)( 59, 63)( 60, 62)( 64,135)
( 65,134)( 66,140)( 67,139)( 68,138)( 69,137)( 70,136)( 71,128)( 72,127)
( 73,133)( 74,132)( 75,131)( 76,130)( 77,129)( 78,142)( 79,141)( 80,147)
( 81,146)( 82,145)( 83,144)( 84,143)( 85,156)( 86,155)( 87,161)( 88,160)
( 89,159)( 90,158)( 91,157)( 92,149)( 93,148)( 94,154)( 95,153)( 96,152)
( 97,151)( 98,150)( 99,163)(100,162)(101,168)(102,167)(103,166)(104,165)
(105,164)(106,177)(107,176)(108,182)(109,181)(110,180)(111,179)(112,178)
(113,170)(114,169)(115,175)(116,174)(117,173)(118,172)(119,171)(120,184)
(121,183)(122,189)(123,188)(124,187)(125,186)(126,185);
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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
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*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