Polytope of Type {4,6,22}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,22}*1056a
Also Known As : {{4,6|2},{6,22|2}}. if this polytope has another name.
Group : SmallGroup(1056,926)
Rank : 4
Schlafli Type : {4,6,22}
Number of vertices, edges, etc : 4, 12, 66, 22
Order of s0s1s2s3 : 132
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 : {2,6,22}*528
3-fold quotients : {4,2,22}*352
6-fold quotients : {4,2,11}*176, {2,2,22}*176
11-fold quotients : {4,6,2}*96a
12-fold quotients : {2,2,11}*88
22-fold quotients : {2,6,2}*48
33-fold quotients : {4,2,2}*32
44-fold quotients : {2,3,2}*24
66-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 67,100)( 68,101)( 69,102)( 70,103)( 71,104)( 72,105)( 73,106)( 74,107)
( 75,108)( 76,109)( 77,110)( 78,111)( 79,112)( 80,113)( 81,114)( 82,115)
( 83,116)( 84,117)( 85,118)( 86,119)( 87,120)( 88,121)( 89,122)( 90,123)
( 91,124)( 92,125)( 93,126)( 94,127)( 95,128)( 96,129)( 97,130)( 98,131)
( 99,132);;
s1 := ( 1, 67)( 2, 68)( 3, 69)( 4, 70)( 5, 71)( 6, 72)( 7, 73)( 8, 74)
( 9, 75)( 10, 76)( 11, 77)( 12, 89)( 13, 90)( 14, 91)( 15, 92)( 16, 93)
( 17, 94)( 18, 95)( 19, 96)( 20, 97)( 21, 98)( 22, 99)( 23, 78)( 24, 79)
( 25, 80)( 26, 81)( 27, 82)( 28, 83)( 29, 84)( 30, 85)( 31, 86)( 32, 87)
( 33, 88)( 34,100)( 35,101)( 36,102)( 37,103)( 38,104)( 39,105)( 40,106)
( 41,107)( 42,108)( 43,109)( 44,110)( 45,122)( 46,123)( 47,124)( 48,125)
( 49,126)( 50,127)( 51,128)( 52,129)( 53,130)( 54,131)( 55,132)( 56,111)
( 57,112)( 58,113)( 59,114)( 60,115)( 61,116)( 62,117)( 63,118)( 64,119)
( 65,120)( 66,121);;
s2 := ( 1, 12)( 2, 22)( 3, 21)( 4, 20)( 5, 19)( 6, 18)( 7, 17)( 8, 16)
( 9, 15)( 10, 14)( 11, 13)( 24, 33)( 25, 32)( 26, 31)( 27, 30)( 28, 29)
( 34, 45)( 35, 55)( 36, 54)( 37, 53)( 38, 52)( 39, 51)( 40, 50)( 41, 49)
( 42, 48)( 43, 47)( 44, 46)( 57, 66)( 58, 65)( 59, 64)( 60, 63)( 61, 62)
( 67, 78)( 68, 88)( 69, 87)( 70, 86)( 71, 85)( 72, 84)( 73, 83)( 74, 82)
( 75, 81)( 76, 80)( 77, 79)( 90, 99)( 91, 98)( 92, 97)( 93, 96)( 94, 95)
(100,111)(101,121)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)
(108,114)(109,113)(110,112)(123,132)(124,131)(125,130)(126,129)(127,128);;
s3 := ( 1, 2)( 3, 11)( 4, 10)( 5, 9)( 6, 8)( 12, 13)( 14, 22)( 15, 21)
( 16, 20)( 17, 19)( 23, 24)( 25, 33)( 26, 32)( 27, 31)( 28, 30)( 34, 35)
( 36, 44)( 37, 43)( 38, 42)( 39, 41)( 45, 46)( 47, 55)( 48, 54)( 49, 53)
( 50, 52)( 56, 57)( 58, 66)( 59, 65)( 60, 64)( 61, 63)( 67, 68)( 69, 77)
( 70, 76)( 71, 75)( 72, 74)( 78, 79)( 80, 88)( 81, 87)( 82, 86)( 83, 85)
( 89, 90)( 91, 99)( 92, 98)( 93, 97)( 94, 96)(100,101)(102,110)(103,109)
(104,108)(105,107)(111,112)(113,121)(114,120)(115,119)(116,118)(122,123)
(124,132)(125,131)(126,130)(127,129);;
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*s0*s1*s0*s1*s0*s1,
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,
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*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(132)!( 67,100)( 68,101)( 69,102)( 70,103)( 71,104)( 72,105)( 73,106)
( 74,107)( 75,108)( 76,109)( 77,110)( 78,111)( 79,112)( 80,113)( 81,114)
( 82,115)( 83,116)( 84,117)( 85,118)( 86,119)( 87,120)( 88,121)( 89,122)
( 90,123)( 91,124)( 92,125)( 93,126)( 94,127)( 95,128)( 96,129)( 97,130)
( 98,131)( 99,132);
s1 := Sym(132)!( 1, 67)( 2, 68)( 3, 69)( 4, 70)( 5, 71)( 6, 72)( 7, 73)
( 8, 74)( 9, 75)( 10, 76)( 11, 77)( 12, 89)( 13, 90)( 14, 91)( 15, 92)
( 16, 93)( 17, 94)( 18, 95)( 19, 96)( 20, 97)( 21, 98)( 22, 99)( 23, 78)
( 24, 79)( 25, 80)( 26, 81)( 27, 82)( 28, 83)( 29, 84)( 30, 85)( 31, 86)
( 32, 87)( 33, 88)( 34,100)( 35,101)( 36,102)( 37,103)( 38,104)( 39,105)
( 40,106)( 41,107)( 42,108)( 43,109)( 44,110)( 45,122)( 46,123)( 47,124)
( 48,125)( 49,126)( 50,127)( 51,128)( 52,129)( 53,130)( 54,131)( 55,132)
( 56,111)( 57,112)( 58,113)( 59,114)( 60,115)( 61,116)( 62,117)( 63,118)
( 64,119)( 65,120)( 66,121);
s2 := Sym(132)!( 1, 12)( 2, 22)( 3, 21)( 4, 20)( 5, 19)( 6, 18)( 7, 17)
( 8, 16)( 9, 15)( 10, 14)( 11, 13)( 24, 33)( 25, 32)( 26, 31)( 27, 30)
( 28, 29)( 34, 45)( 35, 55)( 36, 54)( 37, 53)( 38, 52)( 39, 51)( 40, 50)
( 41, 49)( 42, 48)( 43, 47)( 44, 46)( 57, 66)( 58, 65)( 59, 64)( 60, 63)
( 61, 62)( 67, 78)( 68, 88)( 69, 87)( 70, 86)( 71, 85)( 72, 84)( 73, 83)
( 74, 82)( 75, 81)( 76, 80)( 77, 79)( 90, 99)( 91, 98)( 92, 97)( 93, 96)
( 94, 95)(100,111)(101,121)(102,120)(103,119)(104,118)(105,117)(106,116)
(107,115)(108,114)(109,113)(110,112)(123,132)(124,131)(125,130)(126,129)
(127,128);
s3 := Sym(132)!( 1, 2)( 3, 11)( 4, 10)( 5, 9)( 6, 8)( 12, 13)( 14, 22)
( 15, 21)( 16, 20)( 17, 19)( 23, 24)( 25, 33)( 26, 32)( 27, 31)( 28, 30)
( 34, 35)( 36, 44)( 37, 43)( 38, 42)( 39, 41)( 45, 46)( 47, 55)( 48, 54)
( 49, 53)( 50, 52)( 56, 57)( 58, 66)( 59, 65)( 60, 64)( 61, 63)( 67, 68)
( 69, 77)( 70, 76)( 71, 75)( 72, 74)( 78, 79)( 80, 88)( 81, 87)( 82, 86)
( 83, 85)( 89, 90)( 91, 99)( 92, 98)( 93, 97)( 94, 96)(100,101)(102,110)
(103,109)(104,108)(105,107)(111,112)(113,121)(114,120)(115,119)(116,118)
(122,123)(124,132)(125,131)(126,130)(127,129);
poly := sub<Sym(132)|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*s0*s1*s0*s1*s0*s1, 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,
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*s2*s3 >;
References : None.
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