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# Polytope of Type {66,4}

Atlas Canonical Name : {66,4}*528a
Also Known As : {66,4|2}. if this polytope has another name.
Group : SmallGroup(528,150)
Rank : 3
Schlafli Type : {66,4}
Number of vertices, edges, etc : 66, 132, 4
Order of s0s1s2 : 132
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{66,4,2} of size 1056
Vertex Figure Of :
{2,66,4} of size 1056
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {66,2}*264
3-fold quotients : {22,4}*176
4-fold quotients : {33,2}*132
6-fold quotients : {22,2}*88
11-fold quotients : {6,4}*48a
12-fold quotients : {11,2}*44
22-fold quotients : {6,2}*24
33-fold quotients : {2,4}*16
44-fold quotients : {3,2}*12
66-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {132,4}*1056a, {66,8}*1056
3-fold covers : {198,4}*1584a, {66,12}*1584b, {66,12}*1584c
Permutation Representation (GAP) :
```s0 := (  2, 11)(  3, 10)(  4,  9)(  5,  8)(  6,  7)( 12, 23)( 13, 33)( 14, 32)
( 15, 31)( 16, 30)( 17, 29)( 18, 28)( 19, 27)( 20, 26)( 21, 25)( 22, 24)
( 35, 44)( 36, 43)( 37, 42)( 38, 41)( 39, 40)( 45, 56)( 46, 66)( 47, 65)
( 48, 64)( 49, 63)( 50, 62)( 51, 61)( 52, 60)( 53, 59)( 54, 58)( 55, 57)
( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 78, 89)( 79, 99)( 80, 98)
( 81, 97)( 82, 96)( 83, 95)( 84, 94)( 85, 93)( 86, 92)( 87, 91)( 88, 90)
(101,110)(102,109)(103,108)(104,107)(105,106)(111,122)(112,132)(113,131)
(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)(121,123);;
s1 := (  1, 13)(  2, 12)(  3, 22)(  4, 21)(  5, 20)(  6, 19)(  7, 18)(  8, 17)
(  9, 16)( 10, 15)( 11, 14)( 23, 24)( 25, 33)( 26, 32)( 27, 31)( 28, 30)
( 34, 46)( 35, 45)( 36, 55)( 37, 54)( 38, 53)( 39, 52)( 40, 51)( 41, 50)
( 42, 49)( 43, 48)( 44, 47)( 56, 57)( 58, 66)( 59, 65)( 60, 64)( 61, 63)
( 67,112)( 68,111)( 69,121)( 70,120)( 71,119)( 72,118)( 73,117)( 74,116)
( 75,115)( 76,114)( 77,113)( 78,101)( 79,100)( 80,110)( 81,109)( 82,108)
( 83,107)( 84,106)( 85,105)( 86,104)( 87,103)( 88,102)( 89,123)( 90,122)
( 91,132)( 92,131)( 93,130)( 94,129)( 95,128)( 96,127)( 97,126)( 98,125)
( 99,124);;
s2 := (  1, 67)(  2, 68)(  3, 69)(  4, 70)(  5, 71)(  6, 72)(  7, 73)(  8, 74)
(  9, 75)( 10, 76)( 11, 77)( 12, 78)( 13, 79)( 14, 80)( 15, 81)( 16, 82)
( 17, 83)( 18, 84)( 19, 85)( 20, 86)( 21, 87)( 22, 88)( 23, 89)( 24, 90)
( 25, 91)( 26, 92)( 27, 93)( 28, 94)( 29, 95)( 30, 96)( 31, 97)( 32, 98)
( 33, 99)( 34,100)( 35,101)( 36,102)( 37,103)( 38,104)( 39,105)( 40,106)
( 41,107)( 42,108)( 43,109)( 44,110)( 45,111)( 46,112)( 47,113)( 48,114)
( 49,115)( 50,116)( 51,117)( 52,118)( 53,119)( 54,120)( 55,121)( 56,122)
( 57,123)( 58,124)( 59,125)( 60,126)( 61,127)( 62,128)( 63,129)( 64,130)
( 65,131)( 66,132);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1,
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*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*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*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*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(132)!(  2, 11)(  3, 10)(  4,  9)(  5,  8)(  6,  7)( 12, 23)( 13, 33)
( 14, 32)( 15, 31)( 16, 30)( 17, 29)( 18, 28)( 19, 27)( 20, 26)( 21, 25)
( 22, 24)( 35, 44)( 36, 43)( 37, 42)( 38, 41)( 39, 40)( 45, 56)( 46, 66)
( 47, 65)( 48, 64)( 49, 63)( 50, 62)( 51, 61)( 52, 60)( 53, 59)( 54, 58)
( 55, 57)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 78, 89)( 79, 99)
( 80, 98)( 81, 97)( 82, 96)( 83, 95)( 84, 94)( 85, 93)( 86, 92)( 87, 91)
( 88, 90)(101,110)(102,109)(103,108)(104,107)(105,106)(111,122)(112,132)
(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)
(121,123);
s1 := Sym(132)!(  1, 13)(  2, 12)(  3, 22)(  4, 21)(  5, 20)(  6, 19)(  7, 18)
(  8, 17)(  9, 16)( 10, 15)( 11, 14)( 23, 24)( 25, 33)( 26, 32)( 27, 31)
( 28, 30)( 34, 46)( 35, 45)( 36, 55)( 37, 54)( 38, 53)( 39, 52)( 40, 51)
( 41, 50)( 42, 49)( 43, 48)( 44, 47)( 56, 57)( 58, 66)( 59, 65)( 60, 64)
( 61, 63)( 67,112)( 68,111)( 69,121)( 70,120)( 71,119)( 72,118)( 73,117)
( 74,116)( 75,115)( 76,114)( 77,113)( 78,101)( 79,100)( 80,110)( 81,109)
( 82,108)( 83,107)( 84,106)( 85,105)( 86,104)( 87,103)( 88,102)( 89,123)
( 90,122)( 91,132)( 92,131)( 93,130)( 94,129)( 95,128)( 96,127)( 97,126)
( 98,125)( 99,124);
s2 := 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, 78)( 13, 79)( 14, 80)( 15, 81)
( 16, 82)( 17, 83)( 18, 84)( 19, 85)( 20, 86)( 21, 87)( 22, 88)( 23, 89)
( 24, 90)( 25, 91)( 26, 92)( 27, 93)( 28, 94)( 29, 95)( 30, 96)( 31, 97)
( 32, 98)( 33, 99)( 34,100)( 35,101)( 36,102)( 37,103)( 38,104)( 39,105)
( 40,106)( 41,107)( 42,108)( 43,109)( 44,110)( 45,111)( 46,112)( 47,113)
( 48,114)( 49,115)( 50,116)( 51,117)( 52,118)( 53,119)( 54,120)( 55,121)
( 56,122)( 57,123)( 58,124)( 59,125)( 60,126)( 61,127)( 62,128)( 63,129)
( 64,130)( 65,131)( 66,132);
poly := sub<Sym(132)|s0,s1,s2>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1,
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*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*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*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*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