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# Polytope of Type {4,6,6}

Atlas Canonical Name : {4,6,6}*1152e
if this polytope has a name.
Group : SmallGroup(1152,157559)
Rank : 4
Schlafli Type : {4,6,6}
Number of vertices, edges, etc : 16, 48, 72, 6
Order of s0s1s2s3 : 12
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 : {4,6,6}*576b
3-fold quotients : {4,6,2}*384b
4-fold quotients : {4,6,6}*288b, {4,3,6}*288, {4,6,6}*288e, {4,6,6}*288f
6-fold quotients : {4,6,2}*192
8-fold quotients : {4,3,6}*144, {2,6,6}*144c
12-fold quotients : {4,6,2}*96a, {4,3,2}*96, {4,6,2}*96b, {4,6,2}*96c
16-fold quotients : {2,3,6}*72
24-fold quotients : {4,3,2}*48, {2,6,2}*48
36-fold quotients : {4,2,2}*32
48-fold quotients : {2,3,2}*24
72-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73,111)( 74,112)( 75,109)( 76,110)
( 77,115)( 78,116)( 79,113)( 80,114)( 81,119)( 82,120)( 83,117)( 84,118)
( 85,123)( 86,124)( 87,121)( 88,122)( 89,127)( 90,128)( 91,125)( 92,126)
( 93,131)( 94,132)( 95,129)( 96,130)( 97,135)( 98,136)( 99,133)(100,134)
(101,139)(102,140)(103,137)(104,138)(105,143)(106,144)(107,141)(108,142);;
s1 := (  1, 73)(  2, 74)(  3, 76)(  4, 75)(  5, 81)(  6, 82)(  7, 84)(  8, 83)
(  9, 77)( 10, 78)( 11, 80)( 12, 79)( 13, 97)( 14, 98)( 15,100)( 16, 99)
( 17,105)( 18,106)( 19,108)( 20,107)( 21,101)( 22,102)( 23,104)( 24,103)
( 25, 85)( 26, 86)( 27, 88)( 28, 87)( 29, 93)( 30, 94)( 31, 96)( 32, 95)
( 33, 89)( 34, 90)( 35, 92)( 36, 91)( 37,109)( 38,110)( 39,112)( 40,111)
( 41,117)( 42,118)( 43,120)( 44,119)( 45,113)( 46,114)( 47,116)( 48,115)
( 49,133)( 50,134)( 51,136)( 52,135)( 53,141)( 54,142)( 55,144)( 56,143)
( 57,137)( 58,138)( 59,140)( 60,139)( 61,121)( 62,122)( 63,124)( 64,123)
( 65,129)( 66,130)( 67,132)( 68,131)( 69,125)( 70,126)( 71,128)( 72,127);;
s2 := (  1, 17)(  2, 20)(  3, 19)(  4, 18)(  5, 13)(  6, 16)(  7, 15)(  8, 14)
(  9, 21)( 10, 24)( 11, 23)( 12, 22)( 25, 29)( 26, 32)( 27, 31)( 28, 30)
( 34, 36)( 37, 53)( 38, 56)( 39, 55)( 40, 54)( 41, 49)( 42, 52)( 43, 51)
( 44, 50)( 45, 57)( 46, 60)( 47, 59)( 48, 58)( 61, 65)( 62, 68)( 63, 67)
( 64, 66)( 70, 72)( 73, 89)( 74, 92)( 75, 91)( 76, 90)( 77, 85)( 78, 88)
( 79, 87)( 80, 86)( 81, 93)( 82, 96)( 83, 95)( 84, 94)( 97,101)( 98,104)
( 99,103)(100,102)(106,108)(109,125)(110,128)(111,127)(112,126)(113,121)
(114,124)(115,123)(116,122)(117,129)(118,132)(119,131)(120,130)(133,137)
(134,140)(135,139)(136,138)(142,144);;
s3 := (  5,  9)(  6, 10)(  7, 11)(  8, 12)( 17, 21)( 18, 22)( 19, 23)( 20, 24)
( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 69)( 66, 70)( 67, 71)( 68, 72)
( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 89, 93)( 90, 94)( 91, 95)( 92, 96)
(101,105)(102,106)(103,107)(104,108)(113,117)(114,118)(115,119)(116,120)
(125,129)(126,130)(127,131)(128,132)(137,141)(138,142)(139,143)(140,144);;
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,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(144)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73,111)( 74,112)( 75,109)
( 76,110)( 77,115)( 78,116)( 79,113)( 80,114)( 81,119)( 82,120)( 83,117)
( 84,118)( 85,123)( 86,124)( 87,121)( 88,122)( 89,127)( 90,128)( 91,125)
( 92,126)( 93,131)( 94,132)( 95,129)( 96,130)( 97,135)( 98,136)( 99,133)
(100,134)(101,139)(102,140)(103,137)(104,138)(105,143)(106,144)(107,141)
(108,142);
s1 := Sym(144)!(  1, 73)(  2, 74)(  3, 76)(  4, 75)(  5, 81)(  6, 82)(  7, 84)
(  8, 83)(  9, 77)( 10, 78)( 11, 80)( 12, 79)( 13, 97)( 14, 98)( 15,100)
( 16, 99)( 17,105)( 18,106)( 19,108)( 20,107)( 21,101)( 22,102)( 23,104)
( 24,103)( 25, 85)( 26, 86)( 27, 88)( 28, 87)( 29, 93)( 30, 94)( 31, 96)
( 32, 95)( 33, 89)( 34, 90)( 35, 92)( 36, 91)( 37,109)( 38,110)( 39,112)
( 40,111)( 41,117)( 42,118)( 43,120)( 44,119)( 45,113)( 46,114)( 47,116)
( 48,115)( 49,133)( 50,134)( 51,136)( 52,135)( 53,141)( 54,142)( 55,144)
( 56,143)( 57,137)( 58,138)( 59,140)( 60,139)( 61,121)( 62,122)( 63,124)
( 64,123)( 65,129)( 66,130)( 67,132)( 68,131)( 69,125)( 70,126)( 71,128)
( 72,127);
s2 := Sym(144)!(  1, 17)(  2, 20)(  3, 19)(  4, 18)(  5, 13)(  6, 16)(  7, 15)
(  8, 14)(  9, 21)( 10, 24)( 11, 23)( 12, 22)( 25, 29)( 26, 32)( 27, 31)
( 28, 30)( 34, 36)( 37, 53)( 38, 56)( 39, 55)( 40, 54)( 41, 49)( 42, 52)
( 43, 51)( 44, 50)( 45, 57)( 46, 60)( 47, 59)( 48, 58)( 61, 65)( 62, 68)
( 63, 67)( 64, 66)( 70, 72)( 73, 89)( 74, 92)( 75, 91)( 76, 90)( 77, 85)
( 78, 88)( 79, 87)( 80, 86)( 81, 93)( 82, 96)( 83, 95)( 84, 94)( 97,101)
( 98,104)( 99,103)(100,102)(106,108)(109,125)(110,128)(111,127)(112,126)
(113,121)(114,124)(115,123)(116,122)(117,129)(118,132)(119,131)(120,130)
(133,137)(134,140)(135,139)(136,138)(142,144);
s3 := Sym(144)!(  5,  9)(  6, 10)(  7, 11)(  8, 12)( 17, 21)( 18, 22)( 19, 23)
( 20, 24)( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 41, 45)( 42, 46)( 43, 47)
( 44, 48)( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 69)( 66, 70)( 67, 71)
( 68, 72)( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 89, 93)( 90, 94)( 91, 95)
( 92, 96)(101,105)(102,106)(103,107)(104,108)(113,117)(114,118)(115,119)
(116,120)(125,129)(126,130)(127,131)(128,132)(137,141)(138,142)(139,143)
(140,144);
poly := sub<Sym(144)|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, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 >;

```
References : None.
to this polytope