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# Polytope of Type {4,6,6}

Atlas Canonical Name : {4,6,6}*1152d
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}*576a
3-fold quotients : {4,6,2}*384b
4-fold quotients : {4,6,6}*288a, {4,6,6}*288d
6-fold quotients : {4,6,2}*192
8-fold quotients : {2,6,6}*144a
12-fold quotients : {4,2,6}*96, {4,6,2}*96a, {4,3,2}*96, {4,6,2}*96b, {4,6,2}*96c
24-fold quotients : {4,2,3}*48, {4,3,2}*48, {2,2,6}*48, {2,6,2}*48
36-fold quotients : {4,2,2}*32
48-fold quotients : {2,2,3}*24, {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, 77)(  6, 78)(  7, 80)(  8, 79)
(  9, 81)( 10, 82)( 11, 84)( 12, 83)( 13, 97)( 14, 98)( 15,100)( 16, 99)
( 17,101)( 18,102)( 19,104)( 20,103)( 21,105)( 22,106)( 23,108)( 24,107)
( 25, 85)( 26, 86)( 27, 88)( 28, 87)( 29, 89)( 30, 90)( 31, 92)( 32, 91)
( 33, 93)( 34, 94)( 35, 96)( 36, 95)( 37,109)( 38,110)( 39,112)( 40,111)
( 41,113)( 42,114)( 43,116)( 44,115)( 45,117)( 46,118)( 47,120)( 48,119)
( 49,133)( 50,134)( 51,136)( 52,135)( 53,137)( 54,138)( 55,140)( 56,139)
( 57,141)( 58,142)( 59,144)( 60,143)( 61,121)( 62,122)( 63,124)( 64,123)
( 65,125)( 66,126)( 67,128)( 68,127)( 69,129)( 70,130)( 71,132)( 72,131);;
s2 := (  1, 13)(  2, 16)(  3, 15)(  4, 14)(  5, 21)(  6, 24)(  7, 23)(  8, 22)
(  9, 17)( 10, 20)( 11, 19)( 12, 18)( 26, 28)( 29, 33)( 30, 36)( 31, 35)
( 32, 34)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 57)( 42, 60)( 43, 59)
( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)( 62, 64)( 65, 69)( 66, 72)
( 67, 71)( 68, 70)( 73, 85)( 74, 88)( 75, 87)( 76, 86)( 77, 93)( 78, 96)
( 79, 95)( 80, 94)( 81, 89)( 82, 92)( 83, 91)( 84, 90)( 98,100)(101,105)
(102,108)(103,107)(104,106)(109,121)(110,124)(111,123)(112,122)(113,129)
(114,132)(115,131)(116,130)(117,125)(118,128)(119,127)(120,126)(134,136)
(137,141)(138,144)(139,143)(140,142);;
s3 := (  1,  5)(  2,  6)(  3,  7)(  4,  8)( 13, 17)( 14, 18)( 15, 19)( 16, 20)
( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 37, 41)( 38, 42)( 39, 43)( 40, 44)
( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 61, 65)( 62, 66)( 63, 67)( 64, 68)
( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 85, 89)( 86, 90)( 87, 91)( 88, 92)
( 97,101)( 98,102)( 99,103)(100,104)(109,113)(110,114)(111,115)(112,116)
(121,125)(122,126)(123,127)(124,128)(133,137)(134,138)(135,139)(136,140);;
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,
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*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, 77)(  6, 78)(  7, 80)
(  8, 79)(  9, 81)( 10, 82)( 11, 84)( 12, 83)( 13, 97)( 14, 98)( 15,100)
( 16, 99)( 17,101)( 18,102)( 19,104)( 20,103)( 21,105)( 22,106)( 23,108)
( 24,107)( 25, 85)( 26, 86)( 27, 88)( 28, 87)( 29, 89)( 30, 90)( 31, 92)
( 32, 91)( 33, 93)( 34, 94)( 35, 96)( 36, 95)( 37,109)( 38,110)( 39,112)
( 40,111)( 41,113)( 42,114)( 43,116)( 44,115)( 45,117)( 46,118)( 47,120)
( 48,119)( 49,133)( 50,134)( 51,136)( 52,135)( 53,137)( 54,138)( 55,140)
( 56,139)( 57,141)( 58,142)( 59,144)( 60,143)( 61,121)( 62,122)( 63,124)
( 64,123)( 65,125)( 66,126)( 67,128)( 68,127)( 69,129)( 70,130)( 71,132)
( 72,131);
s2 := Sym(144)!(  1, 13)(  2, 16)(  3, 15)(  4, 14)(  5, 21)(  6, 24)(  7, 23)
(  8, 22)(  9, 17)( 10, 20)( 11, 19)( 12, 18)( 26, 28)( 29, 33)( 30, 36)
( 31, 35)( 32, 34)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 57)( 42, 60)
( 43, 59)( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)( 62, 64)( 65, 69)
( 66, 72)( 67, 71)( 68, 70)( 73, 85)( 74, 88)( 75, 87)( 76, 86)( 77, 93)
( 78, 96)( 79, 95)( 80, 94)( 81, 89)( 82, 92)( 83, 91)( 84, 90)( 98,100)
(101,105)(102,108)(103,107)(104,106)(109,121)(110,124)(111,123)(112,122)
(113,129)(114,132)(115,131)(116,130)(117,125)(118,128)(119,127)(120,126)
(134,136)(137,141)(138,144)(139,143)(140,142);
s3 := Sym(144)!(  1,  5)(  2,  6)(  3,  7)(  4,  8)( 13, 17)( 14, 18)( 15, 19)
( 16, 20)( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 37, 41)( 38, 42)( 39, 43)
( 40, 44)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 61, 65)( 62, 66)( 63, 67)
( 64, 68)( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 85, 89)( 86, 90)( 87, 91)
( 88, 92)( 97,101)( 98,102)( 99,103)(100,104)(109,113)(110,114)(111,115)
(112,116)(121,125)(122,126)(123,127)(124,128)(133,137)(134,138)(135,139)
(136,140);
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, 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*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 >;

```
References : None.
to this polytope