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# Polytope of Type {4,6,6}

Atlas Canonical Name : {4,6,6}*576a
if this polytope has a name.
Group : SmallGroup(576,8659)
Rank : 4
Schlafli Type : {4,6,6}
Number of vertices, edges, etc : 8, 24, 36, 6
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,6,6,2} of size 1152
{4,6,6,3} of size 1728
Vertex Figure Of :
{2,4,6,6} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,6,6}*288d
3-fold quotients : {4,6,2}*192
4-fold quotients : {2,6,6}*144a
6-fold quotients : {4,3,2}*96, {4,6,2}*96b, {4,6,2}*96c
12-fold quotients : {4,3,2}*48, {2,2,6}*48, {2,6,2}*48
24-fold quotients : {2,2,3}*24, {2,3,2}*24
36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,12,6}*1152e, {4,6,12}*1152a, {4,6,6}*1152d, {4,12,6}*1152h, {8,6,6}*1152b, {8,6,6}*1152d
3-fold covers : {4,6,18}*1728, {4,18,6}*1728a, {4,6,6}*1728b, {4,6,6}*1728c, {12,6,6}*1728a, {12,6,6}*1728b
Permutation Representation (GAP) :
```s0 := (  1, 75)(  2, 76)(  3, 73)(  4, 74)(  5, 79)(  6, 80)(  7, 77)(  8, 78)
(  9, 83)( 10, 84)( 11, 81)( 12, 82)( 13, 87)( 14, 88)( 15, 85)( 16, 86)
( 17, 91)( 18, 92)( 19, 89)( 20, 90)( 21, 95)( 22, 96)( 23, 93)( 24, 94)
( 25, 99)( 26,100)( 27, 97)( 28, 98)( 29,103)( 30,104)( 31,101)( 32,102)
( 33,107)( 34,108)( 35,105)( 36,106)( 37,111)( 38,112)( 39,109)( 40,110)
( 41,115)( 42,116)( 43,113)( 44,114)( 45,119)( 46,120)( 47,117)( 48,118)
( 49,123)( 50,124)( 51,121)( 52,122)( 53,127)( 54,128)( 55,125)( 56,126)
( 57,131)( 58,132)( 59,129)( 60,130)( 61,135)( 62,136)( 63,133)( 64,134)
( 65,139)( 66,140)( 67,137)( 68,138)( 69,143)( 70,144)( 71,141)( 72,142);;
s1 := (  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)( 17, 29)
( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 38, 39)
( 42, 43)( 46, 47)( 49, 61)( 50, 63)( 51, 62)( 52, 64)( 53, 65)( 54, 67)
( 55, 66)( 56, 68)( 57, 69)( 58, 71)( 59, 70)( 60, 72)( 74, 75)( 78, 79)
( 82, 83)( 85, 97)( 86, 99)( 87, 98)( 88,100)( 89,101)( 90,103)( 91,102)
( 92,104)( 93,105)( 94,107)( 95,106)( 96,108)(110,111)(114,115)(118,119)
(121,133)(122,135)(123,134)(124,136)(125,137)(126,139)(127,138)(128,140)
(129,141)(130,143)(131,142)(132,144);;
s2 := (  1, 25)(  2, 28)(  3, 27)(  4, 26)(  5, 33)(  6, 36)(  7, 35)(  8, 34)
(  9, 29)( 10, 32)( 11, 31)( 12, 30)( 14, 16)( 17, 21)( 18, 24)( 19, 23)
( 20, 22)( 37, 61)( 38, 64)( 39, 63)( 40, 62)( 41, 69)( 42, 72)( 43, 71)
( 44, 70)( 45, 65)( 46, 68)( 47, 67)( 48, 66)( 50, 52)( 53, 57)( 54, 60)
( 55, 59)( 56, 58)( 73, 97)( 74,100)( 75, 99)( 76, 98)( 77,105)( 78,108)
( 79,107)( 80,106)( 81,101)( 82,104)( 83,103)( 84,102)( 86, 88)( 89, 93)
( 90, 96)( 91, 95)( 92, 94)(109,133)(110,136)(111,135)(112,134)(113,141)
(114,144)(115,143)(116,142)(117,137)(118,140)(119,139)(120,138)(122,124)
(125,129)(126,132)(127,131)(128,130);;
s3 := (  1, 41)(  2, 42)(  3, 43)(  4, 44)(  5, 37)(  6, 38)(  7, 39)(  8, 40)
(  9, 45)( 10, 46)( 11, 47)( 12, 48)( 13, 53)( 14, 54)( 15, 55)( 16, 56)
( 17, 49)( 18, 50)( 19, 51)( 20, 52)( 21, 57)( 22, 58)( 23, 59)( 24, 60)
( 25, 65)( 26, 66)( 27, 67)( 28, 68)( 29, 61)( 30, 62)( 31, 63)( 32, 64)
( 33, 69)( 34, 70)( 35, 71)( 36, 72)( 73,113)( 74,114)( 75,115)( 76,116)
( 77,109)( 78,110)( 79,111)( 80,112)( 81,117)( 82,118)( 83,119)( 84,120)
( 85,125)( 86,126)( 87,127)( 88,128)( 89,121)( 90,122)( 91,123)( 92,124)
( 93,129)( 94,130)( 95,131)( 96,132)( 97,137)( 98,138)( 99,139)(100,140)
(101,133)(102,134)(103,135)(104,136)(105,141)(106,142)(107,143)(108,144);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope