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# Polytope of Type {6,6,6}

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

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

```
References : None.
to this polytope