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# Polytope of Type {4,12,6}

Atlas Canonical Name : {4,12,6}*576f
if this polytope has a name.
Group : SmallGroup(576,8312)
Rank : 4
Schlafli Type : {4,12,6}
Number of vertices, edges, etc : 4, 24, 36, 6
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,12,6,2} of size 1152
{4,12,6,3} of size 1728
Vertex Figure Of :
{2,4,12,6} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,6,6}*288d
3-fold quotients : {4,12,2}*192c
6-fold quotients : {4,6,2}*96c
12-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,12,12}*1152f, {4,12,6}*1152e
3-fold covers : {4,12,18}*1728d, {4,36,6}*1728e, {4,12,6}*1728g, {4,12,6}*1728m
Permutation Representation (GAP) :
```s0 := (  1, 39)(  2, 40)(  3, 37)(  4, 38)(  5, 43)(  6, 44)(  7, 41)(  8, 42)
(  9, 47)( 10, 48)( 11, 45)( 12, 46)( 13, 51)( 14, 52)( 15, 49)( 16, 50)
( 17, 55)( 18, 56)( 19, 53)( 20, 54)( 21, 59)( 22, 60)( 23, 57)( 24, 58)
( 25, 63)( 26, 64)( 27, 61)( 28, 62)( 29, 67)( 30, 68)( 31, 65)( 32, 66)
( 33, 71)( 34, 72)( 35, 69)( 36, 70)( 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 := (  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)( 73,109)( 74,111)
( 75,110)( 76,112)( 77,113)( 78,115)( 79,114)( 80,116)( 81,117)( 82,119)
( 83,118)( 84,120)( 85,133)( 86,135)( 87,134)( 88,136)( 89,137)( 90,139)
( 91,138)( 92,140)( 93,141)( 94,143)( 95,142)( 96,144)( 97,121)( 98,123)
( 99,122)(100,124)(101,125)(102,127)(103,126)(104,128)(105,129)(106,131)
(107,130)(108,132);;
s2 := (  1, 97)(  2,100)(  3, 99)(  4, 98)(  5,105)(  6,108)(  7,107)(  8,106)
(  9,101)( 10,104)( 11,103)( 12,102)( 13, 85)( 14, 88)( 15, 87)( 16, 86)
( 17, 93)( 18, 96)( 19, 95)( 20, 94)( 21, 89)( 22, 92)( 23, 91)( 24, 90)
( 25, 73)( 26, 76)( 27, 75)( 28, 74)( 29, 81)( 30, 84)( 31, 83)( 32, 82)
( 33, 77)( 34, 80)( 35, 79)( 36, 78)( 37,133)( 38,136)( 39,135)( 40,134)
( 41,141)( 42,144)( 43,143)( 44,142)( 45,137)( 46,140)( 47,139)( 48,138)
( 49,121)( 50,124)( 51,123)( 52,122)( 53,129)( 54,132)( 55,131)( 56,130)
( 57,125)( 58,128)( 59,127)( 60,126)( 61,109)( 62,112)( 63,111)( 64,110)
( 65,117)( 66,120)( 67,119)( 68,118)( 69,113)( 70,116)( 71,115)( 72,114);;
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, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope