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# Polytope of Type {6,12,4}

Atlas Canonical Name : {6,12,4}*1152a
if this polytope has a name.
Group : SmallGroup(1152,99266)
Rank : 4
Schlafli Type : {6,12,4}
Number of vertices, edges, etc : 6, 72, 48, 8
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 : {6,12,4}*576b
3-fold quotients : {2,12,4}*384a
4-fold quotients : {6,12,2}*288b, {6,6,4}*288b
6-fold quotients : {2,12,4}*192a
8-fold quotients : {6,6,2}*144b
9-fold quotients : {2,4,4}*128
12-fold quotients : {2,12,2}*96, {2,6,4}*96a
16-fold quotients : {6,3,2}*72
18-fold quotients : {2,4,4}*64
24-fold quotients : {2,6,2}*48
36-fold quotients : {2,2,4}*32, {2,4,2}*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, 37)(  2, 42)(  3, 44)(  4, 40)(  5, 45)(  6, 38)(  7, 43)(  8, 39)
(  9, 41)( 10, 46)( 11, 51)( 12, 53)( 13, 49)( 14, 54)( 15, 47)( 16, 52)
( 17, 48)( 18, 50)( 19, 55)( 20, 60)( 21, 62)( 22, 58)( 23, 63)( 24, 56)
( 25, 61)( 26, 57)( 27, 59)( 28, 64)( 29, 69)( 30, 71)( 31, 67)( 32, 72)
( 33, 65)( 34, 70)( 35, 66)( 36, 68)( 73,109)( 74,114)( 75,116)( 76,112)
( 77,117)( 78,110)( 79,115)( 80,111)( 81,113)( 82,118)( 83,123)( 84,125)
( 85,121)( 86,126)( 87,119)( 88,124)( 89,120)( 90,122)( 91,127)( 92,132)
( 93,134)( 94,130)( 95,135)( 96,128)( 97,133)( 98,129)( 99,131)(100,136)
(101,141)(102,143)(103,139)(104,144)(105,137)(106,142)(107,138)(108,140);;
s1 := (  1, 74)(  2, 73)(  3, 75)(  4, 80)(  5, 79)(  6, 81)(  7, 77)(  8, 76)
(  9, 78)( 10, 83)( 11, 82)( 12, 84)( 13, 89)( 14, 88)( 15, 90)( 16, 86)
( 17, 85)( 18, 87)( 19, 92)( 20, 91)( 21, 93)( 22, 98)( 23, 97)( 24, 99)
( 25, 95)( 26, 94)( 27, 96)( 28,101)( 29,100)( 30,102)( 31,107)( 32,106)
( 33,108)( 34,104)( 35,103)( 36,105)( 37,110)( 38,109)( 39,111)( 40,116)
( 41,115)( 42,117)( 43,113)( 44,112)( 45,114)( 46,119)( 47,118)( 48,120)
( 49,125)( 50,124)( 51,126)( 52,122)( 53,121)( 54,123)( 55,128)( 56,127)
( 57,129)( 58,134)( 59,133)( 60,135)( 61,131)( 62,130)( 63,132)( 64,137)
( 65,136)( 66,138)( 67,143)( 68,142)( 69,144)( 70,140)( 71,139)( 72,141);;
s2 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)
( 73, 91)( 74, 93)( 75, 92)( 76, 97)( 77, 99)( 78, 98)( 79, 94)( 80, 96)
( 81, 95)( 82,100)( 83,102)( 84,101)( 85,106)( 86,108)( 87,107)( 88,103)
( 89,105)( 90,104)(109,127)(110,129)(111,128)(112,133)(113,135)(114,134)
(115,130)(116,132)(117,131)(118,136)(119,138)(120,137)(121,142)(122,144)
(123,143)(124,139)(125,141)(126,140);;
s3 := ( 19, 28)( 20, 29)( 21, 30)( 22, 31)( 23, 32)( 24, 33)( 25, 34)( 26, 35)
( 27, 36)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)( 60, 69)( 61, 70)
( 62, 71)( 63, 72)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)( 96,105)
( 97,106)( 98,107)( 99,108)(127,136)(128,137)(129,138)(130,139)(131,140)
(132,141)(133,142)(134,143)(135,144);;
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, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s3*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope