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# Polytope of Type {4,6,12}

Atlas Canonical Name : {4,6,12}*576b
if this polytope has a name.
Group : SmallGroup(576,6956)
Rank : 4
Schlafli Type : {4,6,12}
Number of vertices, edges, etc : 4, 12, 36, 12
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,6,12,2} of size 1152
Vertex Figure Of :
{2,4,6,12} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,6,12}*288b, {4,6,6}*288c
3-fold quotients : {4,2,12}*192
4-fold quotients : {4,6,3}*144, {2,6,6}*144b
6-fold quotients : {2,2,12}*96, {4,2,6}*96
8-fold quotients : {2,6,3}*72
9-fold quotients : {4,2,4}*64
12-fold quotients : {4,2,3}*48, {2,2,6}*48
18-fold quotients : {2,2,4}*32, {4,2,2}*32
24-fold quotients : {2,2,3}*24
36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,12,12}*1152c, {8,6,12}*1152c, {4,6,24}*1152c
3-fold covers : {4,6,36}*1728b, {4,6,12}*1728b, {12,6,12}*1728e, {4,6,12}*1728h
Permutation Representation (GAP) :
```s0 := (  1, 73)(  2, 74)(  3, 75)(  4, 76)(  5, 77)(  6, 78)(  7, 79)(  8, 80)
(  9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)( 16, 88)
( 17, 89)( 18, 90)( 19, 91)( 20, 92)( 21, 93)( 22, 94)( 23, 95)( 24, 96)
( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,101)( 30,102)( 31,103)( 32,104)
( 33,105)( 34,106)( 35,107)( 36,108)( 37,109)( 38,110)( 39,111)( 40,112)
( 41,113)( 42,114)( 43,115)( 44,116)( 45,117)( 46,118)( 47,119)( 48,120)
( 49,121)( 50,122)( 51,123)( 52,124)( 53,125)( 54,126)( 55,127)( 56,128)
( 57,129)( 58,130)( 59,131)( 60,132)( 61,133)( 62,134)( 63,135)( 64,136)
( 65,137)( 66,138)( 67,139)( 68,140)( 69,141)( 70,142)( 71,143)( 72,144);;
s1 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)( 47, 48)
( 50, 51)( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 65, 66)( 68, 69)( 71, 72)
( 73, 82)( 74, 84)( 75, 83)( 76, 85)( 77, 87)( 78, 86)( 79, 88)( 80, 90)
( 81, 89)( 91,100)( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)
( 98,108)( 99,107)(109,118)(110,120)(111,119)(112,121)(113,123)(114,122)
(115,124)(116,126)(117,125)(127,136)(128,138)(129,137)(130,139)(131,141)
(132,140)(133,142)(134,144)(135,143);;
s2 := (  1,  2)(  4,  8)(  5,  7)(  6,  9)( 10, 11)( 13, 17)( 14, 16)( 15, 18)
( 19, 20)( 22, 26)( 23, 25)( 24, 27)( 28, 29)( 31, 35)( 32, 34)( 33, 36)
( 37, 56)( 38, 55)( 39, 57)( 40, 62)( 41, 61)( 42, 63)( 43, 59)( 44, 58)
( 45, 60)( 46, 65)( 47, 64)( 48, 66)( 49, 71)( 50, 70)( 51, 72)( 52, 68)
( 53, 67)( 54, 69)( 73, 74)( 76, 80)( 77, 79)( 78, 81)( 82, 83)( 85, 89)
( 86, 88)( 87, 90)( 91, 92)( 94, 98)( 95, 97)( 96, 99)(100,101)(103,107)
(104,106)(105,108)(109,128)(110,127)(111,129)(112,134)(113,133)(114,135)
(115,131)(116,130)(117,132)(118,137)(119,136)(120,138)(121,143)(122,142)
(123,144)(124,140)(125,139)(126,141);;
s3 := (  1, 40)(  2, 42)(  3, 41)(  4, 37)(  5, 39)(  6, 38)(  7, 43)(  8, 45)
(  9, 44)( 10, 49)( 11, 51)( 12, 50)( 13, 46)( 14, 48)( 15, 47)( 16, 52)
( 17, 54)( 18, 53)( 19, 58)( 20, 60)( 21, 59)( 22, 55)( 23, 57)( 24, 56)
( 25, 61)( 26, 63)( 27, 62)( 28, 67)( 29, 69)( 30, 68)( 31, 64)( 32, 66)
( 33, 65)( 34, 70)( 35, 72)( 36, 71)( 73,112)( 74,114)( 75,113)( 76,109)
( 77,111)( 78,110)( 79,115)( 80,117)( 81,116)( 82,121)( 83,123)( 84,122)
( 85,118)( 86,120)( 87,119)( 88,124)( 89,126)( 90,125)( 91,130)( 92,132)
( 93,131)( 94,127)( 95,129)( 96,128)( 97,133)( 98,135)( 99,134)(100,139)
(101,141)(102,140)(103,136)(104,138)(105,137)(106,142)(107,144)(108,143);;
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,
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*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*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

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

```
References : None.
to this polytope