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# Polytope of Type {4,6,12}

Atlas Canonical Name : {4,6,12}*576c
if this polytope has a name.
Group : SmallGroup(576,7174)
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 : {4,6,6}*288b, {2,6,12}*288c
3-fold quotients : {4,6,4}*192a
4-fold quotients : {2,6,6}*144c
6-fold quotients : {2,6,4}*96a, {4,6,2}*96a
8-fold quotients : {2,3,6}*72
9-fold quotients : {4,2,4}*64
12-fold quotients : {2,6,2}*48
18-fold quotients : {2,2,4}*32, {4,2,2}*32
24-fold quotients : {2,3,2}*24
36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,12,12}*1152a, {8,6,12}*1152a, {4,6,24}*1152a
3-fold covers : {4,18,12}*1728b, {4,6,12}*1728c, {12,6,12}*1728d, {12,6,12}*1728g, {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)( 37,118)( 38,119)( 39,120)( 40,121)
( 41,122)( 42,123)( 43,124)( 44,125)( 45,126)( 46,109)( 47,110)( 48,111)
( 49,112)( 50,113)( 51,114)( 52,115)( 53,116)( 54,117)( 55,136)( 56,137)
( 57,138)( 58,139)( 59,140)( 60,141)( 61,142)( 62,143)( 63,144)( 64,127)
( 65,128)( 66,129)( 67,130)( 68,131)( 69,132)( 70,133)( 71,134)( 72,135);;
s1 := (  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, 82)( 74, 84)( 75, 83)( 76, 88)( 77, 90)( 78, 89)( 79, 85)( 80, 87)
( 81, 86)( 91,100)( 92,102)( 93,101)( 94,106)( 95,108)( 96,107)( 97,103)
( 98,105)( 99,104)(109,118)(110,120)(111,119)(112,124)(113,126)(114,125)
(115,121)(116,123)(117,122)(127,136)(128,138)(129,137)(130,142)(131,144)
(132,143)(133,139)(134,141)(135,140);;
s2 := (  1, 41)(  2, 40)(  3, 42)(  4, 38)(  5, 37)(  6, 39)(  7, 44)(  8, 43)
(  9, 45)( 10, 50)( 11, 49)( 12, 51)( 13, 47)( 14, 46)( 15, 48)( 16, 53)
( 17, 52)( 18, 54)( 19, 59)( 20, 58)( 21, 60)( 22, 56)( 23, 55)( 24, 57)
( 25, 62)( 26, 61)( 27, 63)( 28, 68)( 29, 67)( 30, 69)( 31, 65)( 32, 64)
( 33, 66)( 34, 71)( 35, 70)( 36, 72)( 73,122)( 74,121)( 75,123)( 76,119)
( 77,118)( 78,120)( 79,125)( 80,124)( 81,126)( 82,113)( 83,112)( 84,114)
( 85,110)( 86,109)( 87,111)( 88,116)( 89,115)( 90,117)( 91,140)( 92,139)
( 93,141)( 94,137)( 95,136)( 96,138)( 97,143)( 98,142)( 99,144)(100,131)
(101,130)(102,132)(103,128)(104,127)(105,129)(106,134)(107,133)(108,135);;
s3 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 37, 55)( 38, 57)( 39, 56)( 40, 58)
( 41, 60)( 42, 59)( 43, 61)( 44, 63)( 45, 62)( 46, 64)( 47, 66)( 48, 65)
( 49, 67)( 50, 69)( 51, 68)( 52, 70)( 53, 72)( 54, 71)( 74, 75)( 77, 78)
( 80, 81)( 83, 84)( 86, 87)( 89, 90)( 92, 93)( 95, 96)( 98, 99)(101,102)
(104,105)(107,108)(109,127)(110,129)(111,128)(112,130)(113,132)(114,131)
(115,133)(116,135)(117,134)(118,136)(119,138)(120,137)(121,139)(122,141)
(123,140)(124,142)(125,144)(126,143);;
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*s3*s2*s3*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
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)( 37,118)( 38,119)( 39,120)
( 40,121)( 41,122)( 42,123)( 43,124)( 44,125)( 45,126)( 46,109)( 47,110)
( 48,111)( 49,112)( 50,113)( 51,114)( 52,115)( 53,116)( 54,117)( 55,136)
( 56,137)( 57,138)( 58,139)( 59,140)( 60,141)( 61,142)( 62,143)( 63,144)
( 64,127)( 65,128)( 66,129)( 67,130)( 68,131)( 69,132)( 70,133)( 71,134)
( 72,135);
s1 := 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, 82)( 74, 84)( 75, 83)( 76, 88)( 77, 90)( 78, 89)( 79, 85)
( 80, 87)( 81, 86)( 91,100)( 92,102)( 93,101)( 94,106)( 95,108)( 96,107)
( 97,103)( 98,105)( 99,104)(109,118)(110,120)(111,119)(112,124)(113,126)
(114,125)(115,121)(116,123)(117,122)(127,136)(128,138)(129,137)(130,142)
(131,144)(132,143)(133,139)(134,141)(135,140);
s2 := Sym(144)!(  1, 41)(  2, 40)(  3, 42)(  4, 38)(  5, 37)(  6, 39)(  7, 44)
(  8, 43)(  9, 45)( 10, 50)( 11, 49)( 12, 51)( 13, 47)( 14, 46)( 15, 48)
( 16, 53)( 17, 52)( 18, 54)( 19, 59)( 20, 58)( 21, 60)( 22, 56)( 23, 55)
( 24, 57)( 25, 62)( 26, 61)( 27, 63)( 28, 68)( 29, 67)( 30, 69)( 31, 65)
( 32, 64)( 33, 66)( 34, 71)( 35, 70)( 36, 72)( 73,122)( 74,121)( 75,123)
( 76,119)( 77,118)( 78,120)( 79,125)( 80,124)( 81,126)( 82,113)( 83,112)
( 84,114)( 85,110)( 86,109)( 87,111)( 88,116)( 89,115)( 90,117)( 91,140)
( 92,139)( 93,141)( 94,137)( 95,136)( 96,138)( 97,143)( 98,142)( 99,144)
(100,131)(101,130)(102,132)(103,128)(104,127)(105,129)(106,134)(107,133)
(108,135);
s3 := 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)( 37, 55)( 38, 57)( 39, 56)
( 40, 58)( 41, 60)( 42, 59)( 43, 61)( 44, 63)( 45, 62)( 46, 64)( 47, 66)
( 48, 65)( 49, 67)( 50, 69)( 51, 68)( 52, 70)( 53, 72)( 54, 71)( 74, 75)
( 77, 78)( 80, 81)( 83, 84)( 86, 87)( 89, 90)( 92, 93)( 95, 96)( 98, 99)
(101,102)(104,105)(107,108)(109,127)(110,129)(111,128)(112,130)(113,132)
(114,131)(115,133)(116,135)(117,134)(118,136)(119,138)(120,137)(121,139)
(122,141)(123,140)(124,142)(125,144)(126,143);
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*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope