Polytope of Type {12,6,4}

Atlas Canonical Name : {12,6,4}*576c
if this polytope has a name.
Group : SmallGroup(576,7174)
Rank : 4
Schlafli Type : {12,6,4}
Number of vertices, edges, etc : 12, 36, 12, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{12,6,4,2} of size 1152
Vertex Figure Of :
{2,12,6,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,6,4}*288b, {12,6,2}*288c
3-fold quotients : {4,6,4}*192a
4-fold quotients : {6,6,2}*144b
6-fold quotients : {2,6,4}*96a, {4,6,2}*96a
8-fold quotients : {6,3,2}*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 : {12,12,4}*1152a, {12,6,8}*1152a, {24,6,4}*1152a
3-fold covers : {12,18,4}*1728b, {12,6,4}*1728c, {12,6,12}*1728c, {12,6,12}*1728g, {12,6,4}*1728h
Permutation Representation (GAP) :
```s0 := (  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);;
s1 := (  1, 38)(  2, 37)(  3, 39)(  4, 44)(  5, 43)(  6, 45)(  7, 41)(  8, 40)
(  9, 42)( 10, 47)( 11, 46)( 12, 48)( 13, 53)( 14, 52)( 15, 54)( 16, 50)
( 17, 49)( 18, 51)( 19, 56)( 20, 55)( 21, 57)( 22, 62)( 23, 61)( 24, 63)
( 25, 59)( 26, 58)( 27, 60)( 28, 65)( 29, 64)( 30, 66)( 31, 71)( 32, 70)
( 33, 72)( 34, 68)( 35, 67)( 36, 69)( 73,119)( 74,118)( 75,120)( 76,125)
( 77,124)( 78,126)( 79,122)( 80,121)( 81,123)( 82,110)( 83,109)( 84,111)
( 85,116)( 86,115)( 87,117)( 88,113)( 89,112)( 90,114)( 91,137)( 92,136)
( 93,138)( 94,143)( 95,142)( 96,144)( 97,140)( 98,139)( 99,141)(100,128)
(101,127)(102,129)(103,134)(104,133)(105,135)(106,131)(107,130)(108,132);;
s2 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 13)( 11, 15)( 12, 14)( 17, 18)
( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)( 35, 36)
( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)( 53, 54)
( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 67)( 65, 69)( 66, 68)( 71, 72)
( 73, 85)( 74, 87)( 75, 86)( 76, 82)( 77, 84)( 78, 83)( 79, 88)( 80, 90)
( 81, 89)( 91,103)( 92,105)( 93,104)( 94,100)( 95,102)( 96,101)( 97,106)
( 98,108)( 99,107)(109,121)(110,123)(111,122)(112,118)(113,120)(114,119)
(115,124)(116,126)(117,125)(127,139)(128,141)(129,140)(130,136)(131,138)
(132,137)(133,142)(134,144)(135,143);;
s3 := (  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);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope