# Polytope of Type {12,6,4}

Atlas Canonical Name : {12,6,4}*576d
if this polytope has a name.
Group : SmallGroup(576,8313)
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
Non-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}*288d
3-fold quotients : {4,6,4}*192b
6-fold quotients : {2,6,4}*96c
12-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {24,6,4}*1152d, {12,12,4}*1152d, {12,12,4}*1152e, {12,6,4}*1152a
3-fold covers : {36,6,4}*1728c, {12,18,4}*1728c, {12,6,4}*1728d, {12,6,4}*1728j
Permutation Representation (GAP) :
```s0 := (  5,  9)(  6, 10)(  7, 11)(  8, 12)( 17, 21)( 18, 22)( 19, 23)( 20, 24)
( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 69)( 66, 70)( 67, 71)( 68, 72)
( 73,109)( 74,110)( 75,111)( 76,112)( 77,117)( 78,118)( 79,119)( 80,120)
( 81,113)( 82,114)( 83,115)( 84,116)( 85,121)( 86,122)( 87,123)( 88,124)
( 89,129)( 90,130)( 91,131)( 92,132)( 93,125)( 94,126)( 95,127)( 96,128)
( 97,133)( 98,134)( 99,135)(100,136)(101,141)(102,142)(103,143)(104,144)
(105,137)(106,138)(107,139)(108,140);;
s1 := (  1, 77)(  2, 79)(  3, 78)(  4, 80)(  5, 73)(  6, 75)(  7, 74)(  8, 76)
(  9, 81)( 10, 83)( 11, 82)( 12, 84)( 13,101)( 14,103)( 15,102)( 16,104)
( 17, 97)( 18, 99)( 19, 98)( 20,100)( 21,105)( 22,107)( 23,106)( 24,108)
( 25, 89)( 26, 91)( 27, 90)( 28, 92)( 29, 85)( 30, 87)( 31, 86)( 32, 88)
( 33, 93)( 34, 95)( 35, 94)( 36, 96)( 37,113)( 38,115)( 39,114)( 40,116)
( 41,109)( 42,111)( 43,110)( 44,112)( 45,117)( 46,119)( 47,118)( 48,120)
( 49,137)( 50,139)( 51,138)( 52,140)( 53,133)( 54,135)( 55,134)( 56,136)
( 57,141)( 58,143)( 59,142)( 60,144)( 61,125)( 62,127)( 63,126)( 64,128)
( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)( 71,130)( 72,132);;
s2 := (  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 17)(  6, 18)(  7, 20)(  8, 19)
(  9, 21)( 10, 22)( 11, 24)( 12, 23)( 27, 28)( 31, 32)( 35, 36)( 37, 49)
( 38, 50)( 39, 52)( 40, 51)( 41, 53)( 42, 54)( 43, 56)( 44, 55)( 45, 57)
( 46, 58)( 47, 60)( 48, 59)( 63, 64)( 67, 68)( 71, 72)( 73, 85)( 74, 86)
( 75, 88)( 76, 87)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 81, 93)( 82, 94)
( 83, 96)( 84, 95)( 99,100)(103,104)(107,108)(109,121)(110,122)(111,124)
(112,123)(113,125)(114,126)(115,128)(116,127)(117,129)(118,130)(119,132)
(120,131)(135,136)(139,140)(143,144);;
s3 := (  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)
(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143);;
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*s2*s1*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope