# Polytope of Type {3,12,4}

Atlas Canonical Name : {3,12,4}*576
if this polytope has a name.
Group : SmallGroup(576,8355)
Rank : 4
Schlafli Type : {3,12,4}
Number of vertices, edges, etc : 6, 36, 48, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,12,4,2} of size 1152
Vertex Figure Of :
{2,3,12,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,12,2}*288
3-fold quotients : {3,4,4}*192b
4-fold quotients : {3,6,4}*144
6-fold quotients : {3,4,2}*96
8-fold quotients : {3,6,2}*72
12-fold quotients : {3,2,4}*48, {3,4,2}*48
24-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,12,8}*1152, {3,24,4}*1152, {6,12,4}*1152j
3-fold covers : {9,12,4}*1728, {3,12,4}*1728a, {3,12,12}*1728b
Permutation Representation (GAP) :
```s0 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 25)( 14, 27)( 15, 26)
( 16, 28)( 17, 33)( 18, 35)( 19, 34)( 20, 36)( 21, 29)( 22, 31)( 23, 30)
( 24, 32)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 49, 61)( 50, 63)
( 51, 62)( 52, 64)( 53, 69)( 54, 71)( 55, 70)( 56, 72)( 57, 65)( 58, 67)
( 59, 66)( 60, 68)( 74, 75)( 77, 81)( 78, 83)( 79, 82)( 80, 84)( 85, 97)
( 86, 99)( 87, 98)( 88,100)( 89,105)( 90,107)( 91,106)( 92,108)( 93,101)
( 94,103)( 95,102)( 96,104)(110,111)(113,117)(114,119)(115,118)(116,120)
(121,133)(122,135)(123,134)(124,136)(125,141)(126,143)(127,142)(128,144)
(129,137)(130,139)(131,138)(132,140);;
s1 := (  1, 17)(  2, 18)(  3, 20)(  4, 19)(  5, 13)(  6, 14)(  7, 16)(  8, 15)
(  9, 21)( 10, 22)( 11, 24)( 12, 23)( 25, 29)( 26, 30)( 27, 32)( 28, 31)
( 35, 36)( 37, 53)( 38, 54)( 39, 56)( 40, 55)( 41, 49)( 42, 50)( 43, 52)
( 44, 51)( 45, 57)( 46, 58)( 47, 60)( 48, 59)( 61, 65)( 62, 66)( 63, 68)
( 64, 67)( 71, 72)( 73, 89)( 74, 90)( 75, 92)( 76, 91)( 77, 85)( 78, 86)
( 79, 88)( 80, 87)( 81, 93)( 82, 94)( 83, 96)( 84, 95)( 97,101)( 98,102)
( 99,104)(100,103)(107,108)(109,125)(110,126)(111,128)(112,127)(113,121)
(114,122)(115,124)(116,123)(117,129)(118,130)(119,132)(120,131)(133,137)
(134,138)(135,140)(136,139)(143,144);;
s2 := (  1,  4)(  2,  3)(  5, 12)(  6, 11)(  7, 10)(  8,  9)( 13, 16)( 14, 15)
( 17, 24)( 18, 23)( 19, 22)( 20, 21)( 25, 28)( 26, 27)( 29, 36)( 30, 35)
( 31, 34)( 32, 33)( 37, 40)( 38, 39)( 41, 48)( 42, 47)( 43, 46)( 44, 45)
( 49, 52)( 50, 51)( 53, 60)( 54, 59)( 55, 58)( 56, 57)( 61, 64)( 62, 63)
( 65, 72)( 66, 71)( 67, 70)( 68, 69)( 73,112)( 74,111)( 75,110)( 76,109)
( 77,120)( 78,119)( 79,118)( 80,117)( 81,116)( 82,115)( 83,114)( 84,113)
( 85,124)( 86,123)( 87,122)( 88,121)( 89,132)( 90,131)( 91,130)( 92,129)
( 93,128)( 94,127)( 95,126)( 96,125)( 97,136)( 98,135)( 99,134)(100,133)
(101,144)(102,143)(103,142)(104,141)(105,140)(106,139)(107,138)(108,137);;
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)( 37,109)( 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);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope