# Polytope of Type {4,12,3}

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

```
References : None.
to this polytope