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# Polytope of Type {6,12,4}

Atlas Canonical Name : {6,12,4}*1152c
if this polytope has a name.
Group : SmallGroup(1152,99269)
Rank : 4
Schlafli Type : {6,12,4}
Number of vertices, edges, etc : 6, 72, 48, 8
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,12,4}*576c
3-fold quotients : {6,4,4}*384a
4-fold quotients : {6,6,4}*288c, {6,12,2}*288c
6-fold quotients : {6,4,4}*192
8-fold quotients : {3,6,4}*144, {6,6,2}*144c
9-fold quotients : {2,4,4}*128
12-fold quotients : {6,2,4}*96, {6,4,2}*96a
16-fold quotients : {3,6,2}*72
18-fold quotients : {2,4,4}*64
24-fold quotients : {3,2,4}*48, {6,2,2}*48
36-fold quotients : {2,2,4}*32, {2,4,2}*32
48-fold quotients : {3,2,2}*24
72-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1, 37)(  2, 39)(  3, 38)(  4, 43)(  5, 45)(  6, 44)(  7, 40)(  8, 42)
(  9, 41)( 10, 46)( 11, 48)( 12, 47)( 13, 52)( 14, 54)( 15, 53)( 16, 49)
( 17, 51)( 18, 50)( 19, 55)( 20, 57)( 21, 56)( 22, 61)( 23, 63)( 24, 62)
( 25, 58)( 26, 60)( 27, 59)( 28, 64)( 29, 66)( 30, 65)( 31, 70)( 32, 72)
( 33, 71)( 34, 67)( 35, 69)( 36, 68)( 73,109)( 74,111)( 75,110)( 76,115)
( 77,117)( 78,116)( 79,112)( 80,114)( 81,113)( 82,118)( 83,120)( 84,119)
( 85,124)( 86,126)( 87,125)( 88,121)( 89,123)( 90,122)( 91,127)( 92,129)
( 93,128)( 94,133)( 95,135)( 96,134)( 97,130)( 98,132)( 99,131)(100,136)
(101,138)(102,137)(103,142)(104,144)(105,143)(106,139)(107,141)(108,140);;
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, 65)( 20, 64)( 21, 66)( 22, 71)( 23, 70)( 24, 72)
( 25, 68)( 26, 67)( 27, 69)( 28, 56)( 29, 55)( 30, 57)( 31, 62)( 32, 61)
( 33, 63)( 34, 59)( 35, 58)( 36, 60)( 73,110)( 74,109)( 75,111)( 76,116)
( 77,115)( 78,117)( 79,113)( 80,112)( 81,114)( 82,119)( 83,118)( 84,120)
( 85,125)( 86,124)( 87,126)( 88,122)( 89,121)( 90,123)( 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 := (  2,  6)(  3,  8)(  5,  9)( 11, 15)( 12, 17)( 14, 18)( 20, 24)( 21, 26)
( 23, 27)( 29, 33)( 30, 35)( 32, 36)( 38, 42)( 39, 44)( 41, 45)( 47, 51)
( 48, 53)( 50, 54)( 56, 60)( 57, 62)( 59, 63)( 65, 69)( 66, 71)( 68, 72)
( 73, 91)( 74, 96)( 75, 98)( 76, 94)( 77, 99)( 78, 92)( 79, 97)( 80, 93)
( 81, 95)( 82,100)( 83,105)( 84,107)( 85,103)( 86,108)( 87,101)( 88,106)
( 89,102)( 90,104)(109,127)(110,132)(111,134)(112,130)(113,135)(114,128)
(115,133)(116,129)(117,131)(118,136)(119,141)(120,143)(121,139)(122,144)
(123,137)(124,142)(125,138)(126,140);;
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, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s3*s0*s2*s1*s3*s2*s3*s1*s2*s3*s0*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope