# Polytope of Type {6,8,3}

Atlas Canonical Name : {6,8,3}*576
if this polytope has a name.
Group : SmallGroup(576,8340)
Rank : 4
Schlafli Type : {6,8,3}
Number of vertices, edges, etc : 6, 48, 24, 6
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,8,3,2} of size 1152
Vertex Figure Of :
{2,6,8,3} of size 1152
{3,6,8,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,4,3}*288
3-fold quotients : {2,8,3}*192
6-fold quotients : {2,4,3}*96
8-fold quotients : {6,2,3}*72
12-fold quotients : {2,4,3}*48
16-fold quotients : {3,2,3}*36
24-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,8,3}*1152, {6,8,6}*1152a
3-fold covers : {18,8,3}*1728, {6,8,9}*1728, {6,24,3}*1728a, {6,24,3}*1728b
Permutation Representation (GAP) :
```s0 := (  9, 17)( 10, 18)( 11, 19)( 12, 20)( 13, 21)( 14, 22)( 15, 23)( 16, 24)
( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)
( 57, 65)( 58, 66)( 59, 67)( 60, 68)( 61, 69)( 62, 70)( 63, 71)( 64, 72)
( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)( 88, 96)
(105,113)(106,114)(107,115)(108,116)(109,117)(110,118)(111,119)(112,120)
(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144);;
s1 := (  1, 85)(  2, 86)(  3, 88)(  4, 87)(  5, 82)(  6, 81)(  7, 83)(  8, 84)
(  9, 77)( 10, 78)( 11, 80)( 12, 79)( 13, 74)( 14, 73)( 15, 75)( 16, 76)
( 17, 93)( 18, 94)( 19, 96)( 20, 95)( 21, 90)( 22, 89)( 23, 91)( 24, 92)
( 25,109)( 26,110)( 27,112)( 28,111)( 29,106)( 30,105)( 31,107)( 32,108)
( 33,101)( 34,102)( 35,104)( 36,103)( 37, 98)( 38, 97)( 39, 99)( 40,100)
( 41,117)( 42,118)( 43,120)( 44,119)( 45,114)( 46,113)( 47,115)( 48,116)
( 49,133)( 50,134)( 51,136)( 52,135)( 53,130)( 54,129)( 55,131)( 56,132)
( 57,125)( 58,126)( 59,128)( 60,127)( 61,122)( 62,121)( 63,123)( 64,124)
( 65,141)( 66,142)( 67,144)( 68,143)( 69,138)( 70,137)( 71,139)( 72,140);;
s2 := (  3,  6)(  4,  5)(  7,  8)( 11, 14)( 12, 13)( 15, 16)( 19, 22)( 20, 21)
( 23, 24)( 25, 49)( 26, 50)( 27, 54)( 28, 53)( 29, 52)( 30, 51)( 31, 56)
( 32, 55)( 33, 57)( 34, 58)( 35, 62)( 36, 61)( 37, 60)( 38, 59)( 39, 64)
( 40, 63)( 41, 65)( 42, 66)( 43, 70)( 44, 69)( 45, 68)( 46, 67)( 47, 72)
( 48, 71)( 73, 74)( 75, 77)( 76, 78)( 81, 82)( 83, 85)( 84, 86)( 89, 90)
( 91, 93)( 92, 94)( 97,122)( 98,121)( 99,125)(100,126)(101,123)(102,124)
(103,127)(104,128)(105,130)(106,129)(107,133)(108,134)(109,131)(110,132)
(111,135)(112,136)(113,138)(114,137)(115,141)(116,142)(117,139)(118,140)
(119,143)(120,144);;
s3 := (  1, 49)(  2, 50)(  3, 55)(  4, 56)(  5, 54)(  6, 53)(  7, 51)(  8, 52)
(  9, 57)( 10, 58)( 11, 63)( 12, 64)( 13, 62)( 14, 61)( 15, 59)( 16, 60)
( 17, 65)( 18, 66)( 19, 71)( 20, 72)( 21, 70)( 22, 69)( 23, 67)( 24, 68)
( 27, 31)( 28, 32)( 29, 30)( 35, 39)( 36, 40)( 37, 38)( 43, 47)( 44, 48)
( 45, 46)( 73,122)( 74,121)( 75,128)( 76,127)( 77,125)( 78,126)( 79,124)
( 80,123)( 81,130)( 82,129)( 83,136)( 84,135)( 85,133)( 86,134)( 87,132)
( 88,131)( 89,138)( 90,137)( 91,144)( 92,143)( 93,141)( 94,142)( 95,140)
( 96,139)( 97, 98)( 99,104)(100,103)(105,106)(107,112)(108,111)(113,114)
(115,120)(116,119);;
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*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s3*s2*s1*s3*s2*s1*s2*s1*s3*s2*s1*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope