# Polytope of Type {6,4,6}

Atlas Canonical Name : {6,4,6}*576a
if this polytope has a name.
Group : SmallGroup(576,8659)
Rank : 4
Schlafli Type : {6,4,6}
Number of vertices, edges, etc : 6, 24, 24, 12
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,4,6,2} of size 1152
Vertex Figure Of :
{2,6,4,6} of size 1152
{3,6,4,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,4,3}*288
3-fold quotients : {2,4,6}*192
4-fold quotients : {6,2,6}*144
6-fold quotients : {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
8-fold quotients : {3,2,6}*72, {6,2,3}*72
12-fold quotients : {2,4,3}*48, {2,2,6}*48, {6,2,2}*48
16-fold quotients : {3,2,3}*36
24-fold quotients : {2,2,3}*24, {3,2,2}*24
36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,4,12}*1152b, {12,4,6}*1152c, {6,4,6}*1152a, {6,4,12}*1152d, {6,8,6}*1152a, {6,8,6}*1152c
3-fold covers : {18,4,6}*1728a, {6,4,18}*1728b, {6,12,6}*1728a, {6,12,6}*1728e, {6,12,6}*1728f, {6,12,6}*1728h
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)
( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 89, 93)( 90, 94)( 91, 95)( 92, 96)
(101,105)(102,106)(103,107)(104,108)(113,117)(114,118)(115,119)(116,120)
(125,129)(126,130)(127,131)(128,132)(137,141)(138,142)(139,143)(140,144);;
s1 := (  1, 79)(  2, 80)(  3, 77)(  4, 78)(  5, 75)(  6, 76)(  7, 73)(  8, 74)
(  9, 83)( 10, 84)( 11, 81)( 12, 82)( 13, 91)( 14, 92)( 15, 89)( 16, 90)
( 17, 87)( 18, 88)( 19, 85)( 20, 86)( 21, 95)( 22, 96)( 23, 93)( 24, 94)
( 25,103)( 26,104)( 27,101)( 28,102)( 29, 99)( 30,100)( 31, 97)( 32, 98)
( 33,107)( 34,108)( 35,105)( 36,106)( 37,115)( 38,116)( 39,113)( 40,114)
( 41,111)( 42,112)( 43,109)( 44,110)( 45,119)( 46,120)( 47,117)( 48,118)
( 49,127)( 50,128)( 51,125)( 52,126)( 53,123)( 54,124)( 55,121)( 56,122)
( 57,131)( 58,132)( 59,129)( 60,130)( 61,139)( 62,140)( 63,137)( 64,138)
( 65,135)( 66,136)( 67,133)( 68,134)( 69,143)( 70,144)( 71,141)( 72,142);;
s2 := (  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)( 17, 29)
( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 38, 39)
( 42, 43)( 46, 47)( 49, 61)( 50, 63)( 51, 62)( 52, 64)( 53, 65)( 54, 67)
( 55, 66)( 56, 68)( 57, 69)( 58, 71)( 59, 70)( 60, 72)( 74, 75)( 78, 79)
( 82, 83)( 85, 97)( 86, 99)( 87, 98)( 88,100)( 89,101)( 90,103)( 91,102)
( 92,104)( 93,105)( 94,107)( 95,106)( 96,108)(110,111)(114,115)(118,119)
(121,133)(122,135)(123,134)(124,136)(125,137)(126,139)(127,138)(128,140)
(129,141)(130,143)(131,142)(132,144);;
s3 := (  1, 61)(  2, 64)(  3, 63)(  4, 62)(  5, 65)(  6, 68)(  7, 67)(  8, 66)
(  9, 69)( 10, 72)( 11, 71)( 12, 70)( 13, 49)( 14, 52)( 15, 51)( 16, 50)
( 17, 53)( 18, 56)( 19, 55)( 20, 54)( 21, 57)( 22, 60)( 23, 59)( 24, 58)
( 25, 37)( 26, 40)( 27, 39)( 28, 38)( 29, 41)( 30, 44)( 31, 43)( 32, 42)
( 33, 45)( 34, 48)( 35, 47)( 36, 46)( 73,133)( 74,136)( 75,135)( 76,134)
( 77,137)( 78,140)( 79,139)( 80,138)( 81,141)( 82,144)( 83,143)( 84,142)
( 85,121)( 86,124)( 87,123)( 88,122)( 89,125)( 90,128)( 91,127)( 92,126)
( 93,129)( 94,132)( 95,131)( 96,130)( 97,109)( 98,112)( 99,111)(100,110)
(101,113)(102,116)(103,115)(104,114)(105,117)(106,120)(107,119)(108,118);;
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,
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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)( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 89, 93)( 90, 94)( 91, 95)
( 92, 96)(101,105)(102,106)(103,107)(104,108)(113,117)(114,118)(115,119)
(116,120)(125,129)(126,130)(127,131)(128,132)(137,141)(138,142)(139,143)
(140,144);
s1 := Sym(144)!(  1, 79)(  2, 80)(  3, 77)(  4, 78)(  5, 75)(  6, 76)(  7, 73)
(  8, 74)(  9, 83)( 10, 84)( 11, 81)( 12, 82)( 13, 91)( 14, 92)( 15, 89)
( 16, 90)( 17, 87)( 18, 88)( 19, 85)( 20, 86)( 21, 95)( 22, 96)( 23, 93)
( 24, 94)( 25,103)( 26,104)( 27,101)( 28,102)( 29, 99)( 30,100)( 31, 97)
( 32, 98)( 33,107)( 34,108)( 35,105)( 36,106)( 37,115)( 38,116)( 39,113)
( 40,114)( 41,111)( 42,112)( 43,109)( 44,110)( 45,119)( 46,120)( 47,117)
( 48,118)( 49,127)( 50,128)( 51,125)( 52,126)( 53,123)( 54,124)( 55,121)
( 56,122)( 57,131)( 58,132)( 59,129)( 60,130)( 61,139)( 62,140)( 63,137)
( 64,138)( 65,135)( 66,136)( 67,133)( 68,134)( 69,143)( 70,144)( 71,141)
( 72,142);
s2 := Sym(144)!(  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)
( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)
( 38, 39)( 42, 43)( 46, 47)( 49, 61)( 50, 63)( 51, 62)( 52, 64)( 53, 65)
( 54, 67)( 55, 66)( 56, 68)( 57, 69)( 58, 71)( 59, 70)( 60, 72)( 74, 75)
( 78, 79)( 82, 83)( 85, 97)( 86, 99)( 87, 98)( 88,100)( 89,101)( 90,103)
( 91,102)( 92,104)( 93,105)( 94,107)( 95,106)( 96,108)(110,111)(114,115)
(118,119)(121,133)(122,135)(123,134)(124,136)(125,137)(126,139)(127,138)
(128,140)(129,141)(130,143)(131,142)(132,144);
s3 := Sym(144)!(  1, 61)(  2, 64)(  3, 63)(  4, 62)(  5, 65)(  6, 68)(  7, 67)
(  8, 66)(  9, 69)( 10, 72)( 11, 71)( 12, 70)( 13, 49)( 14, 52)( 15, 51)
( 16, 50)( 17, 53)( 18, 56)( 19, 55)( 20, 54)( 21, 57)( 22, 60)( 23, 59)
( 24, 58)( 25, 37)( 26, 40)( 27, 39)( 28, 38)( 29, 41)( 30, 44)( 31, 43)
( 32, 42)( 33, 45)( 34, 48)( 35, 47)( 36, 46)( 73,133)( 74,136)( 75,135)
( 76,134)( 77,137)( 78,140)( 79,139)( 80,138)( 81,141)( 82,144)( 83,143)
( 84,142)( 85,121)( 86,124)( 87,123)( 88,122)( 89,125)( 90,128)( 91,127)
( 92,126)( 93,129)( 94,132)( 95,131)( 96,130)( 97,109)( 98,112)( 99,111)
(100,110)(101,113)(102,116)(103,115)(104,114)(105,117)(106,120)(107,119)
(108,118);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope