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# Polytope of Type {6,8,2}

Atlas Canonical Name : {6,8,2}*576
if this polytope has a name.
Group : SmallGroup(576,8380)
Rank : 4
Schlafli Type : {6,8,2}
Number of vertices, edges, etc : 18, 72, 24, 2
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,8,2,2} of size 1152
{6,8,2,3} of size 1728
Vertex Figure Of :
{2,6,8,2} of size 1152
{3,6,8,2} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,4,2}*288
4-fold quotients : {6,4,2}*144
9-fold quotients : {2,8,2}*64
18-fold quotients : {2,4,2}*32
36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,8,2}*1152a, {6,8,4}*1152a, {6,16,2}*1152
3-fold covers : {6,8,2}*1728a, {6,24,2}*1728d, {6,24,2}*1728e, {6,8,6}*1728a, {6,8,2}*1728b, {6,24,2}*1728g, {6,24,2}*1728h
Permutation Representation (GAP) :
```s0 := (  1, 55)(  2, 57)(  3, 56)(  4, 61)(  5, 63)(  6, 62)(  7, 58)(  8, 60)
(  9, 59)( 10, 64)( 11, 66)( 12, 65)( 13, 70)( 14, 72)( 15, 71)( 16, 67)
( 17, 69)( 18, 68)( 19, 46)( 20, 48)( 21, 47)( 22, 52)( 23, 54)( 24, 53)
( 25, 49)( 26, 51)( 27, 50)( 28, 37)( 29, 39)( 30, 38)( 31, 43)( 32, 45)
( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 73,127)( 74,129)( 75,128)( 76,133)
( 77,135)( 78,134)( 79,130)( 80,132)( 81,131)( 82,136)( 83,138)( 84,137)
( 85,142)( 86,144)( 87,143)( 88,139)( 89,141)( 90,140)( 91,118)( 92,120)
( 93,119)( 94,124)( 95,126)( 96,125)( 97,121)( 98,123)( 99,122)(100,109)
(101,111)(102,110)(103,115)(104,117)(105,116)(106,112)(107,114)(108,113);;
s1 := (  1,  4)(  2,  5)(  3,  6)( 10, 13)( 11, 14)( 12, 15)( 19, 31)( 20, 32)
( 21, 33)( 22, 28)( 23, 29)( 24, 30)( 25, 34)( 26, 35)( 27, 36)( 37, 49)
( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 43, 52)( 44, 53)( 45, 54)
( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)( 66, 69)( 73, 94)( 74, 95)
( 75, 96)( 76, 91)( 77, 92)( 78, 93)( 79, 97)( 80, 98)( 81, 99)( 82,103)
( 83,104)( 84,105)( 85,100)( 86,101)( 87,102)( 88,106)( 89,107)( 90,108)
(109,139)(110,140)(111,141)(112,136)(113,137)(114,138)(115,142)(116,143)
(117,144)(118,130)(119,131)(120,132)(121,127)(122,128)(123,129)(124,133)
(125,134)(126,135);;
s2 := (  1, 73)(  2, 76)(  3, 79)(  4, 74)(  5, 77)(  6, 80)(  7, 75)(  8, 78)
(  9, 81)( 10, 82)( 11, 85)( 12, 88)( 13, 83)( 14, 86)( 15, 89)( 16, 84)
( 17, 87)( 18, 90)( 19,100)( 20,103)( 21,106)( 22,101)( 23,104)( 24,107)
( 25,102)( 26,105)( 27,108)( 28, 91)( 29, 94)( 30, 97)( 31, 92)( 32, 95)
( 33, 98)( 34, 93)( 35, 96)( 36, 99)( 37,118)( 38,121)( 39,124)( 40,119)
( 41,122)( 42,125)( 43,120)( 44,123)( 45,126)( 46,109)( 47,112)( 48,115)
( 49,110)( 50,113)( 51,116)( 52,111)( 53,114)( 54,117)( 55,127)( 56,130)
( 57,133)( 58,128)( 59,131)( 60,134)( 61,129)( 62,132)( 63,135)( 64,136)
( 65,139)( 66,142)( 67,137)( 68,140)( 69,143)( 70,138)( 71,141)( 72,144);;
s3 := (145,146);;
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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(146)!(  1, 55)(  2, 57)(  3, 56)(  4, 61)(  5, 63)(  6, 62)(  7, 58)
(  8, 60)(  9, 59)( 10, 64)( 11, 66)( 12, 65)( 13, 70)( 14, 72)( 15, 71)
( 16, 67)( 17, 69)( 18, 68)( 19, 46)( 20, 48)( 21, 47)( 22, 52)( 23, 54)
( 24, 53)( 25, 49)( 26, 51)( 27, 50)( 28, 37)( 29, 39)( 30, 38)( 31, 43)
( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 73,127)( 74,129)( 75,128)
( 76,133)( 77,135)( 78,134)( 79,130)( 80,132)( 81,131)( 82,136)( 83,138)
( 84,137)( 85,142)( 86,144)( 87,143)( 88,139)( 89,141)( 90,140)( 91,118)
( 92,120)( 93,119)( 94,124)( 95,126)( 96,125)( 97,121)( 98,123)( 99,122)
(100,109)(101,111)(102,110)(103,115)(104,117)(105,116)(106,112)(107,114)
(108,113);
s1 := Sym(146)!(  1,  4)(  2,  5)(  3,  6)( 10, 13)( 11, 14)( 12, 15)( 19, 31)
( 20, 32)( 21, 33)( 22, 28)( 23, 29)( 24, 30)( 25, 34)( 26, 35)( 27, 36)
( 37, 49)( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 43, 52)( 44, 53)
( 45, 54)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)( 66, 69)( 73, 94)
( 74, 95)( 75, 96)( 76, 91)( 77, 92)( 78, 93)( 79, 97)( 80, 98)( 81, 99)
( 82,103)( 83,104)( 84,105)( 85,100)( 86,101)( 87,102)( 88,106)( 89,107)
( 90,108)(109,139)(110,140)(111,141)(112,136)(113,137)(114,138)(115,142)
(116,143)(117,144)(118,130)(119,131)(120,132)(121,127)(122,128)(123,129)
(124,133)(125,134)(126,135);
s2 := Sym(146)!(  1, 73)(  2, 76)(  3, 79)(  4, 74)(  5, 77)(  6, 80)(  7, 75)
(  8, 78)(  9, 81)( 10, 82)( 11, 85)( 12, 88)( 13, 83)( 14, 86)( 15, 89)
( 16, 84)( 17, 87)( 18, 90)( 19,100)( 20,103)( 21,106)( 22,101)( 23,104)
( 24,107)( 25,102)( 26,105)( 27,108)( 28, 91)( 29, 94)( 30, 97)( 31, 92)
( 32, 95)( 33, 98)( 34, 93)( 35, 96)( 36, 99)( 37,118)( 38,121)( 39,124)
( 40,119)( 41,122)( 42,125)( 43,120)( 44,123)( 45,126)( 46,109)( 47,112)
( 48,115)( 49,110)( 50,113)( 51,116)( 52,111)( 53,114)( 54,117)( 55,127)
( 56,130)( 57,133)( 58,128)( 59,131)( 60,134)( 61,129)( 62,132)( 63,135)
( 64,136)( 65,139)( 66,142)( 67,137)( 68,140)( 69,143)( 70,138)( 71,141)
( 72,144);
s3 := Sym(146)!(145,146);
poly := sub<Sym(146)|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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```

to this polytope