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# Polytope of Type {2,6,8}

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

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

```

to this polytope