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# Polytope of Type {8,8}

Atlas Canonical Name : {8,8}*256a
if this polytope has a name.
Group : SmallGroup(256,722)
Rank : 3
Schlafli Type : {8,8}
Number of vertices, edges, etc : 16, 64, 16
Order of s0s1s2 : 8
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Dual
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{8,8,2} of size 512
Vertex Figure Of :
{2,8,8} of size 512
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,8}*128a, {8,4}*128a, {8,8}*128a, {8,8}*128b, {8,8}*128c, {8,8}*128d
4-fold quotients : {4,8}*64a, {8,4}*64a, {4,8}*64b, {8,4}*64b, {4,4}*64
8-fold quotients : {4,4}*32, {2,8}*32, {8,2}*32
16-fold quotients : {2,4}*16, {4,2}*16
32-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,16}*512c, {16,8}*512c, {8,16}*512d, {16,8}*512d, {8,16}*512e, {16,8}*512e, {8,16}*512f, {16,8}*512f, {8,8}*512a, {8,8}*512b, {8,8}*512c
3-fold covers : {8,24}*768a, {24,8}*768a
5-fold covers : {8,40}*1280a, {40,8}*1280a
7-fold covers : {8,56}*1792a, {56,8}*1792a
Permutation Representation (GAP) :
```s0 := (  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 69)(  6, 70)(  7, 71)(  8, 72)
(  9, 75)( 10, 76)( 11, 73)( 12, 74)( 13, 79)( 14, 80)( 15, 77)( 16, 78)
( 17, 85)( 18, 86)( 19, 87)( 20, 88)( 21, 81)( 22, 82)( 23, 83)( 24, 84)
( 25, 95)( 26, 96)( 27, 93)( 28, 94)( 29, 91)( 30, 92)( 31, 89)( 32, 90)
( 33, 97)( 34, 98)( 35, 99)( 36,100)( 37,101)( 38,102)( 39,103)( 40,104)
( 41,107)( 42,108)( 43,105)( 44,106)( 45,111)( 46,112)( 47,109)( 48,110)
( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)( 56,116)
( 57,127)( 58,128)( 59,125)( 60,126)( 61,123)( 62,124)( 63,121)( 64,122);;
s1 := (  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 21)( 18, 22)( 19, 23)( 20, 24)
( 25, 30)( 26, 29)( 27, 32)( 28, 31)( 33, 41)( 34, 42)( 35, 43)( 36, 44)
( 37, 45)( 38, 46)( 39, 47)( 40, 48)( 49, 61)( 50, 62)( 51, 63)( 52, 64)
( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 81)( 66, 82)( 67, 83)( 68, 84)
( 69, 85)( 70, 86)( 71, 87)( 72, 88)( 73, 90)( 74, 89)( 75, 92)( 76, 91)
( 77, 94)( 78, 93)( 79, 96)( 80, 95)( 97,123)( 98,124)( 99,121)(100,122)
(101,127)(102,128)(103,125)(104,126)(105,115)(106,116)(107,113)(108,114)
(109,119)(110,120)(111,117)(112,118);;
s2 := (  1, 33)(  2, 34)(  3, 35)(  4, 36)(  5, 37)(  6, 38)(  7, 39)(  8, 40)
(  9, 42)( 10, 41)( 11, 44)( 12, 43)( 13, 46)( 14, 45)( 15, 48)( 16, 47)
( 17, 51)( 18, 52)( 19, 49)( 20, 50)( 21, 55)( 22, 56)( 23, 53)( 24, 54)
( 25, 60)( 26, 59)( 27, 58)( 28, 57)( 29, 64)( 30, 63)( 31, 62)( 32, 61)
( 65, 97)( 66, 98)( 67, 99)( 68,100)( 69,101)( 70,102)( 71,103)( 72,104)
( 73,106)( 74,105)( 75,108)( 76,107)( 77,110)( 78,109)( 79,112)( 80,111)
( 81,115)( 82,116)( 83,113)( 84,114)( 85,119)( 86,120)( 87,117)( 88,118)
( 89,124)( 90,123)( 91,122)( 92,121)( 93,128)( 94,127)( 95,126)( 96,125);;
poly := Group([s0,s1,s2]);;

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

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope