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Polytope of Type {8,8}

Atlas Canonical Name : {8,8}*256b
if this polytope has a name.
Group : SmallGroup(256,5078)
Rank : 3
Schlafli Type : {8,8}
Number of vertices, edges, etc : 16, 64, 16
Order of s0s1s2 : 4
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Dual
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}*128b, {8,4}*128b
4-fold quotients : {4,4}*64
8-fold quotients : {4,4}*32
16-fold quotients : {2,4}*16, {4,2}*16
32-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,8}*512c, {8,8}*512d, {8,8}*512g, {8,8}*512h, {8,8}*512i, {8,8}*512m, {8,8}*512o
3-fold covers : {8,24}*768b, {24,8}*768b
5-fold covers : {8,40}*1280b, {40,8}*1280b
7-fold covers : {8,56}*1792b, {56,8}*1792b
Permutation Representation (GAP) :
```s0 := (  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 69)(  6, 70)(  7, 71)(  8, 72)
(  9, 76)( 10, 75)( 11, 74)( 12, 73)( 13, 80)( 14, 79)( 15, 78)( 16, 77)
( 17, 85)( 18, 86)( 19, 87)( 20, 88)( 21, 81)( 22, 82)( 23, 83)( 24, 84)
( 25, 96)( 26, 95)( 27, 94)( 28, 93)( 29, 92)( 30, 91)( 31, 90)( 32, 89)
( 33, 97)( 34, 98)( 35, 99)( 36,100)( 37,101)( 38,102)( 39,103)( 40,104)
( 41,108)( 42,107)( 43,106)( 44,105)( 45,112)( 46,111)( 47,110)( 48,109)
( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)( 56,116)
( 57,128)( 58,127)( 59,126)( 60,125)( 61,124)( 62,123)( 63,122)( 64,121);;
s1 := (  3,  4)(  7,  8)( 11, 12)( 15, 16)( 17, 21)( 18, 22)( 19, 24)( 20, 23)
( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 33, 41)( 34, 42)( 35, 44)( 36, 43)
( 37, 45)( 38, 46)( 39, 48)( 40, 47)( 49, 62)( 50, 61)( 51, 63)( 52, 64)
( 53, 58)( 54, 57)( 55, 59)( 56, 60)( 65, 81)( 66, 82)( 67, 84)( 68, 83)
( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 89)( 74, 90)( 75, 92)( 76, 91)
( 77, 93)( 78, 94)( 79, 96)( 80, 95)( 97,123)( 98,124)( 99,122)(100,121)
(101,127)(102,128)(103,126)(104,125)(105,115)(106,116)(107,114)(108,113)
(109,119)(110,120)(111,118)(112,117);;
s2 := (  1, 97)(  2, 98)(  3, 99)(  4,100)(  5,101)(  6,102)(  7,103)(  8,104)
(  9,108)( 10,107)( 11,106)( 12,105)( 13,112)( 14,111)( 15,110)( 16,109)
( 17,119)( 18,120)( 19,117)( 20,118)( 21,115)( 22,116)( 23,113)( 24,114)
( 25,126)( 26,125)( 27,128)( 28,127)( 29,122)( 30,121)( 31,124)( 32,123)
( 33, 65)( 34, 66)( 35, 67)( 36, 68)( 37, 69)( 38, 70)( 39, 71)( 40, 72)
( 41, 76)( 42, 75)( 43, 74)( 44, 73)( 45, 80)( 46, 79)( 47, 78)( 48, 77)
( 49, 87)( 50, 88)( 51, 85)( 52, 86)( 53, 83)( 54, 84)( 55, 81)( 56, 82)
( 57, 94)( 58, 93)( 59, 96)( 60, 95)( 61, 90)( 62, 89)( 63, 92)( 64, 91);;
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*s0*s1*s2*s0*s1*s2*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,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
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, 76)( 10, 75)( 11, 74)( 12, 73)( 13, 80)( 14, 79)( 15, 78)
( 16, 77)( 17, 85)( 18, 86)( 19, 87)( 20, 88)( 21, 81)( 22, 82)( 23, 83)
( 24, 84)( 25, 96)( 26, 95)( 27, 94)( 28, 93)( 29, 92)( 30, 91)( 31, 90)
( 32, 89)( 33, 97)( 34, 98)( 35, 99)( 36,100)( 37,101)( 38,102)( 39,103)
( 40,104)( 41,108)( 42,107)( 43,106)( 44,105)( 45,112)( 46,111)( 47,110)
( 48,109)( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)
( 56,116)( 57,128)( 58,127)( 59,126)( 60,125)( 61,124)( 62,123)( 63,122)
( 64,121);
s1 := Sym(128)!(  3,  4)(  7,  8)( 11, 12)( 15, 16)( 17, 21)( 18, 22)( 19, 24)
( 20, 23)( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 33, 41)( 34, 42)( 35, 44)
( 36, 43)( 37, 45)( 38, 46)( 39, 48)( 40, 47)( 49, 62)( 50, 61)( 51, 63)
( 52, 64)( 53, 58)( 54, 57)( 55, 59)( 56, 60)( 65, 81)( 66, 82)( 67, 84)
( 68, 83)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 89)( 74, 90)( 75, 92)
( 76, 91)( 77, 93)( 78, 94)( 79, 96)( 80, 95)( 97,123)( 98,124)( 99,122)
(100,121)(101,127)(102,128)(103,126)(104,125)(105,115)(106,116)(107,114)
(108,113)(109,119)(110,120)(111,118)(112,117);
s2 := Sym(128)!(  1, 97)(  2, 98)(  3, 99)(  4,100)(  5,101)(  6,102)(  7,103)
(  8,104)(  9,108)( 10,107)( 11,106)( 12,105)( 13,112)( 14,111)( 15,110)
( 16,109)( 17,119)( 18,120)( 19,117)( 20,118)( 21,115)( 22,116)( 23,113)
( 24,114)( 25,126)( 26,125)( 27,128)( 28,127)( 29,122)( 30,121)( 31,124)
( 32,123)( 33, 65)( 34, 66)( 35, 67)( 36, 68)( 37, 69)( 38, 70)( 39, 71)
( 40, 72)( 41, 76)( 42, 75)( 43, 74)( 44, 73)( 45, 80)( 46, 79)( 47, 78)
( 48, 77)( 49, 87)( 50, 88)( 51, 85)( 52, 86)( 53, 83)( 54, 84)( 55, 81)
( 56, 82)( 57, 94)( 58, 93)( 59, 96)( 60, 95)( 61, 90)( 62, 89)( 63, 92)
( 64, 91);
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*s0*s1*s2*s0*s1*s2*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,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

```
References : None.
to this polytope