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# Polytope of Type {8,16}

Atlas Canonical Name : {8,16}*256e
if this polytope has a name.
Group : SmallGroup(256,5312)
Rank : 3
Schlafli Type : {8,16}
Number of vertices, edges, etc : 8, 64, 16
Order of s0s1s2 : 16
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{8,16,2} of size 512
Vertex Figure Of :
{2,8,16} of size 512
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {8,8}*128a, {4,16}*128b
4-fold quotients : {4,8}*64a, {8,4}*64b
8-fold quotients : {4,4}*32, {2,8}*32
16-fold quotients : {2,4}*16, {4,2}*16
32-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {16,16}*512d, {16,16}*512l, {8,16}*512c
3-fold covers : {24,16}*768e, {8,48}*768e
5-fold covers : {40,16}*1280e, {8,80}*1280e
7-fold covers : {56,16}*1792e, {8,112}*1792e
Permutation Representation (GAP) :
```s0 := (  1, 33)(  2, 34)(  3, 35)(  4, 36)(  5, 37)(  6, 38)(  7, 39)(  8, 40)
(  9, 44)( 10, 43)( 11, 42)( 12, 41)( 13, 48)( 14, 47)( 15, 46)( 16, 45)
( 17, 51)( 18, 52)( 19, 49)( 20, 50)( 21, 55)( 22, 56)( 23, 53)( 24, 54)
( 25, 58)( 26, 57)( 27, 60)( 28, 59)( 29, 62)( 30, 61)( 31, 64)( 32, 63)
( 65, 97)( 66, 98)( 67, 99)( 68,100)( 69,101)( 70,102)( 71,103)( 72,104)
( 73,108)( 74,107)( 75,106)( 76,105)( 77,112)( 78,111)( 79,110)( 80,109)
( 81,115)( 82,116)( 83,113)( 84,114)( 85,119)( 86,120)( 87,117)( 88,118)
( 89,122)( 90,121)( 91,124)( 92,123)( 93,126)( 94,125)( 95,128)( 96,127);;
s1 := (  5,  6)(  7,  8)(  9, 12)( 10, 11)( 13, 15)( 14, 16)( 17, 21)( 18, 22)
( 19, 23)( 20, 24)( 25, 32)( 26, 31)( 27, 30)( 28, 29)( 33, 41)( 34, 42)
( 35, 43)( 36, 44)( 37, 46)( 38, 45)( 39, 48)( 40, 47)( 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, 86)( 70, 85)( 71, 88)( 72, 87)( 73, 92)( 74, 91)
( 75, 90)( 76, 89)( 77, 95)( 78, 96)( 79, 93)( 80, 94)( 97,123)( 98,124)
( 99,121)(100,122)(101,128)(102,127)(103,126)(104,125)(105,115)(106,116)
(107,113)(108,114)(109,120)(110,119)(111,118)(112,117);;
s2 := (  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 70)(  6, 69)(  7, 72)(  8, 71)
(  9, 75)( 10, 76)( 11, 73)( 12, 74)( 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, 95)( 26, 96)( 27, 93)( 28, 94)( 29, 91)( 30, 92)( 31, 89)( 32, 90)
( 33, 97)( 34, 98)( 35, 99)( 36,100)( 37,102)( 38,101)( 39,104)( 40,103)
( 41,107)( 42,108)( 43,105)( 44,106)( 45,112)( 46,111)( 47,110)( 48,109)
( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)( 56,116)
( 57,127)( 58,128)( 59,125)( 60,126)( 61,123)( 62,124)( 63,121)( 64,122);;
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*s0*s2*s1*s0*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope