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# Polytope of Type {16,8}

Atlas Canonical Name : {16,8}*256d
Also Known As : {16,8|2}. if this polytope has another name.
Group : SmallGroup(256,5304)
Rank : 3
Schlafli Type : {16,8}
Number of vertices, edges, etc : 16, 64, 8
Order of s0s1s2 : 16
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{16,8,2} of size 512
Vertex Figure Of :
{2,16,8} of size 512
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {8,8}*128b, {16,4}*128a
4-fold quotients : {4,8}*64a, {8,4}*64a, {16,2}*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 : {16,16}*512b, {16,16}*512k, {16,8}*512c, {32,8}*512b, {32,8}*512d
3-fold covers : {48,8}*768d, {16,24}*768d
5-fold covers : {80,8}*1280d, {16,40}*1280d
7-fold covers : {112,8}*1792d, {16,56}*1792d
Permutation Representation (GAP) :
```s0 := (  1, 97)(  2, 98)(  3, 99)(  4,100)(  5,102)(  6,101)(  7,104)(  8,103)
(  9,105)( 10,106)( 11,107)( 12,108)( 13,110)( 14,109)( 15,112)( 16,111)
( 17,119)( 18,120)( 19,117)( 20,118)( 21,115)( 22,116)( 23,113)( 24,114)
( 25,127)( 26,128)( 27,125)( 28,126)( 29,123)( 30,124)( 31,121)( 32,122)
( 33, 65)( 34, 66)( 35, 67)( 36, 68)( 37, 70)( 38, 69)( 39, 72)( 40, 71)
( 41, 73)( 42, 74)( 43, 75)( 44, 76)( 45, 78)( 46, 77)( 47, 80)( 48, 79)
( 49, 87)( 50, 88)( 51, 85)( 52, 86)( 53, 83)( 54, 84)( 55, 81)( 56, 82)
( 57, 95)( 58, 96)( 59, 93)( 60, 94)( 61, 91)( 62, 92)( 63, 89)( 64, 90);;
s1 := (  5,  6)(  7,  8)(  9, 11)( 10, 12)( 13, 16)( 14, 15)( 17, 21)( 18, 22)
( 19, 23)( 20, 24)( 25, 31)( 26, 32)( 27, 29)( 28, 30)( 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, 91)( 74, 92)
( 75, 89)( 76, 90)( 77, 96)( 78, 95)( 79, 94)( 80, 93)( 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, 33)(  2, 34)(  3, 35)(  4, 36)(  5, 37)(  6, 38)(  7, 39)(  8, 40)
(  9, 43)( 10, 44)( 11, 41)( 12, 42)( 13, 47)( 14, 48)( 15, 45)( 16, 46)
( 17, 51)( 18, 52)( 19, 49)( 20, 50)( 21, 55)( 22, 56)( 23, 53)( 24, 54)
( 25, 57)( 26, 58)( 27, 59)( 28, 60)( 29, 61)( 30, 62)( 31, 63)( 32, 64)
( 65, 97)( 66, 98)( 67, 99)( 68,100)( 69,101)( 70,102)( 71,103)( 72,104)
( 73,107)( 74,108)( 75,105)( 76,106)( 77,111)( 78,112)( 79,109)( 80,110)
( 81,115)( 82,116)( 83,113)( 84,114)( 85,119)( 86,120)( 87,117)( 88,118)
( 89,121)( 90,122)( 91,123)( 92,124)( 93,125)( 94,126)( 95,127)( 96,128);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope