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# Polytope of Type {4,8}

Atlas Canonical Name : {4,8}*256d
if this polytope has a name.
Group : SmallGroup(256,6665)
Rank : 3
Schlafli Type : {4,8}
Number of vertices, edges, etc : 16, 64, 32
Order of s0s1s2 : 8
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Facet Of :
{4,8,2} of size 512
Vertex Figure Of :
{2,4,8} of size 512
{3,4,8} of size 768
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,8}*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 : {4,8}*512c, {8,8}*512m, {8,8}*512s
3-fold covers : {4,24}*768d, {12,8}*768d
5-fold covers : {4,40}*1280d, {20,8}*1280d
7-fold covers : {4,56}*1792d, {28,8}*1792d
Permutation Representation (GAP) :
```s0 := (  1, 33)(  2, 34)(  3, 36)(  4, 35)(  5, 38)(  6, 37)(  7, 39)(  8, 40)
(  9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 46)( 14, 45)( 15, 47)( 16, 48)
( 17, 53)( 18, 54)( 19, 56)( 20, 55)( 21, 49)( 22, 50)( 23, 52)( 24, 51)
( 25, 62)( 26, 61)( 27, 63)( 28, 64)( 29, 58)( 30, 57)( 31, 59)( 32, 60)
( 65, 97)( 66, 98)( 67,100)( 68, 99)( 69,102)( 70,101)( 71,103)( 72,104)
( 73,105)( 74,106)( 75,108)( 76,107)( 77,110)( 78,109)( 79,111)( 80,112)
( 81,117)( 82,118)( 83,120)( 84,119)( 85,113)( 86,114)( 87,116)( 88,115)
( 89,126)( 90,125)( 91,127)( 92,128)( 93,122)( 94,121)( 95,123)( 96,124);;
s1 := (  5,  7)(  6,  8)( 13, 15)( 14, 16)( 17, 18)( 19, 20)( 21, 24)( 22, 23)
( 25, 26)( 27, 28)( 29, 32)( 30, 31)( 33, 41)( 34, 42)( 35, 43)( 36, 44)
( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 49, 59)( 50, 60)( 51, 57)( 52, 58)
( 53, 61)( 54, 62)( 55, 63)( 56, 64)( 65, 81)( 66, 82)( 67, 83)( 68, 84)
( 69, 87)( 70, 88)( 71, 85)( 72, 86)( 73, 89)( 74, 90)( 75, 91)( 76, 92)
( 77, 95)( 78, 96)( 79, 93)( 80, 94)( 97,126)( 98,125)( 99,128)(100,127)
(101,123)(102,124)(103,121)(104,122)(105,117)(106,118)(107,119)(108,120)
(109,116)(110,115)(111,114)(112,113);;
s2 := (  1, 97)(  2, 98)(  3, 99)(  4,100)(  5,102)(  6,101)(  7,104)(  8,103)
(  9,112)( 10,111)( 11,110)( 12,109)( 13,108)( 14,107)( 15,106)( 16,105)
( 17,118)( 18,117)( 19,120)( 20,119)( 21,114)( 22,113)( 23,116)( 24,115)
( 25,123)( 26,124)( 27,121)( 28,122)( 29,128)( 30,127)( 31,126)( 32,125)
( 33, 65)( 34, 66)( 35, 67)( 36, 68)( 37, 70)( 38, 69)( 39, 72)( 40, 71)
( 41, 80)( 42, 79)( 43, 78)( 44, 77)( 45, 76)( 46, 75)( 47, 74)( 48, 73)
( 49, 86)( 50, 85)( 51, 88)( 52, 87)( 53, 82)( 54, 81)( 55, 84)( 56, 83)
( 57, 91)( 58, 92)( 59, 89)( 60, 90)( 61, 96)( 62, 95)( 63, 94)( 64, 93);;
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*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s0 ];;
poly := F / rels;;

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

```
References : None.
to this polytope