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# Polytope of Type {4,4,8}

Atlas Canonical Name : {4,4,8}*256a
Also Known As : {{4,4|2},{4,8|2}}. if this polytope has another name.
Group : SmallGroup(256,16836)
Rank : 4
Schlafli Type : {4,4,8}
Number of vertices, edges, etc : 4, 8, 16, 8
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,4,8,2} of size 512
Vertex Figure Of :
{2,4,4,8} of size 512
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,4,4}*128, {2,4,8}*128a, {4,2,8}*128
4-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64, {2,2,8}*64
8-fold quotients : {2,2,4}*32, {2,4,2}*32, {4,2,2}*32
16-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,4,8}*512b, {4,8,8}*512a, {4,4,8}*512a, {4,8,8}*512c, {4,8,8}*512e, {4,8,8}*512g, {4,4,8}*512b, {8,4,8}*512d, {4,4,16}*512a, {4,4,16}*512b
3-fold covers : {12,4,8}*768a, {4,12,8}*768a, {4,4,24}*768a
5-fold covers : {20,4,8}*1280a, {4,20,8}*1280a, {4,4,40}*1280a
7-fold covers : {28,4,8}*1792a, {4,28,8}*1792a, {4,4,56}*1792a
Permutation Representation (GAP) :
```s0 := (  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 86)(  7, 87)(  8, 88)
(  9, 89)( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)( 16, 96)
( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)( 24, 72)
( 25, 73)( 26, 74)( 27, 75)( 28, 76)( 29, 77)( 30, 78)( 31, 79)( 32, 80)
( 33,115)( 34,116)( 35,113)( 36,114)( 37,119)( 38,120)( 39,117)( 40,118)
( 41,123)( 42,124)( 43,121)( 44,122)( 45,127)( 46,128)( 47,125)( 48,126)
( 49, 99)( 50,100)( 51, 97)( 52, 98)( 53,103)( 54,104)( 55,101)( 56,102)
( 57,107)( 58,108)( 59,105)( 60,106)( 61,111)( 62,112)( 63,109)( 64,110);;
s1 := (  1, 33)(  2, 34)(  3, 35)(  4, 36)(  5, 37)(  6, 38)(  7, 39)(  8, 40)
(  9, 41)( 10, 42)( 11, 43)( 12, 44)( 13, 45)( 14, 46)( 15, 47)( 16, 48)
( 17, 49)( 18, 50)( 19, 51)( 20, 52)( 21, 53)( 22, 54)( 23, 55)( 24, 56)
( 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)
( 73,105)( 74,106)( 75,107)( 76,108)( 77,109)( 78,110)( 79,111)( 80,112)
( 81,113)( 82,114)( 83,115)( 84,116)( 85,117)( 86,118)( 87,119)( 88,120)
( 89,121)( 90,122)( 91,123)( 92,124)( 93,125)( 94,126)( 95,127)( 96,128);;
s2 := (  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)
( 21, 29)( 22, 30)( 23, 31)( 24, 32)( 33, 37)( 34, 38)( 35, 39)( 36, 40)
( 41, 46)( 42, 45)( 43, 48)( 44, 47)( 49, 61)( 50, 62)( 51, 63)( 52, 64)
( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 73)( 66, 74)( 67, 75)( 68, 76)
( 69, 77)( 70, 78)( 71, 79)( 72, 80)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97,109)( 98,110)( 99,111)(100,112)(101,105)(102,106)(103,107)(104,108)
(113,117)(114,118)(115,119)(116,120)(121,126)(122,125)(123,128)(124,127);;
s3 := (  1, 17)(  2, 18)(  3, 19)(  4, 20)(  5, 21)(  6, 22)(  7, 23)(  8, 24)
(  9, 26)( 10, 25)( 11, 28)( 12, 27)( 13, 30)( 14, 29)( 15, 32)( 16, 31)
( 33, 49)( 34, 50)( 35, 51)( 36, 52)( 37, 53)( 38, 54)( 39, 55)( 40, 56)
( 41, 58)( 42, 57)( 43, 60)( 44, 59)( 45, 62)( 46, 61)( 47, 64)( 48, 63)
( 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)
( 97,113)( 98,114)( 99,115)(100,116)(101,117)(102,118)(103,119)(104,120)
(105,122)(106,121)(107,124)(108,123)(109,126)(110,125)(111,128)(112,127);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(128)!(  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 86)(  7, 87)
(  8, 88)(  9, 89)( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)
( 16, 96)( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)
( 24, 72)( 25, 73)( 26, 74)( 27, 75)( 28, 76)( 29, 77)( 30, 78)( 31, 79)
( 32, 80)( 33,115)( 34,116)( 35,113)( 36,114)( 37,119)( 38,120)( 39,117)
( 40,118)( 41,123)( 42,124)( 43,121)( 44,122)( 45,127)( 46,128)( 47,125)
( 48,126)( 49, 99)( 50,100)( 51, 97)( 52, 98)( 53,103)( 54,104)( 55,101)
( 56,102)( 57,107)( 58,108)( 59,105)( 60,106)( 61,111)( 62,112)( 63,109)
( 64,110);
s1 := Sym(128)!(  1, 33)(  2, 34)(  3, 35)(  4, 36)(  5, 37)(  6, 38)(  7, 39)
(  8, 40)(  9, 41)( 10, 42)( 11, 43)( 12, 44)( 13, 45)( 14, 46)( 15, 47)
( 16, 48)( 17, 49)( 18, 50)( 19, 51)( 20, 52)( 21, 53)( 22, 54)( 23, 55)
( 24, 56)( 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)( 73,105)( 74,106)( 75,107)( 76,108)( 77,109)( 78,110)( 79,111)
( 80,112)( 81,113)( 82,114)( 83,115)( 84,116)( 85,117)( 86,118)( 87,119)
( 88,120)( 89,121)( 90,122)( 91,123)( 92,124)( 93,125)( 94,126)( 95,127)
( 96,128);
s2 := Sym(128)!(  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 25)( 18, 26)( 19, 27)
( 20, 28)( 21, 29)( 22, 30)( 23, 31)( 24, 32)( 33, 37)( 34, 38)( 35, 39)
( 36, 40)( 41, 46)( 42, 45)( 43, 48)( 44, 47)( 49, 61)( 50, 62)( 51, 63)
( 52, 64)( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 73)( 66, 74)( 67, 75)
( 68, 76)( 69, 77)( 70, 78)( 71, 79)( 72, 80)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97,109)( 98,110)( 99,111)(100,112)(101,105)(102,106)(103,107)
(104,108)(113,117)(114,118)(115,119)(116,120)(121,126)(122,125)(123,128)
(124,127);
s3 := Sym(128)!(  1, 17)(  2, 18)(  3, 19)(  4, 20)(  5, 21)(  6, 22)(  7, 23)
(  8, 24)(  9, 26)( 10, 25)( 11, 28)( 12, 27)( 13, 30)( 14, 29)( 15, 32)
( 16, 31)( 33, 49)( 34, 50)( 35, 51)( 36, 52)( 37, 53)( 38, 54)( 39, 55)
( 40, 56)( 41, 58)( 42, 57)( 43, 60)( 44, 59)( 45, 62)( 46, 61)( 47, 64)
( 48, 63)( 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)( 97,113)( 98,114)( 99,115)(100,116)(101,117)(102,118)(103,119)
(104,120)(105,122)(106,121)(107,124)(108,123)(109,126)(110,125)(111,128)
(112,127);
poly := sub<Sym(128)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope