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# Polytope of Type {32,4}

Atlas Canonical Name : {32,4}*256a
Also Known As : {32,4|2}. if this polytope has another name.
Group : SmallGroup(256,6649)
Rank : 3
Schlafli Type : {32,4}
Number of vertices, edges, etc : 32, 64, 4
Order of s0s1s2 : 32
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{32,4,2} of size 512
Vertex Figure Of :
{2,32,4} of size 512
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {16,4}*128a, {32,2}*128
4-fold quotients : {8,4}*64a, {16,2}*64
8-fold quotients : {4,4}*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 : {32,4}*512a, {32,8}*512a, {32,8}*512b, {64,4}*512a, {64,4}*512b
3-fold covers : {32,12}*768a, {96,4}*768a
5-fold covers : {32,20}*1280a, {160,4}*1280a
7-fold covers : {32,28}*1792a, {224,4}*1792a
Permutation Representation (GAP) :
```s0 := (  1, 65)(  2, 66)(  3, 68)(  4, 67)(  5, 71)(  6, 72)(  7, 69)(  8, 70)
(  9, 73)( 10, 74)( 11, 76)( 12, 75)( 13, 79)( 14, 80)( 15, 77)( 16, 78)
( 17, 85)( 18, 86)( 19, 88)( 20, 87)( 21, 81)( 22, 82)( 23, 84)( 24, 83)
( 25, 93)( 26, 94)( 27, 96)( 28, 95)( 29, 89)( 30, 90)( 31, 92)( 32, 91)
( 33, 97)( 34, 98)( 35,100)( 36, 99)( 37,103)( 38,104)( 39,101)( 40,102)
( 41,105)( 42,106)( 43,108)( 44,107)( 45,111)( 46,112)( 47,109)( 48,110)
( 49,117)( 50,118)( 51,120)( 52,119)( 53,113)( 54,114)( 55,116)( 56,115)
( 57,125)( 58,126)( 59,128)( 60,127)( 61,121)( 62,122)( 63,124)( 64,123);;
s1 := (  3,  4)(  5,  7)(  6,  8)( 11, 12)( 13, 15)( 14, 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, 47)( 38, 48)( 39, 45)( 40, 46)( 49, 61)( 50, 62)
( 51, 64)( 52, 63)( 53, 57)( 54, 58)( 55, 60)( 56, 59)( 65, 81)( 66, 82)
( 67, 84)( 68, 83)( 69, 87)( 70, 88)( 71, 85)( 72, 86)( 73, 89)( 74, 90)
( 75, 92)( 76, 91)( 77, 95)( 78, 96)( 79, 93)( 80, 94)( 97,121)( 98,122)
( 99,124)(100,123)(101,127)(102,128)(103,125)(104,126)(105,113)(106,114)
(107,116)(108,115)(109,119)(110,120)(111,117)(112,118);;
s2 := (  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);;
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, 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*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, 65)(  2, 66)(  3, 68)(  4, 67)(  5, 71)(  6, 72)(  7, 69)
(  8, 70)(  9, 73)( 10, 74)( 11, 76)( 12, 75)( 13, 79)( 14, 80)( 15, 77)
( 16, 78)( 17, 85)( 18, 86)( 19, 88)( 20, 87)( 21, 81)( 22, 82)( 23, 84)
( 24, 83)( 25, 93)( 26, 94)( 27, 96)( 28, 95)( 29, 89)( 30, 90)( 31, 92)
( 32, 91)( 33, 97)( 34, 98)( 35,100)( 36, 99)( 37,103)( 38,104)( 39,101)
( 40,102)( 41,105)( 42,106)( 43,108)( 44,107)( 45,111)( 46,112)( 47,109)
( 48,110)( 49,117)( 50,118)( 51,120)( 52,119)( 53,113)( 54,114)( 55,116)
( 56,115)( 57,125)( 58,126)( 59,128)( 60,127)( 61,121)( 62,122)( 63,124)
( 64,123);
s1 := Sym(128)!(  3,  4)(  5,  7)(  6,  8)( 11, 12)( 13, 15)( 14, 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, 47)( 38, 48)( 39, 45)( 40, 46)( 49, 61)
( 50, 62)( 51, 64)( 52, 63)( 53, 57)( 54, 58)( 55, 60)( 56, 59)( 65, 81)
( 66, 82)( 67, 84)( 68, 83)( 69, 87)( 70, 88)( 71, 85)( 72, 86)( 73, 89)
( 74, 90)( 75, 92)( 76, 91)( 77, 95)( 78, 96)( 79, 93)( 80, 94)( 97,121)
( 98,122)( 99,124)(100,123)(101,127)(102,128)(103,125)(104,126)(105,113)
(106,114)(107,116)(108,115)(109,119)(110,120)(111,117)(112,118);
s2 := 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);
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, 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*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