# Polytope of Type {3,8,4}

Atlas Canonical Name : {3,8,4}*384
if this polytope has a name.
Group : SmallGroup(384,17958)
Rank : 4
Schlafli Type : {3,8,4}
Number of vertices, edges, etc : 6, 24, 32, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,8,4,2} of size 768
Vertex Figure Of :
{2,3,8,4} of size 768
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,4,4}*192b, {3,8,2}*192
4-fold quotients : {3,4,2}*96
8-fold quotients : {3,2,4}*48, {3,4,2}*48
16-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,8,8}*768, {6,8,4}*768c
3-fold covers : {9,8,4}*1152, {3,8,12}*1152, {3,24,4}*1152
5-fold covers : {3,8,20}*1920, {15,8,4}*1920
Permutation Representation (GAP) :
```s0 := ( 3, 6)( 4, 5)( 7, 8)( 9,17)(10,18)(11,22)(12,21)(13,20)(14,19)(15,24)
(16,23)(27,30)(28,29)(31,32)(33,41)(34,42)(35,46)(36,45)(37,44)(38,43)(39,48)
(40,47)(49,50)(51,53)(52,54)(57,66)(58,65)(59,69)(60,70)(61,67)(62,68)(63,71)
(64,72)(73,74)(75,77)(76,78)(81,90)(82,89)(83,93)(84,94)(85,91)(86,92)(87,95)
(88,96);;
s1 := ( 1, 9)( 2,10)( 3,12)( 4,11)( 5,15)( 6,16)( 7,13)( 8,14)(19,20)(21,23)
(22,24)(25,33)(26,34)(27,36)(28,35)(29,39)(30,40)(31,37)(32,38)(43,44)(45,47)
(46,48)(49,58)(50,57)(51,59)(52,60)(53,64)(54,63)(55,62)(56,61)(65,66)(69,72)
(70,71)(73,82)(74,81)(75,83)(76,84)(77,88)(78,87)(79,86)(80,85)(89,90)(93,96)
(94,95);;
s2 := ( 1,55)( 2,56)( 3,53)( 4,54)( 5,52)( 6,51)( 7,50)( 8,49)( 9,63)(10,64)
(11,61)(12,62)(13,60)(14,59)(15,58)(16,57)(17,71)(18,72)(19,69)(20,70)(21,68)
(22,67)(23,66)(24,65)(25,79)(26,80)(27,77)(28,78)(29,76)(30,75)(31,74)(32,73)
(33,87)(34,88)(35,85)(36,86)(37,84)(38,83)(39,82)(40,81)(41,95)(42,96)(43,93)
(44,94)(45,92)(46,91)(47,90)(48,89);;
s3 := (49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,81)(58,82)
(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)
(70,94)(71,95)(72,96);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(96)!( 3, 6)( 4, 5)( 7, 8)( 9,17)(10,18)(11,22)(12,21)(13,20)(14,19)
(15,24)(16,23)(27,30)(28,29)(31,32)(33,41)(34,42)(35,46)(36,45)(37,44)(38,43)
(39,48)(40,47)(49,50)(51,53)(52,54)(57,66)(58,65)(59,69)(60,70)(61,67)(62,68)
(63,71)(64,72)(73,74)(75,77)(76,78)(81,90)(82,89)(83,93)(84,94)(85,91)(86,92)
(87,95)(88,96);
s1 := Sym(96)!( 1, 9)( 2,10)( 3,12)( 4,11)( 5,15)( 6,16)( 7,13)( 8,14)(19,20)
(21,23)(22,24)(25,33)(26,34)(27,36)(28,35)(29,39)(30,40)(31,37)(32,38)(43,44)
(45,47)(46,48)(49,58)(50,57)(51,59)(52,60)(53,64)(54,63)(55,62)(56,61)(65,66)
(69,72)(70,71)(73,82)(74,81)(75,83)(76,84)(77,88)(78,87)(79,86)(80,85)(89,90)
(93,96)(94,95);
s2 := Sym(96)!( 1,55)( 2,56)( 3,53)( 4,54)( 5,52)( 6,51)( 7,50)( 8,49)( 9,63)
(10,64)(11,61)(12,62)(13,60)(14,59)(15,58)(16,57)(17,71)(18,72)(19,69)(20,70)
(21,68)(22,67)(23,66)(24,65)(25,79)(26,80)(27,77)(28,78)(29,76)(30,75)(31,74)
(32,73)(33,87)(34,88)(35,85)(36,86)(37,84)(38,83)(39,82)(40,81)(41,95)(42,96)
(43,93)(44,94)(45,92)(46,91)(47,90)(48,89);
s3 := Sym(96)!(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,81)
(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)
(69,93)(70,94)(71,95)(72,96);
poly := sub<Sym(96)|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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 >;

```
References : None.
to this polytope