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# Polytope of Type {3,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4,4}*384b
if this polytope has a name.
Group : SmallGroup(384,17948)
Rank : 4
Schlafli Type : {3,4,4}
Number of vertices, edges, etc : 6, 24, 32, 8
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 4
Special Properties :
Locally Toroidal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,4,4,2} of size 768
Vertex Figure Of :
{2,3,4,4} of size 768
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,4,4}*192a
4-fold quotients : {3,4,2}*96
8-fold quotients : {3,4,2}*48
16-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,8,4}*768a, {3,8,4}*768b, {3,4,4}*768a, {6,4,4}*768b, {3,4,4}*768b, {3,4,4}*768c, {3,8,4}*768e, {3,8,4}*768f, {6,4,4}*768f
3-fold covers : {9,4,4}*1152b, {3,4,12}*1152, {3,12,4}*1152a
5-fold covers : {3,4,20}*1920, {15,4,4}*1920b
Permutation Representation (GAP) :
```s0 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(15,16)(17,33)(18,34)(19,36)(20,35)
(21,41)(22,42)(23,44)(24,43)(25,37)(26,38)(27,40)(28,39)(29,45)(30,46)(31,48)
(32,47)(51,52)(53,57)(54,58)(55,60)(56,59)(63,64)(65,81)(66,82)(67,84)(68,83)
(69,89)(70,90)(71,92)(72,91)(73,85)(74,86)(75,88)(76,87)(77,93)(78,94)(79,96)
(80,95);;
s1 := ( 1,17)( 2,20)( 3,19)( 4,18)( 5,21)( 6,24)( 7,23)( 8,22)( 9,29)(10,32)
(11,31)(12,30)(13,25)(14,28)(15,27)(16,26)(34,36)(38,40)(41,45)(42,48)(43,47)
(44,46)(49,65)(50,68)(51,67)(52,66)(53,69)(54,72)(55,71)(56,70)(57,77)(58,80)
(59,79)(60,78)(61,73)(62,76)(63,75)(64,74)(82,84)(86,88)(89,93)(90,96)(91,95)
(92,94);;
s2 := ( 1,61)( 2,62)( 3,63)( 4,64)( 5,57)( 6,58)( 7,59)( 8,60)( 9,53)(10,54)
(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,77)(18,78)(19,79)(20,80)(21,73)
(22,74)(23,75)(24,76)(25,69)(26,70)(27,71)(28,72)(29,65)(30,66)(31,67)(32,68)
(33,93)(34,94)(35,95)(36,96)(37,89)(38,90)(39,91)(40,92)(41,85)(42,86)(43,87)
(44,88)(45,81)(46,82)(47,83)(48,84);;
s3 := ( 5, 7)( 6, 8)( 9,12)(10,11)(13,14)(15,16)(21,23)(22,24)(25,28)(26,27)
(29,30)(31,32)(37,39)(38,40)(41,44)(42,43)(45,46)(47,48)(53,55)(54,56)(57,60)
(58,59)(61,62)(63,64)(69,71)(70,72)(73,76)(74,75)(77,78)(79,80)(85,87)(86,88)
(89,92)(90,91)(93,94)(95,96);;
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*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s2*s0*s1*s2*s3*s2*s1*s2*s0*s1*s2*s3*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope