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# Polytope of Type {4,24}

Atlas Canonical Name : {4,24}*384d
if this polytope has a name.
Group : SmallGroup(384,18019)
Rank : 3
Schlafli Type : {4,24}
Number of vertices, edges, etc : 8, 96, 48
Order of s0s1s2 : 24
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,24,2} of size 768
Vertex Figure Of :
{2,4,24} of size 768
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,12}*192b
4-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
8-fold quotients : {2,12}*48, {4,3}*48, {4,6}*48b, {4,6}*48c
16-fold quotients : {4,3}*24, {2,6}*24
24-fold quotients : {2,4}*16
32-fold quotients : {2,3}*12
48-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,24}*768j, {8,24}*768l, {4,24}*768i
3-fold covers : {4,72}*1152d, {12,24}*1152q, {12,24}*1152r
5-fold covers : {20,24}*1920d, {4,120}*1920d
Permutation Representation (GAP) :
```s0 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)
(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)
(42,44)(45,47)(46,48)(49,63)(50,64)(51,61)(52,62)(53,67)(54,68)(55,65)(56,66)
(57,71)(58,72)(59,69)(60,70)(73,87)(74,88)(75,85)(76,86)(77,91)(78,92)(79,89)
(80,90)(81,95)(82,96)(83,93)(84,94);;
s1 := ( 1,49)( 2,51)( 3,50)( 4,52)( 5,57)( 6,59)( 7,58)( 8,60)( 9,53)(10,55)
(11,54)(12,56)(13,61)(14,63)(15,62)(16,64)(17,69)(18,71)(19,70)(20,72)(21,65)
(22,67)(23,66)(24,68)(25,85)(26,87)(27,86)(28,88)(29,93)(30,95)(31,94)(32,96)
(33,89)(34,91)(35,90)(36,92)(37,73)(38,75)(39,74)(40,76)(41,81)(42,83)(43,82)
(44,84)(45,77)(46,79)(47,78)(48,80);;
s2 := ( 1, 9)( 2,12)( 3,11)( 4,10)( 6, 8)(13,21)(14,24)(15,23)(16,22)(18,20)
(25,45)(26,48)(27,47)(28,46)(29,41)(30,44)(31,43)(32,42)(33,37)(34,40)(35,39)
(36,38)(49,81)(50,84)(51,83)(52,82)(53,77)(54,80)(55,79)(56,78)(57,73)(58,76)
(59,75)(60,74)(61,93)(62,96)(63,95)(64,94)(65,89)(66,92)(67,91)(68,90)(69,85)
(70,88)(71,87)(72,86);;
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,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1,
s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope