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# Polytope of Type {4,20}

Atlas Canonical Name : {4,20}*320
if this polytope has a name.
Group : SmallGroup(320,368)
Rank : 3
Schlafli Type : {4,20}
Number of vertices, edges, etc : 8, 80, 40
Order of s0s1s2 : 20
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,20,2} of size 640
{4,20,4} of size 1280
{4,20,6} of size 1920
{4,20,6} of size 1920
Vertex Figure Of :
{2,4,20} of size 640
{4,4,20} of size 1280
{6,4,20} of size 1920
{3,4,20} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,20}*160
4-fold quotients : {2,20}*80, {4,10}*80
5-fold quotients : {4,4}*64
8-fold quotients : {2,10}*40
10-fold quotients : {4,4}*32
16-fold quotients : {2,5}*20
20-fold quotients : {2,4}*16, {4,2}*16
40-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,40}*640a, {8,20}*640a, {4,20}*640a, {4,40}*640b, {8,20}*640b
3-fold covers : {12,20}*960a, {4,60}*960a
4-fold covers : {8,40}*1280a, {8,20}*1280a, {8,40}*1280b, {4,40}*1280a, {8,40}*1280c, {8,40}*1280d, {16,20}*1280a, {4,80}*1280a, {16,20}*1280b, {4,80}*1280b, {4,20}*1280a, {4,40}*1280b, {8,20}*1280b, {8,20}*1280c, {8,40}*1280e, {4,40}*1280c, {4,40}*1280d, {8,20}*1280d, {8,40}*1280f, {8,40}*1280g, {8,40}*1280h
5-fold covers : {4,100}*1600, {20,20}*1600a, {20,20}*1600b
6-fold covers : {8,60}*1920a, {4,120}*1920a, {12,40}*1920a, {24,20}*1920a, {4,60}*1920a, {4,120}*1920b, {8,60}*1920b, {12,40}*1920b, {24,20}*1920b, {12,20}*1920a
Permutation Representation (GAP) :
s0 := (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)
(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,76)
(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75);;
s1 := ( 1,41)( 2,45)( 3,44)( 4,43)( 5,42)( 6,46)( 7,50)( 8,49)( 9,48)(10,47)
(11,51)(12,55)(13,54)(14,53)(15,52)(16,56)(17,60)(18,59)(19,58)(20,57)(21,61)
(22,65)(23,64)(24,63)(25,62)(26,66)(27,70)(28,69)(29,68)(30,67)(31,71)(32,75)
(33,74)(34,73)(35,72)(36,76)(37,80)(38,79)(39,78)(40,77);;
s2 := ( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,17)(12,16)(13,20)(14,19)(15,18)(21,22)
(23,25)(26,27)(28,30)(31,37)(32,36)(33,40)(34,39)(35,38)(41,62)(42,61)(43,65)
(44,64)(45,63)(46,67)(47,66)(48,70)(49,69)(50,68)(51,77)(52,76)(53,80)(54,79)
(55,78)(56,72)(57,71)(58,75)(59,74)(60,73);;
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,
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

References : None.
to this polytope