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# Polytope of Type {4,16}

Atlas Canonical Name : {4,16}*128a
Also Known As : {4,16|2}. if this polytope has another name.
Group : SmallGroup(128,916)
Rank : 3
Schlafli Type : {4,16}
Number of vertices, edges, etc : 4, 32, 16
Order of s0s1s2 : 16
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,16,2} of size 256
{4,16,4} of size 512
{4,16,4} of size 512
{4,16,6} of size 768
{4,16,10} of size 1280
{4,16,14} of size 1792
Vertex Figure Of :
{2,4,16} of size 256
{4,4,16} of size 512
{6,4,16} of size 768
{3,4,16} of size 768
{6,4,16} of size 1152
{10,4,16} of size 1280
{14,4,16} of size 1792
{5,4,16} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,8}*64a, {2,16}*64
4-fold quotients : {4,4}*32, {2,8}*32
8-fold quotients : {2,4}*16, {4,2}*16
16-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,16}*256a, {8,16}*256c, {8,16}*256d, {4,32}*256a, {4,32}*256b
3-fold covers : {4,48}*384a, {12,16}*384a
4-fold covers : {4,16}*512a, {8,16}*512a, {16,16}*512a, {16,16}*512b, {16,16}*512g, {16,16}*512h, {8,16}*512c, {4,32}*512a, {4,32}*512b, {8,32}*512a, {8,32}*512b, {8,32}*512c, {8,32}*512d, {4,64}*512a, {4,64}*512b
5-fold covers : {4,80}*640a, {20,16}*640a
6-fold covers : {12,16}*768a, {4,48}*768a, {24,16}*768c, {8,48}*768c, {8,48}*768d, {24,16}*768d, {12,32}*768a, {4,96}*768a, {12,32}*768b, {4,96}*768b
7-fold covers : {4,112}*896a, {28,16}*896a
9-fold covers : {36,16}*1152a, {4,144}*1152a, {12,48}*1152a, {12,48}*1152b, {12,48}*1152c, {4,16}*1152a, {4,48}*1152a, {12,16}*1152a
10-fold covers : {20,16}*1280a, {4,80}*1280a, {40,16}*1280c, {8,80}*1280c, {8,80}*1280d, {40,16}*1280d, {20,32}*1280a, {4,160}*1280a, {20,32}*1280b, {4,160}*1280b
11-fold covers : {44,16}*1408a, {4,176}*1408a
13-fold covers : {52,16}*1664a, {4,208}*1664a
14-fold covers : {28,16}*1792a, {4,112}*1792a, {56,16}*1792c, {8,112}*1792c, {8,112}*1792d, {56,16}*1792d, {28,32}*1792a, {4,224}*1792a, {28,32}*1792b, {4,224}*1792b
15-fold covers : {60,16}*1920a, {4,240}*1920a, {12,80}*1920a, {20,48}*1920a
Permutation Representation (GAP) :
s0 := ( 1,17)( 2,18)( 3,19)( 4,20)( 5,21)( 6,22)( 7,23)( 8,24)( 9,25)(10,26)
(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(33,49)(34,50)(35,51)(36,52)(37,53)
(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)
(48,64);;
s1 := ( 3, 4)( 7, 8)( 9,11)(10,12)(13,15)(14,16)(17,21)(18,22)(19,24)(20,23)
(25,31)(26,32)(27,29)(28,30)(33,41)(34,42)(35,44)(36,43)(37,45)(38,46)(39,48)
(40,47)(49,61)(50,62)(51,64)(52,63)(53,57)(54,58)(55,60)(56,59);;
s2 := ( 1,33)( 2,34)( 3,36)( 4,35)( 5,37)( 6,38)( 7,40)( 8,39)( 9,43)(10,44)
(11,41)(12,42)(13,47)(14,48)(15,45)(16,46)(17,49)(18,50)(19,52)(20,51)(21,53)
(22,54)(23,56)(24,55)(25,59)(26,60)(27,57)(28,58)(29,63)(30,64)(31,61)
(32,62);;
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, 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(64)!( 1,17)( 2,18)( 3,19)( 4,20)( 5,21)( 6,22)( 7,23)( 8,24)( 9,25)
(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(33,49)(34,50)(35,51)(36,52)
(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)
(48,64);
s1 := Sym(64)!( 3, 4)( 7, 8)( 9,11)(10,12)(13,15)(14,16)(17,21)(18,22)(19,24)
(20,23)(25,31)(26,32)(27,29)(28,30)(33,41)(34,42)(35,44)(36,43)(37,45)(38,46)
(39,48)(40,47)(49,61)(50,62)(51,64)(52,63)(53,57)(54,58)(55,60)(56,59);
s2 := Sym(64)!( 1,33)( 2,34)( 3,36)( 4,35)( 5,37)( 6,38)( 7,40)( 8,39)( 9,43)
(10,44)(11,41)(12,42)(13,47)(14,48)(15,45)(16,46)(17,49)(18,50)(19,52)(20,51)
(21,53)(22,54)(23,56)(24,55)(25,59)(26,60)(27,57)(28,58)(29,63)(30,64)(31,61)
(32,62);
poly := sub<Sym(64)|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, 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