Polytope of Type {2,4,4,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,4,4}*256
if this polytope has a name.
Group : SmallGroup(256,27633)
Rank : 5
Schlafli Type : {2,4,4,4}
Number of vertices, edges, etc : 2, 4, 8, 8, 4
Order of s0s1s2s3s4 : 4
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,4,4,4,2} of size 512
Vertex Figure Of :
{2,2,4,4,4} of size 512
{3,2,4,4,4} of size 768
{5,2,4,4,4} of size 1280
{7,2,4,4,4} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,2,4,4}*128, {2,4,4,2}*128, {2,4,2,4}*128
4-fold quotients : {2,2,2,4}*64, {2,2,4,2}*64, {2,4,2,2}*64
8-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,4,4,4}*512, {2,4,8,4}*512a, {2,4,8,4}*512b, {2,4,8,4}*512c, {2,4,8,4}*512d, {2,4,4,8}*512a, {2,8,4,4}*512a, {2,4,4,8}*512b, {2,8,4,4}*512b, {2,4,4,4}*512a, {2,4,4,4}*512b
3-fold covers : {6,4,4,4}*768, {2,4,4,12}*768, {2,12,4,4}*768, {2,4,12,4}*768a
5-fold covers : {10,4,4,4}*1280, {2,4,4,20}*1280, {2,20,4,4}*1280, {2,4,20,4}*1280
7-fold covers : {14,4,4,4}*1792, {2,4,4,28}*1792, {2,28,4,4}*1792, {2,4,28,4}*1792
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)( 9,17)(10,18)(19,27)(20,28)
(21,29)(22,30)(23,31)(24,32)(25,33)(26,34)(35,43)(36,44)(37,45)(38,46)(39,47)
(40,48)(41,49)(42,50)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(57,65)
(58,66);;
s2 := (11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,25)(24,26)(27,33)(28,34)
(29,31)(30,32)(35,39)(36,40)(37,41)(38,42)(51,57)(52,58)(53,55)(54,56)(59,61)
(60,62)(63,65)(64,66);;
s3 := ( 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)(17,33)(18,34)(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)(49,65)
(50,66);;
s4 := ( 3,43)( 4,44)( 5,45)( 6,46)( 7,47)( 8,48)( 9,49)(10,50)(11,35)(12,36)
(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,60)(20,59)(21,62)(22,61)(23,64)
(24,63)(25,66)(26,65)(27,52)(28,51)(29,54)(30,53)(31,56)(32,55)(33,58)
(34,57);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(66)!(1,2);
s1 := Sym(66)!( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)( 9,17)(10,18)(19,27)
(20,28)(21,29)(22,30)(23,31)(24,32)(25,33)(26,34)(35,43)(36,44)(37,45)(38,46)
(39,47)(40,48)(41,49)(42,50)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(57,65)
(58,66);
s2 := Sym(66)!(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,25)(24,26)(27,33)
(28,34)(29,31)(30,32)(35,39)(36,40)(37,41)(38,42)(51,57)(52,58)(53,55)(54,56)
(59,61)(60,62)(63,65)(64,66);
s3 := Sym(66)!( 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)(17,33)(18,34)(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)(49,65)
(50,66);
s4 := Sym(66)!( 3,43)( 4,44)( 5,45)( 6,46)( 7,47)( 8,48)( 9,49)(10,50)(11,35)
(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,60)(20,59)(21,62)(22,61)
(23,64)(24,63)(25,66)(26,65)(27,52)(28,51)(29,54)(30,53)(31,56)(32,55)(33,58)
(34,57);
poly := sub<Sym(66)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope