# Polytope of Type {2,6,6}

Atlas Canonical Name : {2,6,6}*576a
if this polytope has a name.
Group : SmallGroup(576,8659)
Rank : 4
Schlafli Type : {2,6,6}
Number of vertices, edges, etc : 2, 24, 72, 24
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,6,6,2} of size 1152
Vertex Figure Of :
{2,2,6,6} of size 1152
{3,2,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,6,3}*288
3-fold quotients : {2,6,6}*192
4-fold quotients : {2,6,6}*144b
6-fold quotients : {2,3,6}*96, {2,6,3}*96
8-fold quotients : {2,6,3}*72
12-fold quotients : {2,3,3}*48, {2,2,6}*48
24-fold quotients : {2,2,3}*24
36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,6,12}*1152a, {2,12,6}*1152c, {2,6,6}*1152a, {2,6,12}*1152d, {4,6,6}*1152f, {2,12,6}*1152e
3-fold covers : {2,6,18}*1728, {2,6,6}*1728b, {6,6,6}*1728a, {2,6,6}*1728c
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 8, 9)(12,13)(15,27)(16,29)(17,28)(18,30)(19,31)(20,33)(21,32)
(22,34)(23,35)(24,37)(25,36)(26,38)(40,41)(44,45)(48,49)(51,63)(52,65)(53,64)
(54,66)(55,67)(56,69)(57,68)(58,70)(59,71)(60,73)(61,72)(62,74);;
s2 := ( 3,15)( 4,16)( 5,18)( 6,17)( 7,23)( 8,24)( 9,26)(10,25)(11,19)(12,20)
(13,22)(14,21)(29,30)(31,35)(32,36)(33,38)(34,37)(39,51)(40,52)(41,54)(42,53)
(43,59)(44,60)(45,62)(46,61)(47,55)(48,56)(49,58)(50,57)(65,66)(67,71)(68,72)
(69,74)(70,73);;
s3 := ( 3,46)( 4,44)( 5,45)( 6,43)( 7,42)( 8,40)( 9,41)(10,39)(11,50)(12,48)
(13,49)(14,47)(15,70)(16,68)(17,69)(18,67)(19,66)(20,64)(21,65)(22,63)(23,74)
(24,72)(25,73)(26,71)(27,58)(28,56)(29,57)(30,55)(31,54)(32,52)(33,53)(34,51)
(35,62)(36,60)(37,61)(38,59);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :
s0 := Sym(74)!(1,2);
s1 := Sym(74)!( 4, 5)( 8, 9)(12,13)(15,27)(16,29)(17,28)(18,30)(19,31)(20,33)
(21,32)(22,34)(23,35)(24,37)(25,36)(26,38)(40,41)(44,45)(48,49)(51,63)(52,65)
(53,64)(54,66)(55,67)(56,69)(57,68)(58,70)(59,71)(60,73)(61,72)(62,74);
s2 := Sym(74)!( 3,15)( 4,16)( 5,18)( 6,17)( 7,23)( 8,24)( 9,26)(10,25)(11,19)
(12,20)(13,22)(14,21)(29,30)(31,35)(32,36)(33,38)(34,37)(39,51)(40,52)(41,54)
(42,53)(43,59)(44,60)(45,62)(46,61)(47,55)(48,56)(49,58)(50,57)(65,66)(67,71)
(68,72)(69,74)(70,73);
s3 := Sym(74)!( 3,46)( 4,44)( 5,45)( 6,43)( 7,42)( 8,40)( 9,41)(10,39)(11,50)
(12,48)(13,49)(14,47)(15,70)(16,68)(17,69)(18,67)(19,66)(20,64)(21,65)(22,63)
(23,74)(24,72)(25,73)(26,71)(27,58)(28,56)(29,57)(30,55)(31,54)(32,52)(33,53)
(34,51)(35,62)(36,60)(37,61)(38,59);
poly := sub<Sym(74)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 >;

to this polytope