Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C53×#2126 ⋊ C2 |
| Sylow | C53 ⋊ #13577 |
408 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,57,60,58)(55,62,65,72)(56,67,70,73)(59,71,75,74)(61,76,78,82)(63,79,80,64)(66,77,83,81)(68,84,69,85);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,56)(55,61)(57,67)(58,73)(59,68)(60,70)(62,76)(63,66)(64,81)(65,78)(69,75)(71,84)(72,82)(74,85)(77,79)(80,83);(54,55)(56,66,68,82)(57,62)(58,72)(59,64,75,79)(60,65)(61,67,77,84)(63,74,80,71)(69,76,70,83)(73,81,85,78)Polycyclic presentation:
2^8,53|a^2=e,c^2=h,d^2=g,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,i^b=i^52,e^d=egh,f^e=fh704 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 89854);