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×#1812 ⋊ C2 |
| Sylow | C53 ⋊ #11868 |
396 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 91 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)(54,56,68,83,59,70,58,67)(55,61,63,79,65,80,64,66)(57,73,84,82,74,69,72,85)(60,75,62,77,71,81,76,78);(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)(55,60)(56,67)(57,72)(58,68)(61,77)(62,64)(63,76)(65,71)(66,81)(69,73)(70,83)(74,84)(75,79)(78,80);(54,55)(56,66)(57,71,74,60)(58,63)(59,65)(61,67)(62,72,76,84)(64,68)(69,78,73,77)(70,79)(75,85,81,82)(80,83)Polycyclic presentation:
2^8,53|a^2=e,c^2=gh,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,g^a=gh,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,i^b=i^52,d^c=dh,e^d=eh866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 80561);