Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #11524 |
288 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(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)(55,62,65,63)(56,67)(58,73)(61,76,78,66)(64,80,81,79)(68,70)(71,74)(77,82,85,84);(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)(55,61)(57,70)(58,72)(59,68)(60,67)(62,66)(63,76)(64,77)(65,78)(69,73)(71,75)(74,83)(79,82)(80,84)(81,85);(54,55,58,64,59,63,74,80,60,65,73,81,57,62,71,79)(56,66,69,82,68,61,75,77,67,76,72,84,70,78,83,85)Polycyclic presentation:
2^8,53|a^2=e,c^2=f,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,d^b=dh,e^b=efh,e^c=eh,i^c=i^52,e^d=eh716 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 79009);