Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #11769 |
304 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 65 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)(54,56,58,66,59,68,71,80)(55,61,63,76,65,78,64,77)(57,67,70,79,72,82,83,85)(60,73,62,75,69,81,74,84);(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)(55,60)(56,66)(57,70)(58,71)(61,75)(62,64)(63,74)(65,69)(67,82)(68,80)(72,83)(73,76)(77,81)(78,84);(54,55)(56,61)(57,69,72,60)(58,63)(59,65)(62,70,74,83)(64,71)(66,76)(67,81,82,73)(68,78)(75,79,84,85)(77,80)Polycyclic presentation:
2^8,53|a^2=e,c^2=f,d^2=gh,f^2=h,g^2=h,b^a=bd,d^a=df,g^a=gh,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,i^c=i^52,e^d=eh554 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 80101);