Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C53 ⋊ #12772 |
760 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(2,8)(4,17)(5,19)(7,21)(9,23)(10,11)(12,28)(13,15)(14,31)(22,27)(24,26)(25,29)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,3)(2,7)(4,15)(5,14)(6,16)(8,21)(9,25)(10,27)(11,22)(12,26)(13,17)(18,32)(19,31)(20,30)(23,29)(24,28);(1,2,4,9)(3,12,15,7)(5,10,18,23)(6,11,17,26)(8,20,24,19)(13,22,16,25)(14,29,32,21)(27,31,28,30)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)Polycyclic presentation:
2^8,53|a^2=f,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^52,d^b=dh,e^b=egh,f^b=fgh,d^c=dg,e^c=eh,f^c=fh,e^d=eh,f^d=fh1022 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85439);