Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #7490 |
274 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 59 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 — 2 generators on 85 points:
(1,3,5,14,7,17,4,19)(2,8,10,24,12,26,18,30)(6,15,21,31,22,16,20,32)(9,27,29,25,28,13,11,23)(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);(1,2)(3,13,15,30)(4,18)(5,10)(6,11,22,29)(7,12)(8,14,23,31)(9,20,28,21)(16,24,17,27)(19,25,32,26)(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=d,b^2=e,c^2=h,e^2=h,f^2=h,b^a=bc,c^a=cf,f^a=fg,g^a=gh,i^a=i^52,c^b=ch,d^b=df,f^b=fh,g^b=gh,d^c=dgh,f^c=fh,f^d=fh866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57322);