Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #7524 |
134 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 25 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:
(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)(55,60,63,77)(57,69,71,65)(59,73,72,67)(61,64,79,82)(62,76,80,75)(68,85,78,81)(70,83,74,84);(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)(54,55)(56,64)(57,68,70,61)(58,63)(59,62,72,80)(60,65,75,83)(66,82)(67,81,73,85)(69,76,84,77)(71,78,74,79)Polycyclic presentation:
2^8,53|a^2=d,b^2=h,c^2=g,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=cg,d^b=deh,e^b=egh,f^b=fg,i^b=i^52,d^c=dfg,e^c=eg,f^c=fh,e^d=egh,f^d=fh383 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57410);